Nonlinear Water Waves: Cetraro, Italy 2013 [1st ed.] 3319314610, 978-3-319-31461-7, 978-3-319-31462-4, 3319314629

Table of contents :
Front Matter....Pages i-vii
Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows....Pages 1-82
Breaking Water Waves....Pages 83-119
Asymptotic Methods for Weakly Nonlinear and Other Water Waves....Pages 121-196
A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems....Pages 197-228
Back Matter....Pages 229-230

Citation preview

Lecture Notes in Mathematics 2158 CIME Foundation Subseries

Adrian Constantin Joachim Escher Robin Stanley Johnson Gabriele Villari

Nonlinear Water Waves Cetraro, Italy 2013 Adrian Constantin Editor

Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zürich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Catharina Stroppel, Bonn Anna Wienhard, Heidelberg

More information about this series at http://www.springer.com/series/304

2158

Adrian Constantin • Joachim Escher • Robin Stanley Johnson • Gabriele Villari

Nonlinear Water Waves Cetraro, Italy 2013 Adrian Constantin Editor

123

Authors Adrian Constantin Faculty of Mathematics University of Vienna Vienna, Austria

Joachim Escher Inst. for Applied Mathematics Gottfried Wilhelm Leibniz University Niedersachsen Hannover, Germany

Robin Stanley Johnson School of Mathematics and Statistics University of Newcastle Newcastle upon Tyne, United Kingdom

Gabriele Villari Department of Mathematics “Ulisse Dini” University of Florence Florence, Italy

Editor Adrian Constantin Faculty of Mathematics University of Vienna Vienna, Austria

ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-31461-7 DOI 10.1007/978-3-319-31462-4

ISSN 1617-9692 (electronic) ISBN 978-3-319-31462-4 (eBook)

Library of Congress Control Number: 2016941322 Mathematics Subject Classification (2010): 76B15, 35Q35, 34C05 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

The study of water waves involves various disciplines such as mathematics, physics and engineering—to name the obvious—and within this, there are many specific areas of direct or associated interest such as pure mathematics, applied mathematics, modelling, numerical simulation, laboratory experiments, data collection in the field, the design and construction of ships, harbours and offshore platforms, the prediction of natural disasters (e.g. tsunamis), climate studies and so on. We are all familiar with, and probably excited by, the experience of seeing waves in lakes, rivers, oceans and even baths and sinks; they are often beautiful, but sometimes terrifying. They are also mathematically intriguing and susceptible to a number of different, but very particular, theoretical approaches. All these various studies help us to improve, in one way or another, our understanding of wave propagation which, in turn, inevitably leads to better physics and engineering as we work with, and deal with the effects of, waves on water. The meeting held in Cetraro, Italy, June 24–28, 2013, under the auspice of, and supported by, the Centro Internazionale Matematico Estivo (C.I.M.E., the International Mathematical Summer Centre) aimed to present some of the current mathematical research in this area. The summer school provided a vehicle for a selection of the main mathematical avenues to be presented via a series of lectures; in addition, there were short presentations and much discussion, covering other related topics, such as numerical methods and modelling. This volume brings together the four main lecture courses. The intention, through the lectures, was to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. The general background to the mathematical formulation of the classical water-wave problem, and the interplay between what is observed and how we model this using a robust mathematical formulation, appears in ‘Asymptotic methods for weakly nonlinear and other water waves’ by R.S. Johnson. These lectures also covered some aspects of the construction and generalisation of soliton-type equations (including a brief introduction to ‘soliton theory’) and, of particular current interest, the rôle that background vorticity can play in the evolution of the waves and its effects upon their properties. In order to show the wealth of possibilities using an asymptotic approach, periodic waves with v

vi

Preface

vorticity and edge waves are also discussed. The lectures given by A. Constantin, entitled ‘Exact travelling periodic water waves in two-dimensional irrotational flows’, also explain the connection between what is observed (both in the laboratory and in the field) and what we can describe and predict using a mathematical approach. The exact solutions are described by, for example, the properties of the associated particle paths based on harmonic analysis and the theory of elliptic partial differential equations. Thus the very practical essence of the waves and very powerful and rigorous techniques are moulded to produce a comprehensive picture of the types of flows associated with classical water waves. The theme of particle paths is taken up in ‘A survival kit in phase plane analysis: some basic models and problems’ by G. Villari, but the approach here is to use another fundamental mathematical tool: the method of phase-plane analysis. Again, the thrust is to show how a sophisticated and familiar branch of mathematics can relate to, and usefully describe, the details of the complex flow patterns that are observed. The final series of lectures made use of the very powerful ideas that underpin the modern techniques of functional analysis. J. Escher discussed the nature of wave breaking as it applies, mainly, to the solutions of the Camassa-Holm equation in ‘Breaking water waves’. One of the exciting properties of this model equation is that it captures both the non-breaking and breaking wave phenomena of classical water waves. Some details that help to explain the rôle of the initial data in predicting the final development of the wave are provided, producing some important estimates. These four lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of this rapidly developing classical research area. In addition, some of the material can be used by those who are already familiar with one branch of the study of water waves, to learn more about other areas. We therefore commend this collection of lectures to both the novice and the expert. Vienna, Austria Hannover, Germany Newcastle upon Tyne, UK Florence, Italy

A. Constantin J. Escher R.S. Johnson G. Villari

Acknowledgements CIME activity is carried out with the collaboration and financial support of: - INdAM (Istituto Nazionale di Alta Matematica) - MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) - Ente Cassa di Risparmio di Firenze

Contents

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Adrian Constantin Breaking Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Joachim Escher

1 83

Asymptotic Methods for Weakly Nonlinear and Other Water Waves . . . . . 121 Robin Stanley Johnson A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197 Gabriele Villari

vii

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows Adrian Constantin

. . . fluid dynamicists were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed. Sir James Lighthill (1924–1998)

Abstract Most of the waves that are observed on the surface of the world’s oceans, seas and lakes are wind generated. Once initiated, these water waves propagate substantial distances before their energy is dissipated—propagation distances in excess of hundreds or thousands times a wavelength are needed for the occurrence of a significant energy loss. We address some fundamental aspects of water-wave propagation once waves have been generated, within the framework of inviscid twodimensional flow theory and in the absence of underlying currents. The emphasis is placed upon periodic travelling gravity water waves of large amplitude. These wave patterns can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. An in-depth mathematical study is made possible by uncovering the rich structure of the hydrodynamical free-boundary problem under investigation, taking advantage of insights from physical observation, experimental evidence and numerical simulations. The interdisciplinary nature of this classical research subject is also reflected in the fact that its theoretical investigation relies on an interplay between methods and techniques from dynamical systems, complex analysis, functional analysis, topology, harmonic analysis, and the calculus of variations.

1 Introduction Mathematics is the basic language of Physics and Engineering but the three subjects emphasize different approaches to a specific problem in fluid mechanics, even if the respective boundaries overlap considerably. Mathematical techniques and physical

A. Constantin () Department of Mathematics, King’s College London, Strand, London WC2L 2RS, UK Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria e-mail: [email protected] © Springer International Publishing Switzerland 2016 A. Constantin (ed.), Nonlinear Water Waves, Lecture Notes in Mathematics 2158, DOI 10.1007/978-3-319-31462-4_1

1

2

A. Constantin

principles aim to explain and predict natural phenomena, while engineers mostly use experimental tools to probe these phenomena. By adhering to a high standard of rigour, mathematicians working in fluid mechanics primarily expand and elucidate physical arguments that are more heuristic or intuitive, thus contributing to the growth of our understanding. Often this process reveals interesting features that were overlooked, and sometimes it even permits the discovery of new facets that were not within reach of less advanced mathematical techniques. However, while the capacity to grasp, manipulate and develop concepts and tools possibly suffices to define the mathematical value of an approach, it is not enough to validate realworld applications. A coherent explanation of a natural phenomenon involves care in selecting relevant explanatory factors and back-up by empirical tests for the theory. This highlights the importance of engineering expertise in the context of water-wave studies. A significant advance in water waves requires the combination of abstract ideas and techniques with an understanding of the physical reality. While our approach relies on rigorous mathematical considerations, throughout these lectures we will try to support our theoretical claims with field data. There are also a few aspects contingent to our considerations where a mathematical proof appears to remain elusive, in which case we will present some numerical evidence pointing towards a likely conclusion.

2 Preliminaries Natural phenomena are more complex than any model that anyone can make, so that one must necessarily accept less than exhaustive descriptions. An efficient mathematical model of a natural phenomenon uses small amounts of information to produce experimentally validated conclusions. To capture all factors is impossible and not even desirable since one can always expand the problem to the point where it could not be answered (at the current state-of-the-art). Even if the mathematics involved in the study is highly sophisticated, one should always be aware that the model is a simplified idealisation of the real world phenomenon. The adequacy of the model depends on how well it represents the key factors and on how reliable its predictions are (in the physical regime in which it is legitimate to apply it). The degree of allowed complexity should capture the essential physical factors and permit the pursuit of an in-depth analysis that leads to qualitative as well as quantitative predictions.

2.1 Periodic Travelling Waves We will study the most regular water wave patterns: periodic travelling waves that propagate on the surface of water in a given direction and at constant speed. Such waves, termed swell in oceanography, are not generated by the local wind but by a

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

3

direction of wave propagation crest amplitude undisturbed water level height trough

wavelength

Fig. 1 The amplitude of a periodic travelling wave is the maximum deviation of the wave from undisturbed water surface. In contrast to the sinusoidal waves encountered in linear wave theory, swell presents sharper elevations and flatter depressions, so that the distance from the wave crest to the undisturbed water level generally exceeds the distance at which the wave sinks beneath its mean level (Mass conservation ensures that the wave mean level and the undisturbed water level coincide)

distant storm. As the irregular waves generated by the storm propagate away from their source, they start to organise themselves into swell and these regular wave patterns persists over very long distances (in excess of hundreds of km). However, since swell waves are mostly mixed with local wind-generated sea waves, they are difficult to detect with the naked eye if they are not significantly large. Nevertheless, often these regular waves were captured on camera (see e.g. the photographs in [13]), and their generation in wave tanks is a standard procedure. Note also that while these two-dimensional waves present three-dimensional instabilities (known as Benjamin-Feir instabilities [5]), it was experimentally observed that the disintegrated periodic travelling wave continued to develop and eventually re-formed into a new uniform wave train cf. [50, 54]. The fact that the initial wave pattern is (approximately) regained after a while explains the ubiquity of swell in the open sea. The characteristics of these waves are (see Fig. 1): wave propagation direction; wave speed c (constant); wave crest and wave trough—the highest point the water rises to and the lowest point the water sinks to, respectively; wave height—the vertical distance from crest to trough; wavelength L (constant)— the horizontal distance between two consecutive crests or troughs; wave period T D L=c (constant)—the time interval between arrival of consecutive crests at a stationary point. Typical values are L D 50–100 m, c D 30–40 km/h for swell in the North Atlantic, and L D 200–300 m, c  70 km/h for swell in the South Pacific, with typical wave heights in the range 1.5–3 m and 2–4.5 m, respectively. Gale and hurricane winds can produce wave heights from 7:5 to 16 m (Metoffice.gov.uk, 2012) off the coasts of Ireland and Scotland, two key regions for monitoring some of the largest waves on Earth.

4

A. Constantin

3 The Governing Equations An assumption that is very convenient for our purposes is that the water bed is flat (and horizontal). Compelling practical grounds for the study of this type of water flows are the waves propagating in a canal (at the surface of water of relatively constant average depth), as well as surface waves in sea regions with a flat bed.1 We will also make some other simplifying assumptions, referring to [13] for a discussion of their physical relevance: • The density  of the water is constant; • The flow is inviscid—we neglect internal and boundary dissipation; • The influence exerted by the air above the water’s surface is in the form of atmospheric pressure acting on the surface, taken to be constant. • The evolution of the waves is governed by the balance between the restoring force of gravity and the inertia of the system, that is, gravity is the only external force of significance. To describe a two-dimensional water wave it suffices to consider a cross section of the flow in the direction of wave propagation since the motion is identical in any plane parallel to it. Choose Cartesian coordinates .X; Y/ with the X-axis pointing in the direction of wave propagation and the Y-axis pointing vertically upwards, while the origin is located on the mean water level Y D 0 (see Fig. 2).  Let U.X; Y; T/; V.X; Y; T// be the velocity field of the two-dimensional flow propagating in the X-direction over the flat bed Y D d, where d > 0 is the average depth, and let Y D .X; T/ be the water’s free surface with mean water level Y D 0. For water of constant density the equation of mass conservation is

1

For example, large parts of the Pacific Ocean floor are very flat. Relevant for the discussion are abyssal plains—vast sediment-covered regions of the sea floor that are the flattest areas on Earth, with an almost total absence of geographic features, presenting variations in depth in the range of 10–100 cm/km of horizontal distance. Abyssal plains result from the blanketing of a preexisting irregular ocean floor topography by accumulated land-derived sediment. They are found in all major sea and ocean basins, usually adjacent to a continent and at depths between 2 and 6 km, covering overall almost a third of the Earth’s surface (about as much as all the exposed land combined). The largest are hundreds of km wide and thousands of km long. For example, in the North Atlantic the Sohm Plain, located to the south of Newfoundland, has an area of approximately 900,000 km2 . The flat areas of the bed of the Pacific Ocean cover a greater area than those of the Atlantic Ocean, but the proportion due to abyssal plains is considerably smaller since deep trenches (with lengths of thousands of km, and generally hundreds of km wide) near the continents trap most of the sediment before it reaches the open ocean. In the Mediterranean Sea the most extensive abyssal plains are the Balearic Abyssal Plain (at 2800 m depth, with a total area of about 240,000 km2 , its major sources of sediment being the Ebro and Rhone rivers) and the Tyrrhenian Abyssal Plain (at 3500 m depth and with a total area of about 30,000 km2 , it is composed by two plains separated by an undersea ridge, with sediment feeded partly by the Tiber river but mostly composed of volcanic materials—several terrestrial and submarine volcanoes being located in this area).

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

5

Fig. 2 The choice of Cartesian coordinates .X; Y/: X points in the direction of wave propagation and Y points vertically upwards, with Y D 0 being the water mean level

UX C V Y D 0

(1)

throughout the fluid (see [13]). Since the assumption of inviscid flow is physically realistic for water waves, the equation of motion is Euler’s equation 8 1 ˆ ˆ < UT C UUX C VUY D  PX ;  1 ˆ ˆ : VT C UVX C VVY D  PY  g ; 

(2)

cf. [13], where P.X; Y; T/ is the pressure, g is the (constant) acceleration of gravity and  is the (constant) density. We draw attention to the fact that while in a compressible fluid the pressure determines the density, the present incompressible context withholds this effect and P has to be regarded as a reaction to the constraint of incompressibility, playing the role of a Lagrange multiplier in the equation of evolution. The boundary conditions associated to (1) and (2) are VD0

on Y D d ;

(3)

on the flat bed and V D T C UX

on Y D .X; T/;

(4)

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A. Constantin

as well as P D Patm

on Y D .X; T/;

(5)

on the free surface. The kinematic boundary conditions (3) and (4) reflect the fact that both boundaries are interfaces: particles on these boundaries are confined to them at all times (see the discussion in [13]). The dynamic boundary condition (5), in which Patm represents the (constant) atmospheric pressure at the surface, permits the motion of the water to decouple from that of the air above: the interface between the liquid (water) and the gas (air) can be assumed to be free because the air density is about 900 times lower than that of the water (see [13]). The vorticity ! D UY  V X of the water flow beneath the surface wave is a measure of the local spin of a fluid element (see the discussion in [13]). Irrotational flows are characterised by a complete absence of this local spin: !  0 throughout the flow. Since vorticity is the hallmark of the interaction of waves with underlying sheared currents, cf. [13], irrotational flows arise in the absence of non-uniform currents. A flow that is irrotational initially (at time T D 0) remains so at later times. While this feature is valid in the full generality of three-dimensional water flows (see e.g. [13]), it is instructive to confirm it in the simpler setting of two-dimensional flows. Consider a particle located initially at some point .X0 ; Y0 / in the fluid domain, so that Y0 2 Œd; .X0 ; 0/. Its subsequent location due to the action of the water flow is determined by solving the differential system (

X 0 .T/ D U.X.T/; Y.T/; T/ ; Y 0 .T/ D V.X.T/; Y.T/; T/ ;

(6)

with initial data .X.0/; Y.0// D .X0 ; Y0 /. Consequently, the vorticity at the location of this particle, as it moves about, satisfies the relation d !.X.T/; Y.T/; T/ D UYT C UXY U C UYY V  VXT  VXX U  VXY V ; dT in which the right side is evaluated at .X.T/; Y.T/; T/. Differentiating the first and second component of (2) with respect to the Y and X variable, respectively, and takd !.X.T/; Y.T/; T/ D ing advantage of (1), a straightforward calculation yields dT 0. This proves our claim. The simplest setting for the study of water waves is provided by the governing equations (1)–(5) for irrotational flows, that is, with the additional specification that UY  VX D 0 throughout the flow:

(7)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

7

In particular, field data shows that for waves entering a region of still water the assumption of irrotational flow is realistic [45]. Even within this setting, the governing equations represent a very challenging problem. The difficulty lies in the nonlinear character of the governing equations and in the fact that the free boundary Y D .X; T/ is unknown: finding possible wave profiles  is part of the problem. To illustrate the level of complexity that is involved, note that despite intensive studies over more than two centuries, an explicit solution with a non-flat free surface remains elusive. To gain insight two approaches proved successful: 1. The quest of special exact solutions depends on structural properties that, once uncovered, permit a qualitative description of certain type of wave motions although no explicit formulas can be provided. An in-depth analysis is necessary for a thorough understanding of the relevant physical mechanics. In the absence of explicit exact solutions, this approach usually requires an advanced level of mathematical sophistication. 2. The implementation of a perturbative approach depends on the availability of an explicit solution. Upon identification of suitable small parameters that simplify considerably the problem when they vanish (providing, in particular, the explicit solution), the aim is to construct an asymptotic solution that solves the problem up to a small error if these parameters are sufficiently small. The sizes, and relative sizes, of these parameters then govern the type of approximation that one undertakes, being also indicative of the physical regime of validity. We will develop the first approach in the context of the distinguished class of solutions that represent travelling waves. We will only discuss (in Sect. 3.2) some basic ideas that are relevant for the second approach, (2), illustrating in particular the motivation to pursue the alternative, (1).

3.1 Stokes Waves Throughout these notes, a Stokes wave is an irrotational smooth solution to the governing equations for which there exists a period L > 0 and a wave speed c such that the free surface profile , the fluid velocity .U; V/ and the pressure P have period L in the X variable,  depends only on .X  cT/, while U; V, and P depend only on .X  cT/ and Y. Moreover, there is a single crest and trough per period, the wave profile is strictly monotone between successive crests and troughs, and ; U and P are symmetric while V is antisymmetric about the crest line—by a crest line (trough line), we mean the vertical line directly below a crest (trough). This definition excludes flows with a flat free surface, as well as the Stokes wave of greatest height.2 for which the symmetric free surface is not continuously

2

For a prescribed wavelength, this is the highest two-dimensional periodic travelling wave [26]. However, the highest wave is not the fastest nor the most energetic [53].

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A. Constantin

differentiable, with the profile having at the crest a corner with an angle of 120ı . We refrain from discussing the latter flow pattern, as it presents peculiar features and it requires a lot of specific highly technical considerations that are beyond the scope of our discussion; the interested reader is referred to [7, 14, 59]. Section 4 is devoted to a survey of the existence theory for Stokes waves: the existence of Stokes waves of small amplitude is established by means of local bifurcation, while analytic global bifurcation yields the existence of waves of large amplitude. In Sect. 4 we will also discuss the flow beneath a Stokes wave. The points of interest are the behaviour of the velocity field and of the pressure, as well as the particle path pattern. The study of these aspects relies on an interplay between harmonic function theory and the structural properties of the problem.

3.2 Approximate Model Equations for Wave Propagation In our setting, the only available explicit solutions to the governing equations (1)– (2)–(3)–(4)–(5)–(7) are uniform horizontal flows with a flat free surface, in particular, the solution representing water at rest: the free surface   0 is flat, there is no motion in the fluid domain (that is, u  0 and v  0), and the pressure is hydrostatic, P.X; Y; T/ D Patm  gY for d  Y  0. The process of nondimensionalisation ensures the removal of scales associated with a specific problem and permits us to give a well-defined meaning to the concept “small”. The procedure consists in introducing non-dimensional variables, whose original units are suitable for the given problem, thus formulating the problem in the form of numerical relations involving some non-dimensional parameters. Seeking approximations for non-trivial water flows, we consider the perturbation of the pressure relative to the hydrostatic pressure distribution, measuring the change in pressure as a wave moves over the surface of water. Thus we introduce the scalar function p by means of3 P.X; Y; T/ D Patm  gY C gd p.X; Y; T/;

d  Y  .X; T/ :

(8)

To generate a non-dimensional version of the governing equations, we need a suitable scale for time, distance and speed. Given that d is the average depth of the water, if L is a typical wavelength of the wave, the process of non-dimensionalising requires, for example, to replace the dimensional variables .X; Y/ by .Lx; dy/, where x and y are now non-dimensional versions of the physical variables X and Y, respectively. Since the speed scale c0 is a priori not obvious, it has to remain undetermined for the moment. However, using c0 as a speed scale forces upon us the choice of L=c0 as a time scale: real-world observation indicates that in two-

3 The choice of the coefficient gd is due by calibration relative to the standard unit of pressure measurement, so that p takes on numerical values.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

9

dimensional water flow the most dramatic changes occur in the horizontal direction (of wave propagation), and not in the vertical direction. We therefore perform the following change of variables: 8 ˆ 0 (in the non-dimensional setting). With an .x; t/-dependence of uO ; v; O pO , and h of the form .x  cO t/, the

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

13

system (18) takes on the form6 8 ˆ uO .x; y/ C vOy .x; y/ D 0 for  1 < y < 0 ; ˆ ˆ x ˆ ˆ ˆ ˆ cO uO x .x; y/ D pO x .x; y/ for  1 < y < 0 ; ˆ ˆ ˆ 2 ˆ ˆ < cO ı vO x .x; y/ D pO y .x; y/ for  1 < y < 0 ; v.x; O 1/ D 0 on y D 1 ; ˆ ˆ ˆ ˆ v.x; O 0/ D  cO h0 .x/ on y D 0 ; ˆ ˆ ˆ ˆ ˆ pO .x; 0/ D h.x/ on y D 0 ; ˆ ˆ ˆ : uO y .x; y/ D ı 2 vOx .x; y/ for  1 < y < 0 :

(22)

Note that the first and last relation yield ı 2 vOxx C vO yy D 0

for

1 < y < 0:

(23)

Seeking smooth solutions, using the Fourier series expansion X

v.x; O y/ D

˛k .y/ e2ikx ;

1  y  0 ;

k2Znf0g

in (23) leads to ˛k00 .y/  4 2 k2 ı 2 ˛k .y/ D 0 on .1; 0/ for every k 2 Z. For k ¤ 0 we obtain that ˛k .y/ D ak e2kıy C bk e2kıy ;

1  y  0 ;

while for k D 0 we have ˛0 .y/ D a0 y C b0 ; here ak ; bk 2 R are some constants. The boundary condition on y D 1 in (22) forces a0 D b0 and bk D ak e4kı for every k 2 Z n f0g, so that v.x; O y/ D a0 .y C 1/ C

X

2 ak e2kı sinhŒ2kı.y C 1/ e2ikx ;

1  y  0 :

k2Znf0g

Since vO has to be a real-valued function, we must have ak e2kı D ak e2kı for all k 2 Z n f0g, so that, with k D 4ak e2kı , v.x; O y/ D a0 .y C 1/ C

X

k sinhŒ2kı.y C 1/ cos.2kx/ :

(24)

k1

6

Since the time-dependence amounts to a simple translation of the horizontal spatial variable x by ct, it suffices to investigate the problem at time t D 0.

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A. Constantin

From the first and last relation in (22) we now infer that uO .x; y/ D   a0 x  ı

X

k coshŒ2kı.y C 1/ sin.2kx/ ;

(25)

k1

for some constant  2 R. Using now the last equation of (22) in the third relation of (22), we get cO uO y D pO y for y 2 .1; 0/. In combination with the second equation in (22), this means that pO D cO uO C ˇ throughout the strip f.x; y/ W 1  y  0g, for some constant ˇ 2 R. The periodicity of uO .x; y/ in the x-variable now forces a0 D 0, so that X k coshŒ2kı.y C 1/ sin.2kx/ : uO .x; y/ D   ı k1

From the sixth relation in (22) and (25) we obtain h.x/ D cO  C ˇ  cO ı

X

k cosh.2kı/ sin.2kx/ :

k1

Since the average of h over one period should vanish, as it represents the mean water level y D 0, we have cO  C ˇ D 0 and h.x/ D Oc ı

X

k cosh.2kı/ sin.2kx/ :

(26)

k1

On the other hand, the fifth relation in (22) and (24) with a0 D 0 yield  cO h0 .x/ D

X

k sinh.2kı/ cos.2kx/ :

(27)

k1

tanh.2kı/ 2kı whenever k ¤ 0. Since the function s 7! tanh.s/=s is strictly decreasing for s > 0, there is at most one integer k  1 for which k ¤ 0. This means that superpositions of Fourier modes are not possible: with principal unit period, the surface wave is of the form h.x/ D A sin.2x/, obtained from (26) with A D Oc ı 1 cosh.2ı/ by setting k D 0 for k  2. The travelling wave profile h moves in the non-dimensional setting with speed The differentiated form of (26) coincides with (27) only if cO 2 D

r cO D

tanh.2ı/ ; 2ı

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

15

and the full solution to (18) is given by 8 h.x; t/ D A sinŒ2.x  cO t/ ; pO .x; y; t/ D cO ŒOu.x; y; t/    ; ˆ ˆ ˆ ˆ ˆ < A uO .x; y; t/ D  C coshŒ2ı.y C 1/ sinŒ2.x  cO t/ ; cO cosh.2ı/ ˆ ˆ ˆ A ˆ ˆ : v.x; sinhŒ2ı.y C 1/ cosŒ2.x  cO t/ O y; t/ D  ı cO cosh.2ı/

(28)

for some constant  2 R. Due to (8), (9), (11), (12), (13), (16), this non-dimensional solution corresponds to the linear solution 8  2.X  c T/  ˆ ˆ .X; T/ D " d A sin ; ˆ ˆ L ˆ ˆ   2 ˆ ˆ ˆ ˆ .Y C d/ cosh  2.X  c T/  ˆ p " gdA ˆ L ˆ ˆ U.X; Y; T/ D "  gd C sin ;   ˆ 2d ˆ c L ˆ ˆ cosh ˆ ˆ <  L  2 .Y C d/ sinh  2.X  c T/  " gdA ˆ L ˆ cos V.X; Y; T/ D  ; ˆ   ˆ 2 ˆ c L ˆ ˆ cosh ˆ ˆ L ˆ   2 ˆ ˆ ˆ ˆ .Y C d/ cosh  2.X  c T/  ˆ ˆ L ˆ ˆ ;  gY C " gdA P.X; Y; T/ D P sin atm   ˆ 2d ˆ L : cosh L (29) in physical variables, where r cD

 2d  gL tanh : 2 L

(30)

In (29), the constant  measures the strength of an underlying uniform current, cf. the discussion of the current beneath a Stokes wave in Sect. 4. The inadequacy of linear theory becomes apparent. Within the shallow-water regime any smooth periodic wave profile is admissible in (21), which is not a realistic scenario. If we refrain from imposing a shallow-water regime, the solution (29) is always sinusoidal and provides only a very rough idea of a swell pattern: the photographs in [13] clearly show that actual periodic travelling waves in shallow water with a flat bed are not sinusoidal, being almost flat near the trough and with a pronounced elevation near the crest. This indicates that the linearisation procedure is grossly incorrect in some respects. Therefore nonlinear effects should be accounted for. We conclude our discussion by pointing out that the function on the right-hand side of (30) presents a strictly increasing dependence on the wavelength L. This reflects the dispersive effect of linear water waves: within the linear framework the

16

A. Constantin

superposition principle applies7 and the fact that waves of different lengths travel at different speeds ensures that a group of waves of different wavelengths starting together spreads out so that after a while the longer/faster waves  are at the front. gL d D gd tanh.2ı/ In the shallow-water limit ı D L ! 0 we have 2 tanh 2d  L 2ı gd, so that (30) yields the critical shallow water speed c

p gd :

(31)

r

tanh.s/ 2 .0:97 ; 1/ for s < 0:44, the approximation (31) is of s an accuracy within 3 % from the formula (30) for ı < 0:07, which is the standard range characterizing shallow-water waves cf. [45]. The regime of validity of (31) is non-dispersive: within the linear framework, all waves that p are sufficiently large compared to the mean water depth travel at the same speed gd. This fact is often taken advantage of in leisure boat activities: a ride inp a motor boat on a relatively calm water is smoother when the boat speed exceeds gd since the waves created by the displacement of the boat can not overtake it, thus preventing the creation of disturbances in front of the boat. An initial sudden burst of power carries the boat beyond the critical speed before waves ahead of it had the time to form. With g  9:8 m/s2 , the value (31) of the critical speed is easily computed, e.g. for a mean water depth d  10 m, we have c  36 km/h. Since numerically

3.2.2 The Shallow-Water Problem The disenchantment with a linear theory of periodic travelling waves leads us to discuss the other standard approximation, corresponding to ı ! 0 (with " fixed) and represented by the set of equations 8 ˆ uO x C vO y D 0 for  1 < y < " h ; ˆ ˆ ˆ ˆ ˆ uO t C " .OuuO x C vO uO y / D  pO x for  1 < y < " h ; ˆ ˆ ˆ ˆ ˆ ˆ < pO y D 0 for  1 < y < " h ; vO D 0 on y D 1 ; ˆ ˆ ˆ ˆ vO D ht C " uO hx on y D " h ; ˆ ˆ ˆ ˆ ˆ ˆ pO D h on y D " h ; ˆ ˆ : uO y D 0 for  1 < y < " h :

7

(32)

That is, a linear combination of solutions is again a solution, so that the overall wave motion is simply the sum of all its parts. The previous discussion shows that, in general, the speeds of the individual wave components are different, so that the resulting interaction pattern is not a travelling wave.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

17

By the last relation uO .x; y; t/ D uO .x; t/, so that from the first and fourth relation in (32) we can infer that v.x; O y; t/ D .y C 1/ uO x .x; t/. On the other hand, the third and sixth relation in (32) yield pO .x; y; t/ D h.x; t/. Using this information in the second and fifth relation, we see that (32) is equivalent to the shallow-water system (

uO t C " uO uO x C hx D 0 ; ht C " uO hx C " hOux C uO x D 0 :

(33)

Seeking periodic travelling waves of unit spatial period and speed cO > 0 (in the non-dimensional setting), the dependence of uO and h on the .x; t/-variables of the form .x  cO t/ transforms (33) to the system of ordinary differential equations (

Oc uO 0 C " uO uO 0 C h0 D 0 ; Oc h0 C " uO h0 C " hOu0 C uO 0 D 0 :

Integrating each component of the above system yields (

. 2" uO  cO / uO C h D ˛ ; ." uO  cO / h C uO D ˇ ;

for some constants ˛; ˇ 2 R. Inserting the expression for h in terms of uO from the first equation into the second equation leads to "2 3 3 uO  " cO uO 2 C .Oc2  1  " ˛/ uO C ˇ C ˛ cO D 0 : 2 2 If " > 0, then uO must be a constant, and h likewise. Consequently the system (32) fails to model travelling waves, unless " D 0, in which case we are in the setting of (19), with the general solution (20).

3.2.3 Small-Amplitude Nonlinear Shallow-Water Waves The discussion in the previous two subsections showed that individually both the linear regime " ! 0 and the (nonlinear) shallow-water regime ı ! 0 fail to provide insight into the dynamics of travelling waves, due to the scarcity of admissible wave patterns. On the other hand, the regime of linear shallow-water waves (obtained in the double limit " ! 0 and ı ! 0) presents the opposite drawback by allowing for arbitrary travelling wave profiles. In light of this, it appears reasonable to explore the possibility of a regime that describes a balance between the effects of nonlinearity and dispersion. The fact that the non-dimensional scaled version (17) of the governing equations involves only the parameters " and ı 2 is suggestive of

18

A. Constantin

the regime "  ı2

(34)

of small-amplitude shallow water waves. At leading order8 we recover (19) from (17). As discussed in Sect. 3.2.1, at leading order any perturbation of a flat free surface splits up into two components moving in opposite directions. Taking into account terms of order O."/, within the regime (34), one can show that the right-going wave component satisfies the Korteweg-de Vries equation [43] in the following sense (see Sect. 6.3 in [1] for a rigorous justification): given "0 > 0 and a smooth periodic initial wave profile hC 0 .x/ with the associated smooth periodic irrotational velocity field .u0 ; v0 /, if inf

x2R; "2.0;"0 /

f1 C " hC 0 .x/g > 0 ;

then there exists T0 > 0 such that for every " 2 .0; "0 / the system (17) of nondimensional scaled governing equations with initial data .hC 0 ; u0 ; v0 / has a unique family of smooth solutions defined for t 2 Œ0; T0 =". Moreover, if the free surface C profile of this family of solutions is denoted by hC " .x; t/ and if one defines h"; KdV as C hC "; KdV .x; t/ D h .x  t; " t/ ;

(35)

where hC .y; / solves the Korteweg-de Vries equation  C 3 y C

1 yyy D 0 ; 6

(36)

then C 2 jhC " .x; t/  h"; KdV .x; t/j  C " t ;

t 2 Œ0; T0 =" ;

(37)

for some C > 0 independent of " 2 .0; "0 /. A similar estimate holds for the velocity fields. One can show (see e.g. [38] and references therein) that the periodic travelling waves of (36) with zero average over the (minimal) unit period are given by the explicit parameter-dependent family of functions n h   i o .x; t/ D ˛. / 2 cn2 4K. / x  c. / t I  1 ;

8

That is, setting ı D 0, and therefore also " D 0, due to (34).

(38)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

19

where cnŒ I  is the elliptic function with parameter 2 .0; 1/, 16 K 2 . / ; ˛. / D 3

Z

1

K. / D 0

ds ; p 2 .1  s2 /.1  s2 /

and c. / D

16 K 2 . / 2 .  2/ < 0 : 3

(39)

In view of (35) and (37), we deduce that in the regime (34) a good approximation of the periodic travelling waves is provided by the wave profiles h   i o n 2 hC "; KdV .x; t/ D ˛. / 2 cn 4K. / x  t  " c. / t I  1 :

(40)

Due to (9), (12), (13), and (15), this corresponds to n h 4K. /   i o p .X; T/  "d ˛. / 2 cn2 X  .1 C " c. // gd T I  1 L

(41)

in physical variables. At a fixed time T, the wave p crest and wave trough of (41) are to be found at the points X D .1 C " c. // gd T C k L and X D .1 C p " c. // gd T C .2k C 1/ L=2 with k 2 Z, respectively. Moreover, the profile (41), with wave amplitude "d ˛. / and wave height 2 "d ˛. /, is strictly monotone between consecutive crests and troughs, and symmetric. In contrast to linear shallow water wave theory, the crests of the cnoidal wave (41) are sharper, being separated by wide flat troughs. Thus the approximation (40) is more realistic than the sinusoidal wave profiles (29) that p are typical for linear shallow water wave theory. p In view of (39), notice that gd exceeds the propagation speed .1 C " c. // gd of the travelling wave (41). We will see in Sect. 4 that this feature is inherited from the Stokes waves of small amplitude (as their propagation speed is always less than p gd). The above considerations show the practical relevance of the Korteweg-de Vries equation. However, they are limited to the regime (34) of shallow water waves of small amplitude. Moreover, due to (35), the (non-dimensional) region of time and space where KdV-dynamics is relevant corresponds to x  t D O.1/ and "t D O.1/. Recalling (9), the distance and time needed for the wave dynamics to be accurately approximated by KdV-dynamics can be estimated by T D O. "pLgd / and X D O. L" /, provided (34) holds. It is instructive to notice that waves with amplitudes a  1 m and wavelengths L  125 m, in water of mean depth d D 25 m, enter the regime (34) and KdV-dynamics is relevant in about 3 km. However, for the Boxing Day tsunami of December 26, 2004, while the waves generated by the rupture of the ocean floor due to an earthquake propagated across the Indian Ocean/Bay of Bengal with characteristics a  1 m, d  4 km, L  180 km (based on accurate

20

A. Constantin

satellite measurements—see [20]) enter the regime (34) since "  25  105 and ı  2  102 , the distance required for the applicability of KdV-dynamics in this case would be of the order of 105 km. Starting with ı 1 as a specification of the shallow water regime, one might hope that even beyond the realm (34) of waves of small amplitude it might be possible to derive a tractable model equation. The regime " D O.ı/ of shallow water waves of moderate amplitude was recently investigated in [22]: by neglecting terms of order O.ı 2 /, the solutions to t C x C C

3" 3"2 2 3"3 3 ı2 x   x  x C .xxx  xxt / 2 8 16 12

7"ı 2 .xxx C 2x xx / D 0 24

(42)

are good approximations of unidirectional surface wave solutions to the nondimensional scaled governing equations (17). However, non-trivial periodic travelling wave solutions to (42) appear not to be available in closed form so that their existence and qualitative features must be studied abstractly. For shallow-water regimes of larger amplitude, say, " D O.ı 2=3 / with ı 1, even the derivation of a partial differential equation that models the evolution of an approximation to the free surface waves appears to be problematic. These considerations highlight the limitations of the exalted vision that waves of large amplitude are tractable by means of asymptotic analysis. It is therefore necessary to deal directly with the Stokes waves.

4 Existence and Dynamics of Stokes Waves The existence theory of Stokes waves of large amplitude is beyond the scope of constructive methods—such as linearised or asymptotic approximations—since the fact that the only available explicit irrotational flows with a free surface are simple, uniform flows (with a flat surface) precludes any simplifying approximation. Nevertheless, a proper appreciation of experimental data suggests some a priori features that Stokes waves must present (for example, the monotonicity of the symmetric wave profile between successive crests and troughs, as well as the absence of internal stagnation points). These lead to a tidy mathematical formulation, abstracted from its original natural setting and cleared of the inexactnesses inevitable in practice. We will see that an in-depth analysis has the capacity to establish definite conclusions about Stokes waves without any restrictions on the wave characteristics (in particular, with respect to their amplitude), so that one can gain insight into realistic situations for which constructive methods are virtually impossible.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

21

4.1 Structural Properties and Dimensionless Form Taking advantage of the .X; T/-dependence of the form .X  cT/, we can write the governing equations for Stokes waves in the form 8 1 ˆ ˆ .U  c/UX C VUY D  PX for  d  Y  .X  cT/ ; ˆ ˆ  ˆ ˆ ˆ ˆ 1 ˆ ˆ .U  c/VX C VVY D  PY  g for  d  Y  .X  cT/ ; ˆ ˆ ˆ  ˆ ˆ ˆ ˆ < UX C VY D 0 for  d  Y  .X  cT/ ; ˆ UY D VX for  d  Y  .X  cT/ ; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ V D 0 on Y D d ; ˆ ˆ ˆ ˆ V D .U  c/X on Y D .X  cT/ ; ˆ ˆ ˆ ˆ : P D Patm on Y D .X  cT/ : Passing to the moving frame X D X  cT ;

Y DY;

(43)

the above system becomes 8 1 ˆ ˆ .U  c/UX C VUY D  PX for  d  Y  .X / ; ˆ ˆ  ˆ ˆ ˆ ˆ 1 ˆ ˆ .U  c/VX C VVY D  PY  g for  d  Y  .X / ; ˆ ˆ ˆ  ˆ ˆ ˆ ˆ < UX C VY D 0 for  d  Y  .X / ; ˆ UY D VX for  d  Y  .X / ; ˆ ˆ ˆ ˆ ˆ ˆ V D 0 on Y D d ; ˆ ˆ ˆ ˆ ˆ ˆ ˆ V D .U  c/X on Y D .X / ; ˆ ˆ ˆ : P D Patm on Y D .X / :

(44)

Since the system (44) is nonlinear and we aim to investigate solutions that are not merely small-amplitude waves, arising as perturbations of uniform flows with a flat free surface, we have to take advantage of its structural properties to unravel some of the apparent complexity, reducing it to a tractable problem. Some concepts from hydraulics—mass flux, head, flow force—have counterparts for the flow beneath a Stokes wave, in the form of invariants which are conductive to a significant reduction of the number of variables. We will derive these invariants as dimensionally

22

A. Constantin

homogeneous relationships between physical quantities, in the sense that they involve consistent units (that is, each additive term has the same unit). 1. The relative mass flux (flow rate per unit height), relative to the uniform flow at constant speed c, given by Z MD

.X / d

ŒU.X ; Y/  c dY ;

(45)

is a flow invariant. Indeed, using the third, fifth and sixth relation in (44), by differentiation one can check that the right side of (45) is independent on the X -variable. 2. There is also a conservation of energy statement (Bernoulli’s law): the expression E D

.U  c/2 C V 2 C P C g.Y C d/ 2

(46)

is constant throughout the fluid, as one can see by differentiation, using the first, second and fourth relation in (44). In (46) the first term on the right represents kinetic energy per unit volume, the last term is potential energy per unit volume, while the middle term captures the internal energy per unit volume (neglecting viscous effects). We can now express the last relation in (44) as the constraint .U  c/2 C V 2 CY Cd D H 2g

on Y D .X / ;

(47)

for some physical constant H, called head. The terms on the left of (47) have the dimension of length: the first, called the velocity head, represents the elevation needed for the fluid to reach the velocity j.U  c; V/j during frictionless free fall, the second term being the height above the flat bed. Note that (47) unveils the hidden nonlinear character of the dynamic boundary condition (5). 3. The horizontal flow beneath a genuine, non-flat wave is not uniform; indeed, U constant forces V to be also constant and thus PX D PY  g D 0, so that differentiation of the last relation in (44) with respect to the X -variable shows that the free surface can only be flat in such a setting. The non-uniformity means that the fluid is accelerated in the horizontal direction and since it has mass this requires a force, provided by the deviation of the pressure within the fluid from the atmospheric pressure, P  Patm . This pressure force plus the momentum flux (relative mass flux times the velocity relative to the moving frame, U  c) represent the flow force. Since the rate of relative mass flux across any small vertical surface element of area is .U  c/ , we see that the momentum flux across the vertical cross-section of width l at X is  .X  R / given by l ŒU.X ; Y/  c2 dY , while the corresponding pressure force is d

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

23

 .X  R / l ŒP.X ; Y/  Patm  dY . The flow being uniform in the horizontal direction d

orthogonal to X by two-dimensionality, the relevant expression is .X Z /

SD

2

.X Z /

ŒU.X ; Y/  c dY C d

ŒP.X ; Y/  Patm  dY :

(48)

d

The independence of S on the X -variable follows from the first, third and sixth relation in (44), by differentiation. For uniform flows with a flat free surface .X / D 0 > d, corresponding to the constant velocity .U0 ; 0/ with the hydrostatic pressure distribution P0 D Patm  gŒY  0 , we can compute 8 ˆ ˆ M D .U0  c/.0 C d/ ; ˆ ˆ < .c  U0 /2 C .0 C d/ ; HD 2g ˆ ˆ ˆ ˆ : S D .c  U0 /2 .0 C d/ C 1 g.0 C d/2 : 2

(49)

In particular, we see that 2gH D

 M= 2 C 2g.0 C d/ : 0 C d

(50)

Given M, the right side of (50), as a function of  D .0 C d/ > 0 ; has a minimum .2gH/ D 3 ŒgM=2=3 at the critical value   D Œ.M=/2 =g1=3 , being strictly monotone (without an upper bound) on either side of   . Consequently, for every 2gH > .2gH/ there exist two positive roots  , with  <   < C ; while   is the only double root of (50), corresponding to 2gH D .2gH/ . The uniform flows corresponding to the distinct solutions  are called subcritical and supercritical, respectively, and, unless  D   D C , the corresponding values of S; S , are different. Indeed, from (49) and (50), we get n 3 2o ; S D 2g H    4

24

A. Constantin

so that S D SC for  < C holds only if H D 34 . C C /. On the other hand, the fact that  are roots of the cubic polynomial equation in .0 C d/ 2 associated to (50) yields by subtraction that .C C  /H D C C  C C 2 . The equality of the two expressions for H amounts to C D  . Consequently, the uniform flow solutions to (44) are representable by the three invariants .M; H; S/. Given M, this property extends to long waves of small amplitude, provided that H > H is sufficiently close to H (see the discussion in [4, 44]). While the overall interdependence of the possible ranges of M; H and S remains to be elucidated, these considerations nevertheless suggest the introduction of dimensionless variables that give prominence to the rôle of M and H, leaving S as a free parameter and thus providing some degree of freedom in the quest to obtain a large class of solutions that go beyond waves of small amplitude. We adopt this viewpoint rather than the one pursued in Sect. 2, where direct variations in amplitude were pivotal. Concerning the interconnection between the invariants M and H, note that a disproportion between energy propagation and the transport of matter is typical for water waves. Indeed, while waves do cause the water to move, the idea that waves transport water from one location to another is misleading: in essence, waves are the outward manifestation of energy propagating through water—disturbances caused by the movement of energy from a source through a medium (i.e. water), the wave motion being quite different from the motion of the medium itself.9 Also, changes in the values of these invariants induce variations of the corresponding wave amplitudes.10 One advantage of a dimensionless form of the problem is that it provides a simplification since all the physical constants disappear, being replaced by a generally smaller number of dimensionless parameters; the expense being that the often convenient dimensional checking is lost. A dimensionless formulation facilitates analytical investigations and numerical simulations, so that one may gain structural insight. In particular, this procedure makes it manageable to assess the relative importance of various terms, so that one can decide with ease which phenomena, at which scale, one intends to study. Prior to presenting a dimensionless alternative formulation of the governing equations (44), it is desirable to reduce the number of variables without some loss of information. A reduction of the number of unknowns can be achieved by introducing the stream function .X ; Y/, defined up to an additive dimensional constant by X D V ;

Y D U  c :

(51)

9 In particular, the wave speed c should not be confused with the horizontal component U of the fluid velocity: energy is moving at the speed of the wave, but water is not (see the discussion in Sect. 4.4). 10 For example, in linear water wave theory energy is roughly proportional to amplitude squared (see [45]); each linear metre of a wave 2 m above the sea level represents an energy flow of about 25 kW (34 horsepower), cf. [35].

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

25

Note that the fifth and sixth relations in (44) ensure that must be constant on the flat bed Y D d and on the free surface Y D .X /. Since (45) and (51) show that the difference between the constant values of on the free surface and on the flat bed is precisely M=, we can determine uniquely by setting D 0 on the free surface and D M= on the bed. Recalling (47), the governing equations (44) are transformed to the equivalent system 8 ˆ ˆ X X C YY D 0 for  d  Y  .X / ; ˆ ˆ ˆ ˆ < D 0 on Y D .X / ; ˆ ˆ D M= on Y D d ; ˆ ˆ ˆ ˆ : 2 X C Y2 C 2g.Y C d/ D 2gH on Y D .X / :

(52)

Instead of the four unknowns .X /; U.X ; Y/; V.X ; Y/; P.X ; Y/, of the system (44), (52) involves only two functions that have to be determined, .X / and .X ; Y/, which are both L-periodic and symmetric in the X -variable. Note that .U  c/ and V are easily found by means of (51), if is known, with the caveat that U appears in the moving frame (43) only in the combination .U  c/. The non-trivial issue of splitting .U  c/, to determine U and the wave speed c, involves a discussion of the possible presence of underlying currents (see Sect. 4.2). Note that the pressure P is altogether absent from (52). It can be recovered using the dynamic boundary condition—the last relation in (44)—and the invariance of the expression on the right of (46), by setting P D Patm C gH  g.Y C d/  

X2 C Y2 : 2

(53)

Thus Euler’s equation, expressed componentwise by the first two equations in (44), becomes simply an alternative way to express the gradient of P. Throughout our discussion we consider a fixed flat bed Y D d, but we do not assume that there is a fixed overlying fluid volume; in particular, if Stokes waves propagate on the surface of the water, we do not impose the condition that the mean depth is d (see also the discussion related to relation (59) below). To nondimensionalise the system (52), given M and H, we use the length scale  (the depth of the uniform subcritical flow corresponding to these invariants); we do not introduce a separate scale for horizontal and vertical motions—as we did in Sect. 3—because we would like to preserve the harmonicity of the dimensionless counterpart of . The set of non-dimensional variables is given by: 8 X Y Cd ˆ ˆ yD ; 0 M2

(57)

is a dimensionless control parameter.11 Due to (54), the unknown functions .x/ and .x; y/ in (56) have to be periodic in the x-variable, with period `D

L : 

(58)

Shifting x by a suitable constant, if necessary, we may assume that the wave crests/troughs are located at x D j` and x D .2j C 1/`=2 with j 2 Z, respectively. Glancing at (56), it is tempting to expect x 7! .x/ to have zero mean, by analogy to the considerations in Sect. 2. However, anticipating the results presented in Sect. 4.2,

11

This terminology is used since variations of such parameters bring about qualitative changes in the flow regime even for fixed geometry and boundary conditions. If .U0  c; 0/ is the uniform subcritical flow with relative mass flux M and head H, then M D .U0  c/ , so that 1 D .U0 c/2 g

p0 cj . In hydraulics the range Fr > 1 defines is the square of the Froude number Fr D jU g the supercritical uniform open channel flows .U0  c; 0/, to be avoided in design, if possible, since they are highly unstable.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

27

we point out that actually Z 0

`

.x/ dx > 0 ;

(59)

for all solutions to (56) that represent genuine waves (that is, with a non-flat free surface) in a flow without an underlying current. This outcome highlights the fact that the solutions of the dimensionless system (56) do not arise as perturbations of the still state of a water body with a fixed volume.

4.2 Stagnation Points and Underlying Currents An important observation is that, in the absence of underlying currents, the horizontal fluid velocity U in the flow beneath a Stokes propagating at wave speed c > 0 can never attain the value c: U.X ; Y/ < c

 d  Y  .X / :

for

(60)

To verify the validity of the claim (60), we first discuss an important aspect concerning the possible presence of underlying currents. In the setting of periodic waves travelling at constant speed at the surface of water in irrotational flow over a flat bed, the mean flow at any depth beneath the trough level must be uniform. Indeed, given y0 2 .0; 1 C .`=2/, we get Z

` 0

Z

Z

y0

yy .x; y/ dydx D 

0

y0

Z

0

` 0

xx .x; y/ dxdy

D 0;

in view of (56) and the periodicity property in the x-variable. Thus Z

`

0

Z y .x; 0/ dx D

`

y .x; y0 / dx ;

0

or, equivalently, using (51) and (54) to pass to the physical moving frame, Z

Z

L 0

U.X ; d/ dX D

L 0

U.X ; Y0 / dX ;

(61)

for any horizontal level Y D Y0 below the trough level Y D .L=2/. Consequently the dimensional constant 1 D L

Z

L 0

M 1 U.X ; d/ dX D c C  `

Z

` 0

y .x; 0/ dx

28

A. Constantin

represents the strength of the (uniform) underlying current and we can characterise the absence of underlying currents—a setting which corresponds to swell entering a region of still water12 —by the constraint Z Z 1 L M 1 ` Y .X ; d/ dX ; (62) cD y .x; 0/ dx D   ` 0 L 0 defining the (dimensional) wave speed associated to a solution of (56). A solution of (56) is a harmonic function throughout the closure D of the dimensionless fluid domain D D f.x; y/ W 0 < y < 1 C .x/g: Due to the periodicity in the x-variable, the weak maximum principle [37] ensures that the maximum and minimum values of the harmonic function throughout ˝, where ˝ is the periodicity cell containing the origin, ˝ D f.x; y/ W x 2 .`=2; `=2/ ; 0 < y < 1 C .x/g ; are attained only on the upper or lower boundary, unless is constant. The second and third relation in (56) therefore ensure that the minimum and maximum of are attained only on y D 0 and y D 1 C .x/, respectively. Moreover, at any point on the these boundaries, Hopf’s maximum principle [37] yields y > 0. But y .x; y/ is itself harmonic and periodic in the x-variable throughout D. Consequently y > 0 throughout D, by the weak maximum principle. Since c > 0, (62) shows that M < 0:

(63)

This, in combination with (54) and the fact that y > 0 in D, proves (60). We are now in a position to prove the validity of (59). For this, in view of (54), it suffices to establish that min

X 2ŒL=2;L=2

f.X / C dg D .L=2/ C d >  :

(64)

Let us first show that U.L=2; Y/ decreases strictly as Y raises from  d to .L=2/ : Indeed, using its periodicity and anti-symmetry, we see that any solution is such that

12

(65) to (56)

x .0; y/

D0

for y 2 Œ0; 1 C .0/ ;

(66)

x .`=2; y/

D0

for y 2 Œ0; 1 C .`=2/ ;

(67)

For a discussion of the effects of a current (that is, ¤ 0) we refer to [23].

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows x .x; 0/

D0

for x 2 Œ0; `=2 :

29

(68)

Moreover, since  0 .x/ < 0 for x 2 .0; `=2/ and y > 0 throughout D, by differentiating the second relation in (56) with respect to the x-variable we deduce that x .x; 1

C .x// > 0

for x 2 .0; `=2/ :

(69)

Consequently, the strict minimum of the harmonic function x on the boundary of the region f.x; y/ W x 2 .0; `=2/; 0 < y < 1 C .x/g is attained all along f.`; y/ W 0 < y < 1 C .`=2/. By Hopf’s maximum principle we therefore know that xx .`=2; y/ < 0 for all y 2 .0; 1 C .`=2//. Since is harmonic, this means that yy .`=2; y/ > 0 for y 2 .0; 1 C .`=2//, and, in view of (54) and (63), we see that (65) holds true. Recalling now the notation of Sect. 4.1, if   .L=2/ C d, the fact that the  M= 2 C 2g is decreasing for lengths    yields expression  

 M= 2 M= 2 C 2gŒ.L=2/ C d  C 2g D 2gH ; .L=2/ C d 

taking (55) into account. On the other hand, V.L=2; 0/ D 0 by periodicity and antisymmetry in the x-variable, so that evaluating (47) at the wave trough Y D .L=2/, we obtain Y2 .L=2; .L=2// C 2gŒ.L=2/ C d D 2gH : From the last two displayed relations we would infer  2 ŒM=2  .L=2/ C d Y2 .L=2; .L=2// ; which is impossible since (45) evaluated at X D L=2, and the fact that Y .L=2; .L=2// D U.L=2; .L=2//  c < 0 is a strict minimum of the expression Y .L=2; Y/ as Y runs from d to .L=2/, cf. (65), ensure Z 0 > M= D

.L=2/ d

  Y .L=2; Y/ dY > .L=2/ C d Y .L=2; .L=2// :

The obtained contradiction validates (59). The dimensionless form of (60), stating that y .x; y/

> 0 for 0  y  1 C .x/ ;

(70)

30

A. Constantin

permits one to use methods from the regularity theory of elliptic boundary value problems (see [19]) or, alternatively, from complex analysis (see [59]), to show that for a (smooth) Stokes wave the free surface and both components of the velocity field are real analytic. This ensures the existence of local power series expansions which provide useful information even though the involved algebraic complexity prohibits us from going beyond the leading corrections.

4.3 Existence Theory Early on, starting with Stokes in the nineteenth century (see [56] and the discussion in [57]), computational approaches to Stokes waves were made available and even nowadays the engineering literature abounds of power series approximations (e.g. formal expansions up to fifth order in the wave steepness). However, these have serious drawbacks. The first issue is that the series may not converge in the region where the solution is needed, accurate estimates for the radius of convergence being out of reach. Furthermore, even for waves of small amplitude it turns out that if the water is shallow, then the contribution of the higher-order terms in the power series expansion will tend to dominate, and results obtained by truncation at a certain order will not be accurate (see the discussion in [33]). To avoid these issues, the modern existence theory for Stokes waves relies on bifurcation methods, and indepth qualitative studies offset the incertitudes associated with formal quantitative considerations. After providing a summary of some of the ideas and directions which were developed in abstract bifurcation theory, we sketch the use of the procedures in the context of Stokes waves. For this, a suitable variational reformulation of the governing equations, aimed at reducing the number of unknowns and at identifying structural properties which ease the computational undertaking, will be presented. The mathematical study of these alternative formulations of the governing equations for Stokes waves broadens our understanding of the physical phenomenon. While certain steps in operating with mathematical objects might be devoid of an outright physical interpretation, often the conclusions reached have a natural counterpart.

4.3.1 Abstract Bifurcation Theory Every model of a physical phenomenon contains physical parameters which may vary over certain specified sets. In vague terms, the values of the parameters where a drastic change in the behaviour takes place are called bifurcation values. Bifurcation is a paradigm for non-uniqueness, and in the context of Stokes waves a relevant scenario is the following: starting from a parameter-dependent family of uniform irrotational flows (trivial waves, with a flat free surface), critical values of the parameter bring about the possibility of the creation of genuine waves—the flat state might no longer be observed, and the flow takes on a new state that “bifurcates” from

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

31

f solution set of F( , f ) = 0 bifurcation point ( *, f 0 ) curve of known solutions

*

Fig. 3 A typical one-parameter bifurcation diagram: the real scalar  belongs to some open interval and the function f belongs to an open set in a suitable function space

the continuous branch of flows with a flat free surface. One distinguishes between a local theory, which describes the bifurcation diagram in a neighborhood of the bifurcation point, and a global theory, where the continuation of the local non-trivial solution branch beyond that neighbourhood is investigated. In abstract terms, one-parameter bifurcation theory is the study of functional equations F.; f / D 0, where the parameter  is a real scalar,13 f is an unknown function and F a known operator. A bifurcation point of a known solution set .; f0 .// is a point . ; f0 . // from which another parameter-dependent solution f1 ./ branches, that is, f1 . / D f0 . / but f1 ./ ¤ f0 ./ for all  in some open interval having  as an endpoint (Fig. 3). To be technically more specific, it is advantageous to treat the map F.; f / as a whole, de-emphasizing the special role of the parameter : let X and Y be real Banach spaces and F a continuous map from an open subset O of RX to Y, having a continuous Fréchet derivative with respect to f ; Ff .; f /, which is continuous in O. If .0 ; f0 / 2 O is such that F.0 ; f0 / D 0 and if Ff .0 ; f0 / is an isomorphism of X onto Y, then, by the implicit function theorem (see [7]), there is " > 0 and a unique continuous map f W .0  "; 0 C "/ ! X such that f .0 / D f0 ; .; f .// 2 O and F.; f .// D 0 for all  2 .0  "; 0 C "/. If F is of class Cp for some p  1, then the map  7! f ./ is of class Cp . Moreover, if F is real analytic, then so is the solution f ./ because it is obtained, by means of the contraction principle, as the unique limit of iterates, all of which are real analytic. This fact permits us to deduce that the possible locations for bifurcation points are where Ff .0 ; f0 / fails to be an isomorphism, or, using the open mapping theorem and the fact that Ff .0 ; f0 / is a bounded linear operator from X to Y, points where Ff .0 ; f0 / fails to

Allowing the parameter  to be multi-dimensional corresponds to “multiparameter bifurcation theory”.

13

32

A. Constantin

be a bijection. In particular, this draws attention to points .0 ; f0 / where the kernel N ŒFf .0 ; f0 / is not trivial. While this necessary condition is generally not sufficient (see the discussion in [7, 13]), additional requirements for sufficiency are available in many guises. The following result on bifurcation at simple eigenvalues has wide applicability, even if it does not exhaust all possible bifurcations: The Crandall–Rabinowitz Local Bifurcation Theorem Let X; Y be real Banach spaces and let F 2 Ck .O; Y/ with k  2, where O RX is an open set containing the point . ; 0/, satisfy: (i) F.; 0/ D 0 for all .; 0/ 2 O; (ii) L D @f F. ; 0/ 2 L.X; Y/ is a linear Fredholm operator of index zero and one-dimensional kernel N .L/ generated by some f  2 X n f0g; (iii) the transversality condition Œ@2;f F. ; 0/ .1; f  / 62 R.L/ holds, where ˇ @2;f F. ; 0/ D @ Œ@f F.; 0/ˇD 2 L.R; L.X; Y// D L.R  X; Y/; here R.L/ stands for the range of the linear operator L. Then there exist " > 0 and a branch of solutions f.; f / D ..s/; s .s// W s 2 R; jsj < "g R  X of F.; f / D 0 with .0/ D  ; .0/ D f  , and such that s 7! .s/ 2 R; s 7! s.s/ 2 X are of class Ck1 on ."; "/. Furthermore, there exists an open set O0

O with . ; 0/ 2 O0 and f.; f / 2 O0 W F.; f / D 0; f ¤ 0g D f..s/; s .s// W 0 < jsj < "g: If F is real analytic, then  and  are real analytic on ."; "/. We refer to [7, 27] for a proof of the above result and for a discussion of the relevance of the transversality condition and of the requirement that @f F. ; 0/ 2 L.X; Y/ is a Fredholm operator of index zero (in the context of a simple eigenvalue, this means that the range of the operator is closed and has a one-dimensional complement). The Crandall-Rabinowitz theorem elucidates the local structure of the solution set near the bifurcation point . ; 0/ by linearisation, and will prove adequate in our context for waves of small amplitude. With the aim of describing waves of large amplitude, it is essential to deal with the existence in-the-large of solutions. Relatively sophisticated topological techniques, especially degree theory, proved very powerful in establishing a quite general framework for the existence of large connected sets of solutions, which capture nonlinear phenomena that are not merely small perturbations of linear processes. However, in contrast to the structure of the solution set near the bifurcation points, even when the operators involved are smooth, these global solutions sets, while connected, fail to be path-connected. This gap can be closed by using the fact that analytic sets (sets defined by the vanishing

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows Fig. 4 The analytic variety representing the set of zeros of the real-analytic function f .x; y/ D .x2 C y2 /.x  2/.y  2/Œ.x  2/2 C .y  2/2  1

33

y

2

0

2

x

of an analytic function) have a nice, distinguished structure. Indeed, in contrast to the case of smooth (C1 ) functions f W Rn ! R, in which case any closed subset of Rn could be the set of zeros, the set of zeros of a real-analytic function f W Rn ! R, called a real-analytic variety, can be stratified into submanifolds of dimensions 0; 1; : : : ; N; for some N  0; see the discussion in [7, 13, 29]. This feature permits to show that, under fairly mild hypotheses, in a deleted neighbourhood of a singular point, the solution set of F.; f / D 0, where F is a real-analytic map from an open subset O of RX to Y, consists either of no points (isolated singularity) or of a finite even number of branches that can be matched in unambiguous pairs. In particular, it is not possible to have an odd number of solution branches in a neighbourhood of a singular point (see Fig. 4). Intuitively, to obtain a path-connected global solution set, one can start at the bifurcation point along the non-trivial local solution curve provided by the Crandall-Rabinowiz theorem, continue until reaching a singular point of the solution set, and at this stage continue along the unique branch that is coupled to the one used to reach the singular point. The following version of the global bifurcation theorem for real-analytic operators is discussed in [7, 25]: the result presented in [7] being slightly inaccurate as stated there, a corrected version is provided in [25]. Analytic Bifurcation Theorem Let X and Y be real Banach spaces, O be an open subset of R  X containing the point . ; 0/, and F W O ! Y be a real-analytic function. Suppose that .H1 / F.; 0/ D 0 for all .; 0/ 2 O; .H2 / @f F.; f / is a Fredholm operator of index zero whenever F.; f / D 0; .H3 / the kernel N Œ@f F. ; 0/ and the complement of the range RŒ@f F. ; 0/ are both one-dimensional, with the kernel generated by f  , and the transversality condition Œ@2f ; F. ; 0/ .1; f  / 62 R.@f F. ; 0// holds; .H4 / in R  X, all bounded and closed subsets of f.; f / 2 O W F.; f / D 0g are compact. Then there exists in O a continuous curve K D f..s/; f .s// W s 2 Rg of solutions to F.; f / D 0 such that:

34

A. Constantin

.C1 / ..0/; u.0// D . ; 0/; u.s/ D sf  C o.s/ for jsj sufficiently small, and f.; f / 2 W W f ¤ 0; F.; f / D 0g D f..s/; f .s// W 0 < jsj < "g, for some neighbourhood W of . ; 0/ and some " > 0 sufficiently small; .C2 / K has a real-analytic parametrization locally around each of its points; .C3 / One of the following alternatives occurs: (˛) as s ! 1; ..s/; f .s// eventually leaves every bounded closed subset B of O; (ˇ) there exists some T > 0 such that ..s C T/; f .s C T// D ..s/; f .s// for all s 2 R. Moreover, a curve of solutions to F.; f / D 0 having the properties .C1 /  .C3 / is unique (up to reparametrization). Note that, in view of the real-analytic version of the Crandall-Rabinowitz local bifurcation theorem, the assumptions .H1 / and .H3 / ensure the existence of a realanalytic local bifurcating curve Kloc D f..s/; f .s// W s 2 ."; "/g of solutions to F.; f / D 0 with the properties stated in .C1 /. The curve K exhausts all possibilities of extending the local bifurcation curve Kloc by real-analytic arcs in such a way that K has a real-analytic parametrization around each of its points (see Fig. 5). However,

Fig. 5 The global bifurcation curve K R  X consists of distinguished real-analytic open arcs Cj that end in the branch points Bj1 and Bj if j > 0, or in the branch points Bj and BjC1 if j < 0, with B0 D . ; 0/ and where the local bifurcation curve Kloc is a subset of C1 [ fB0 g [ C1 . A point on Cj corresponds to a non-singular solution (in a neighbourhood of which the implicit function theorem applies), while each Bi arises as the unique intersection point of the closures of a finite even number of open one-dimensional real-analytic varieties. A distinguished arc Cj can be uniquely continued across Bj by choosing an outgoing branch CjC1 if j > 0, or Cj1 if j < 0, so that each curve Cj [ fBj g [ CjC1 if j > 0, or Cj [ fBj g [ Cj1 if j < 0, admits a local uniformizing real-analytic parametrization near Bj

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

35

the global bifurcation curve K is not necessarily a maximal connected subset of the solution set f.; f / 2 O W F.; f / D 0g.

4.3.2 Alternative Formulations of the Free-Boundary Problem Prior to the outline of a bifurcation approach showing the existence of a global curve of solutions to the problem of the existence of Stokes waves, we present an advantageous formulation of the Stokes-wave problem as a quasi-linear pseudodifferential operator equation, drawing upon theory developed in the studies [2, 8, 9, 60] of irrotational travelling water waves of infinite depth. Notwithstanding the fact that some technical considerations are unavoidable since the approach mixes complex analysis, harmonic analysis and the calculus of variations in an essential way, we try to provide a conceptual understanding without immersing ourselves in technicalities.

Re-formulation by Means of Conformal Mappings In this subsection we show that, by means of a suitable conformal mapping, one can re-formulate the free-boundary problem (56) as a fully nonlinear pseudo-differential equation for a periodic function of a single variable. The condition of irrotational flow, expressed by the fourth relation in (56), allows us to introduce the velocity potential .x; y/ uniquely up to an additive constant by x D

y

;

y D 

x

;

.x; y/ 2 D ;

(71)

using the notation introduced in Sect. 4.1. Setting Z .x; y/ D

Z

x 0

y .l; 0/ dl

y

 0

x .x; s/ ds ;

x 2 R;

0  y  1 C .x/ ;

we have .0; y/ D 0

for y 2 Œ0; 1 C .0/ :

Defining 1 c0 D `

Z 0

`

y .x; 0/ dx ;

(72)

36

A. Constantin

since Z` .x C `; y/  .x; y/ D

y .s; 0/ ds ; 0

we see that .x; y/ 7! .x; y/  c0 x is odd and periodic with period ` in the x  variable : The stream function and the velocity potential  are harmonic conjugate functions, with the complex mapping .x C iy/ 7! .x; y/ C i .x; y/ analytic throughout the domain D : In view of the second and third relation in (56), and since x > 0 throughout D by (70), one can see that, setting wD

2 ; `c0

(73)

the orientation-preserving conformal change of variables 

a D w .x; y/; b D w .x; y/ ;

(74)

is a global diffeomorphism from D to the closure of the strip Rw D f.a; b/ W a 2 R ; w < b < 0g : The scaling in (74) was chosen to ensure that an increase of x by ` leaves b unchanged and increases a by 2. A first benefit of the hodograph transform (74) is that it permits us to re-formulate the nonlinear free-boundary problem (56) as a nonlinear boundary problem in the fixed strip Rw of width w, for the unknown height D above the flat bed, D.a; b/ D y : Indeed, we have ( @a D Db @x C Da @y ; @b D Da @x C Db @y ;

(

@x D w

(75)

y

@y D w

@a C w

x

x @a C w

@b ; y

@b ;

(76)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

where 8 ˆ ˆ Da D 
0 warranting the existence of a conformal mapping ˛ C iˇ from a horizontal strip Rw of with w to D, which admits an extension as a homeomorphism between the closures of these domains and satisfies the properties (80). Indeed, since .aCib/ 7! ei.aCib/ is a conformal diffeomorphism from the periodicity cell f.a; b/ W 0  a < 2; w  b  0g to the annulus fz 2 C W 1  jzj  ew g, we see that the claim is equivalent to a statement about the conformal equivalence of two such annuli (in the sense of the existence of a bijective analytic map between them); it is known (see [52]) that two annuli are conformally equivalent if and only if they have the same ratio of outer radius to inner radius (Fig. 6). By means of the Dirichlet-Neumann operator, which delivers normal derivatives (“Neumann data”) given boundary measurements (“Dirichlet data”), we can further reduce the dimension of the boundary-value problem (78), showing that it is equivalent to a quasilinear pseudo-differential equation for the 2-periodic function of a single variable h.a/ D ˇ.a; 0/ D D.a; 0/ :

(81)

38

A. Constantin

a=0

x=0

a=2

x=

Fig. 6 The conformal parametrisation of the (dimensionless) fluid domain

Note that by (80) we have Z @b

2 0

Z ˇ.a; b/ da D

2 0

Z ˇb .a; b/ da D

2 0

˛a .a; b/ da D `

for all b 2 .w; 0/. Since ˇ.a; w/ D 0 for all a 2 R, we see that Z

2 0

ˇ.a; b/ da D `.b C w/ ;

b 2 Œw; 0 I

in particular, Z

2 0

h.a/ da D `w :

(82)

To proceed we need some results from harmonic analysis, about conjugation and the Hilbert transform acting on periodic functions defined on a strip; for proofs of the stated results we refer to [24, 25]. For our purposes it is convenient to introduce, given an integer n  1 and a constant s 2 .0; 1/, the n;s class C2 .R/ of Banach spaces, consisting of functions f W R ! R which are 2-periodic and n-times continuouslyn differentiable,owith derivative f .n/ satj f .n/ .a2 /f .n/ .a1 /j < 1. Moreover, let isfying the Hölder condition sup ja2 a1 js 0 a1 0 and f D .q; f/ 2 X, by  F1 .; f / D 2q.1  w/ C

 1  1 C w  0 C f .f /   q C w 2 w2 22 w2

 o 1 C w  n 0 0 0  qC f C .f /  ŒŒf C .f / C C .ff / ; w w w 22 w2

(109)

and F2 .; f / D

hhn

2q.1  w/ C

1 2 w2

  2 oii 1 C w  on 1  2q C 2 2 f .f0 /2 C  C Cw .f0 /  2: w w

(110)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

47

If .; f / D .; .q; f// is such that F.; f / D 0, then, since w is fixed, (105) gives the corresponding dimensionless wavelength ` and mean depth ŒŒh, (106) determines the corresponding dimensionless control parameter , and (104) provides us with the dimensionless wave profile h. Concerning the choice of function spaces, it is foolhardy to seek directly smooth (C1 ) or real-analytic solutions to (107) since it is easier to first consider solutions in a larger class of functions, tailored to optimise the analytical considerations, and then proving that such solutions are necessarily more regular than other members of the class. In our setting, the form of (109)–(110) suggests that f should be at least differentiable, and it turns out that already in the pursuit of waves of small amplitude it is convenient to avail of the second derivative of f (see the next subsection). On the other hand, for a merely continuous 2-periodic function f, while Cw .f/ is welldefined as a function in Lp Œ0; 2 for any p 2 .1; 1/, it need not be continuous and may even be unbounded at some points.16 However, controlling the modulus 0;s of continuity of f, by requiring f to belong to C2 .R/ for some s 2 .0; 1/, will 0;s redress this shortcoming since Cw .f/ 2 C2 .R/ is then granted (see [7]). These considerations justify the specific choice of the function space X, with Y emerging by accounting for the loss of one derivative inherent to (109). As for regularity, note that standard results which characterise the behaviour of conformal mappings 1;s at the boundary (see [51] and the discussion in [24]) show that if h 2 C2 .R/ for some s 2 .0; 1/ solves (89), then the conformal mapping ˛ C iˇ from the strip Rw to the dimensionless fluid domain D, given by (79)–(80), with ˇ.a; 0/ D h.a/, 1;s cf. (81), will admit a C2 .Rw / extension. The conformal property ensures ˛a C iˇa ¤ 0 in Rw . Moreover, since the harmonic function ˇ satisfies ˇ > 0 in Rw and ˇ.a; w/ D 0 for all a 2 R, from Hopf’s maximum principle it follows that 0 < ˇb .a; 0/ D ˛a .a; 0/ for all a 2 R. On the other hand, (81), (88), (89), in combination with the Cauchy-Riemann equation ˛a D ˇb , ensure ˛a2 C ˇa2 > 0 on the upper boundary b D 0 of Rw , so that ˛a2 C ˇa2 > 0 in

Rw :

This permits us to write ˛a C iˇa D

q

˛a2 C ˇa2 ei ;

(111)

0;s where  2 C2 .R p w / is such that .a; 0/ D 0 for all a 2 R and the complex map .a C ib/ 7! log ˛a2 .a; b/ C ˇa2 .a; b/ C i.a; b/ is analytic in Rw . Setting

0 .a/ D .a; 0/ ;

a 2 R;

(112)

16 Since Cw is the sum of the periodic Hilbert transform C and the operator of convolution with a specific smooth 2-periodic function, cf. [24], this replicates a feature which is well-established for C (see [18]). By the same token, Cw shows some “traces of continuity”, in that it has the intermediate mean-value property on the set in which it is finite, as it is the case for C , cf. [66].

48

A. Constantin

(81), (88) and the Cauchy-Riemann equation ˛a D ˇb yield Cw .0 / D log or

p p   .h0 /2 C .Gw .h//2  log .h0 /2 C .Gw .h//2 ;

 p p   : 0 D Cw1 log .h0 /2 C .Gw .h//2  log .h0 /2 C .Gw .h//2

(113)

Since 1 C 2  2h.a/ > 0 for all a 2 R, we may write (89) as p .h0 /2 C .Gw .h//2 D

w2

1 : p 1 C 2  2h

1;s In combination with (113), this shows that 0 2 C2 .R/. In view of (81), from (111) we now get

h0 D

w2

sin 0 1;s 2 C2 p .R/ ; 1 C 2  2h

2;s 1Cn;s .R/. Repeating the procedure n times yields h 2 C2 .R/. so that h 2 C2 Consequently h is smooth. As pointed out earlier, h is actually real-analytic. For 2;s this, using elliptic regularity theory (see [19]), one can show that if h 2 C2 .R/, then  is real-analytic, while standard regularity results for conformal mappings up to the boundary [51] now ensure that ˛ C iˇ has an analytic extension17 to a neighbourhood of the real axis b D 0. In particular, this shows that h is real-analytic on R. Changing to the dimensionless variables and  and subsequently to the physical variables, by means of (54), confirms that the free surface profile function and the velocity field have maximal regularity.

Variational Formulation In this subsection we digress from the main path to discuss a variational derivation of (109)–(110). This approach distinguishes the physically relevant conformal maps on the strip Rw among all conceivable maps by associating a functional having a stationary point at a physically relevant input. The motivation for the variational point of view is conceptual. Newtonian mechanics describes the motion of a system in terms of the response of the individual component particles to the action of forces. This intrinsic particle-byparticle viewpoint is in contrast to the variational approach which relies on aggregate quantities associated with the motion of the system as a whole. In the context of Stokes waves these correspond to the difference of the kinetic energy (per 17 Due to the third relation in (80), the analytic extension to a neighbourhood of the line b D w is a consequence of Schwarz’s reflection principle.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

49

unit volume) and the potential energy (per unit volume), with an additive factor accounting for the total mass (weighted by the head), more precisely W.O; / D

“   jr j2  g.Y C d/ C gH dX dY ;  2 O

(114)

where O D f.X ; Y/ W L=2 < X < L=2 ; d < Y < .X /g: Note that while the potential energy is characteristic of the arrangement of the particles, and the kinetic energy is determined by the velocities of the particles, neither depends on the specification of those positions and velocities—the functional is independent on the choice of the coordinate system. While not important for our subsequent considerations, the variational formulation is pleasing since enlarging the physical reality by virtual possibilities and then selecting the physically realistic possibilities as critical points of the functional induces a sense of purpose to natural phenomena, adding another dimension to the causal description. The non-dimensional form of (114) is W.O; / D

M2 W.˝; / ; 

where “  jr j2 1  .y  1/ C dxdy ; W.˝; / D 2 2 ˝

(115)

in view of (54), (55) and (57); the domain ˝ was introduced in Sect. 4.2. We now express (115) as a functional of h D f C w with ŒŒf D 0. Since the conjugation operator Cw , which is fundamental for the conformal parametrization, acts on functions of mean zero, it is convenient to identify h with the pair .; f/, where  D w D ŒŒh

(116)

and regard  and  as parameters appropriate for the containment, in the nondimensional setting, of the wavelength and of the energy, respectively. Green’s theorem and the first three relations in (56) yield “ Z `=2 jr j2 dxdy D y .x; 0/ dx ˝

Z

D D

`=2 

ˇb .a; w/ 2 2  wŒˇa .a; w/ C ˇb .a; w/ Z  

2 2 1 da D D ; w w 

ˇb .a; w/ da

50

A. Constantin

using (76)–(77) to obtain the third integral and the last relation in (80) to get the fourth integral. On the other hand, in view of (76), (81), (86), (88), (104), (105), we get “ Z `=2 Z  1 dxdy D f1 C .x/g dx D ˇ.a; 0/ ˇb .a; 0/ da ˝

Z D

`=2  



n on   o  C f.a/  C Cw .f0 / .a/ da Z

D 2 C

 

  f.a/ Cw .f0 / .a/ da ;

and “

Z

˝

.y  1/ dxdy D Z D Z D

`=2

n Œ1 C .x/2 2

`=2   

n ˇ 2 .a; 0/

o  ˇ.a; 0/ ˇb .a; 0/ da

2 n ΠC f.a/2 2



o  1  .x/ dx

on   o    f.a/  C Cw .f0 / .a/ da Z

2

D   2 C .  1/  C 2

Z



1 f2 .a/ da C 2 

Z



 



  f.a/ Cw .f0 / .a/ da

  f2 .a/ Cw .f0 / .a/ da :

Consequently W.˝; / can be expressed as W.; f/ D

n o  C  1 C 2  2   Z   n 1o  f.a/ Cw .f0 / .a/ da C .1  / C 2  Z  Z      2  f2 .a/ Cw .f0 / .a/ da  f .a/ da : 2  2 

(117)

The domain of definition of the functional W is the space of pairs .; f/ with  > 1 2;s and f 2 C2;o .R/ for some s 2 .0; 1/. We claim that: Any critical point18 of the functional W, corresponding to variations taken with respect to the function h and with respect to the scalar w, satisfies Eq. (109) and the scalar constraint (110). 18

Generally, in the calculus of variations, a change in the geometric or topological type of the level sets of the functional is revealed by the existence of a critical point.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

51

To validate this variational formulation of the system (109)–(110), we compute the first variation of W, considering in turn the variations of f and . 2;s For  > 1; f 2 C2;o .R/ and g smooth, 2-periodic, and of zero mean, the variation W.; f C "g/  W.; f/ ıW .; f/ g D lim "!0 ıf "

(118)

with respect to f is Z Z  n     o 1 on  .1  / C g.a/ Cw .f0 / .a/ da C f.a/ Cw .g0 / .a/ da 2   Z  Z  n     o   2f.a/g.a/ Cw .f0 / .a/ da C f2 .a/ Cw .g0 / .a/ da 2   Z  f.a/g.a/ da :  

Since the operator f 7! Cw .f0 / is self-adjoint, we can write the above as Z .2 C 1/ nZ 

 



 

  g.a/ Cw .f0 / .a/ da

Z   f.a/g.a/ Cw .f0 / .a/ da C



Z 





  o g.a/ Cw .ff0 / .a/ da

f.a/g.a/ da :

At a critical point we have

ıW .; f/ g D 0, so that ıf Z

 

g.a/ .a/ da D 0 ;

(119)

where     .a/ D  f.a/   f.a/ Cw .f0 / .a/   Cw .ff0 / .a/  n o C 2.1  / C 1 Cw .f0 / .a/ : The validity of (119) for all smooth 2-periodic functions g of zero mean implies that  is a constant. The value of this constant is found by taking averages: since ŒŒf D ŒŒCw .f0 / D ŒŒCw .ff0 / D 0, we obtain  D  ŒŒf Cw .f0 / :

52

A. Constantin

Note that the above equation is precisely (109). We will take advantage of this fact in our subsequent variational considerations. To compute the variation of W with respect to , note first that the representations h.a/ D ŒŒh C

1   X an cos.na/ C bn sin.na/ ; nD1

  Cw .h0 / .a/ D

1 X

n coth

nD1

 n    an cos.na/ C bn sin.na/ ; 

obtained by reading off (85), yield 1  d     n i  X n2 h Cw .h0 / .a/ D 1  coth2 an cos.na/ C bn sin.na/ : d   nD1

Therefore d 1 1 Cw .h0 / D  h00  Cw2 .h00 / ; d  

2;s h 2 C2 .R/ :

(120)

From (117) we now compute Z  n   ıW 1 C o 0   .; f/ D 2.1  /   f.a/ C .f / .a/ da w ı 22  Z n   o 1n 1o   .1  / C f.a/ f00 .a/ C Cw2 .f00 / .a/ da  2  Z  n   o  f2 .a/ f00 .a/ C Cw2 .f00 / .a/ da : C 2  Clearly Z



Z

00

ff da D 





0 2



.f / da ;

Z



Z

2 00

f f da D 2







f .f0 /2 da :

On the other hand, since f ! Cw .f0 / is self-adjoint, we have Z Z



  

f Cw2 .f00 / da D f2 Cw .f0 / da D D

Z



Z

 

Z

  

 2 Cw .f0 / da n

o f2  ŒŒf2  Cw .f0 / da

Z   f Cw .f2  ŒŒf2 /0 da D 2

 

f Cw .ff0 / da ;

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

Z



2

f 

Cw2 .f00 / da

Z



Z

 

D D



n

53

o f2  ŒŒf2  Cw2 .f00 / da

Z   Cw .f2  ŒŒf2 /0 Cw .f0 / da D 2

 

Cw .ff0 / Cw .f0 / da :

Furthermore, multiplying (109) by Cw .f0 / and integrating the outcome on Œ; , we get Z 

 

0

0

n

Cw .ff / Cw .f / da D 2.1  / C 1 Z 

 

oZ

 

 2 Cw .f0 / da

f Cw .f0 / da  

Z

 

 2 f Cw .f0 / da :

Using the above relations, we obtain Z  n 1 C o ıW  2 .w; f/ D 2.1  /   f Cw .f0 / da ıw 22  Z 2 1n 1o   C .1  / C Cw .f0 / da  2  Z n 1o  0 2 1 C .1  / C .f / da  2  Z Z 2      f .f0 /2 da  f Cw .f0 / da ;      which is precisely (110) multiplied by =. This corroborates the variational formulation.

Waves of Small Amplitude By applying the Crandall-Rabinowitz local bifurcation theorem to (107) we will now establish the existence of Stokes waves of small amplitude. Clearly F is real-analytic on .0; 1/  R, with F.; .0; 0// D 0 for all  > 0. We can easily compute    1  1 C w 0 2 @.q;f/ F.; .0; 0// .; '/ D C .' /   ' ; 2.1  w/  : w 2 w2 22 w2

54

A. Constantin

The Fourier series expansion19 '.x/ D

1 X

an cos.na/

nD1 2;s .R/ and (85) lead us to the representation of the function ' 2 C2;ı;e



 @.q;f/ F.; .0; 0// .; '/ D

(121)

1 X  w.1 C w/ o an n 2 nw coth.nw/  cos.na/ ; 2.1  w/  2Y 2 w3 2 nD1

for .; '/ 2 X. Since the function t 7! t coth.t/ is strictly increasing on .0; 1/, with range .1; 1/, we see from (121) that @.q;f/ F.; .0; 0// is an isomorphism from X to Y whenever  > 0 is not a solution of w.1 C w/ D 2nw coth.nw/

(122)

for some integer n  0, with the understanding that for n D 0 the right side of (122) is assigned the value 2. The positive solutions of (122) are given by 0

1 D w

and

n

D

1 C

p 1 C 8nw coth.nw/ 2w

with

n  1:

(123)

Note that the sequence fn gn0 is strictly increasing. Moreover, for each integer n  0, the kernel of the operator Ln D @.q;f/ F.n ; .0; 0// is one-dimensional, being generated by fn D .0; cos.na// 2 X for n  1, and by f0 D .1; 0/ 2 X for n D 0, respectively. On the other hand, for n  1 the range R.Ln / of Ln is the closed subspace of Y formed by the elements .˚; t/ 2 Y with t 2 R arbitrary and Z

2 0

˚.a/ cos.na/ da D 0 ;

1;s while R.L0 / is the closed subspace f.˚; 0/ W ˚ 2 C2;ı;e .R/g of Y. Thus, for  any integer n  0, @.q;f/ F.n ; .0; 0// is a Fredholm operator of index zero, with a

19

For the relevant basic results in Fourier analysis we refer to the discussion in [18].

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

55

one-dimensional kernel. From (121) we now compute h

 i  1 1 .cos.na/; 0/ 62 R.Ln / C @2; .q;f/ F.n ; .0; 0// .1; fn / D n w 2.n /2 w2

for n  1, while h i @2; .q;f/ F.0 ; .0; 0// .1; f0 / D .0; 20 / 62 R.Ln / : Consequently, the hypotheses of the Crandall-Rabinowitz theorem are fulfilled and .n ; .0; 0// is a bifurcation point for every integer n  0: (107) admits non-trivial solutions .n ./;  n .// with n W ."n ; "n / ! R and n W ."n ; "n / ! X real analytic for some "n > 0, and with n .0/ D n ; n .0/ D fn . For n  1, since n ./ D .q./; f.; a// D .0;  cos.na// C o. 2 / ;

(124)

2;s in the space R  C2;ı;e .R/, we deduce that for jj > 0 sufficiently small the profile f is not constant. While this already proves the existence of genuine waves of small amplitude, a detailed study gives more insight. Firstly, since the function a 7!  cos.na/ oscillates n times over a period 2, with amplitude 2, (124) ensures that for n  2 and for jj small enough, the function a 7! f.; a/ will have more than one strict local maximum on .0; 2/, and therefore, after passing to physical variables by undoing the non-dimensionalisation (54), it does not correspond to our definition of a Stokes wave. The case n D 1 contrasts with this. Indeed, note that since a 7! f.; a/ is 2-periodic andpeven, we have @a f.; 0/ D @a f.; /pD 0. On the other hand, since sin.a/ p  1= 2 for a 2 Œ=4; 3=4; cos.a/  1= 2 for a 2 Œ0; =4; cos.a/  1= 2 for a 2 Œ3=4; , and (124) yields

@a f.; a/ D  sin.a/ C o. 2 / @aa f.; a/ D  cos.a/ C o. 2 /

uniformly on Œ=4; 3=4 ; uniformly on

Œ0; =4 [ Œ3=4;  ;

we deduce that for  > 0 small enough, @a f.; a/ < 0 on Œ=4; 3=4, while @aa f.; a/= < 0 on .0; =4/ and @aa f.; a/ > 0 on .3=4; /. Consequently, for  > 0 small enough, the value of @a f.; a/ decreases from zero at a D 0 to a negative value at a D =4, then remains negative for a 2 Œ=4; 3=4, and subsequently increases strictly towards @a f.; / D 0 as a runs from 3=4 to . Overall, this means that for  > 0 small enough we will have @a f.; a/ < 0 for a 2 .0; /, so that the profile of the corresponding wave is strictly decreasing between the wave crest and a successive trough. With symmetry granted by the functional-analytic setup, we conclude that near the bifurcation point .1 ; .0; 0//, all points on the local bifurcation curve, with exception of the bifurcation point, correspond to genuine small-amplitude Stokes waves.

56

A. Constantin q

0

*

0

*

*

1

2

*

3

Fig. 8 The curve of non-trivial solutions to (107) which bifurcates at . 0 ; .0; 0// from the curve of trivial solutions f.; .0; 0// W  > 0g lies entirely in the two-dimensional subspace f.; q; 0/ W ; q 2 Rg of R  X. It corresponds to precisely one uniform flow with a flat free surface, with infinitely many different representations in the .; .q; f//-variables

Let us now investigate the situation near the bifurcation point .0 ; .0; 0//. Since D 1=w means ŒŒh D 1 by (105), the bifurcation point represents the solution20 to the following peculiar form of (56): 0

8

D 0 in 0 < y < 1 ; ˆ ˆ ˆ ˆ ˆ < D 0 on y D 1 ; ˆ ˆ ˆ ˆ ˆ :

D 1

on y D 0 ;

jr j2 D 1

(125)

on y D 1 :

This is the only case in which  is a dummy parameter, to which we may assign an arbitrary value. Due to (106), we expect that q D   1 could also be arbitrary. These considerations suggest that perhaps q 7! .0 ; .q; 0// is the solution curve that bifurcates from the curve f.; .0; 0// W  > 0g of trivial solutions to (107), at .0 ; .0; 0//. This hypothesis being easily confirmed by inspection, we conclude that the local bifurcation at .0 ; .0; 0// does not produce new solutions of (56)—the previous solutions to (107), while different in terms of the .; .q; f//-variables, all correspond to the unique solution to (125) already represented by the trivial solution .0 ; .0; 0// of (107) (Fig. 8). The trivial solution corresponding to the bifurcation point .1 ; .0; 0// is, in physical variables, a uniform flow at speed p  2gM 1=3 g cD p D h1  h1 .1 C h1 /

20

It is easy to see that the unique solution of (125) is

.x; y/ D y  1.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

57

in water of uniform depth d D h1  D

 h .1 C h /M2 1=3 1 1 ; 2g2

(126)

in view of (101), (102) and (54); here h1 D 1 w. Using (58) and (105), we get wD

2d h1 ;  D 1 L

(127)

so that (99) and (122) lead to the dispersion relation (30): s cD

g D h21 

s

gL 2 D  2 1 .1 C 1 w/

r

 2d  gL tanh : 2 L

(128)

Since for h  1 the range of the possible values of the head (103) of the uniform flat3  M2 1=3 surface flows with mass flux M consists of all values larger than , we 2 g2 can summarize the previous results on Stokes waves of small amplitude as follows: Theorem 1 Given w > 0, for any fixed value M < 0 of the (relative) mass flux there exists a one-parameter family of uniform flows with a flat free surface and (relative) mass flux M, indexed by the corresponding possible values of the head H

1=3 3 2 M =.g2 / : 2

The specific value H D

 M2 1=3 n 2g2

1=3 o 2=3  1 C 2w coth.w/ 2w coth.w/

of the head triggers the appearance of periodic travelling waves of small amplitude, with conformal mean depth w, which have a smooth profile with one crest and one trough per period, are strictly monotone between successive crests and troughs, and are symmetric about the crest line. Moreover, H is the unique value with this property. The above statement about the range of possible values of the head H for the uniform flat-surface flows follows from (103) and the fact that the map h 7! 1=3  2=3  C h.h C 1/ is strictly monotone for h  1. Instead of indexing h.h C 1/ these flows by the values of H, one could rely on the associated mean depth (126), that varies when H does so. While explicit formulas can be made available for this alternative, they are excessively complicated. Indeed, this alternative classification amounts to finding h > 1 from the assignment of a value Z > 3  22=3 to the

58

A. Constantin

 2=3  1=3 expression h.h C 1/ C h.h C 1/ . In terms of z D h.h C 1/, we have to find the root of the cubic polynomial z3  Zz2 C 1. Since for any Z > 3  22=3 this cubic polynomial is positive at z D 0 and negative at z D 2, we see that there are three real roots: one negative, one in .0; 2/, and a physically relevant root in .2; 1/. By Cardano’s classical formula (see [47]), these roots are given explicitly by s

Z zD C 3 s C

3

r  Z 3 1 2  Z 2 3 Z3 1  C   27 2 27 2 9 r  Z 3 1 2  Z 2 3 Z3 1     ; 27 2 27 2 9 3

(129)

where21 the two square roots are chosen to be the same and the two cubic roots are chosen such that their product is Z 2 =9 (this giving three choices). We can alleviate slightly the intricate nature of (129) by using an alternative interpretation related to the geometry of the cubic (see [47]), which provides the three roots in the form zk D

1  2Z 27  2k  Z C cos arccos 1  3 C ; 3 3 3 2Z 3

k D 0; 1; 2 :

(130)

Selecting from (130) the unique root z > 2, it remains to observe that the quadratic polynomial equation h1 .h1 C 1/ D z has the unique positive root h1 D

1 C

p 1 C 4z > 2; 2

with the corresponding mean depth provided by (126). These computational considerations explain why in Theorem 1 we give preference to the head instead of the mean depth. Some flow characteristics of the Stokes waves described in Theorem 1 are already built-in our existence approach: all these flows have (relative) mass flux M and conformal depth w. For other characteristics we have to contemplate possible variations. In particular, we can not expect the multi-parameter .; q/ to be constant along the bifurcation curve, so that we do not fix the wavelength L and the mean depth, which depend on the values of  and q, respectively. Note that q involves , and therefore  , variations of which mean that the value of the head H changes, since M is fixed. The existence theory being perturbative, we can find reference values for the wave speed, for the wavelength and for the mean depth of the waves of small amplitude by computing those of the bifurcating uniform flat-surface flow corresponding to the trivial solution .1 ; .0; 0//. For example, the reference value

3

Note that the expression under the square root in (129) is 14  Z27 < 0, so that imaginary numbers are involved although all three roots are real.

21

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

59

for the spatial period is obtained from (126)–(127) for 1 solving (122) with n D 1, L D 2

 coth.w/M2 1=3 g2 w2

;

with (126) and (128) providing the reference values for the mean depth and the wave speed, respectively. Pertaining to the practical relevance of the Stokes waves of small amplitude, let us point out that most predictions of water-wave propagation are based on the equations for irrotational motion linearized about the state of a uniform flow beneath a flat free surface. While linear theory gives a reasonable approximation for deepwater waves, the nonlinearity of the governing equations is most apparent in water depths that are small compared with a wave’s length,22 i.e. in “shallow water”, cf. [50]. Since case studies show that surface waves can be sustained as high as 70 % of the water’s depth [63], we are motivated to go beyond the confines of linear analysis. This technically difficult endeavour enhances the utility of the theory and presents many unresolved questions.

Waves of Large Amplitude We now present a study of Stokes waves of large amplitude, at the interface of the so-called “soft” and “hard” analysis. The abstract (“soft”) setting is a quite general context in which to work with a specific problem, providing an adequate framework for a priori estimates and controls required to explain and predict physical phenomena (which is the “hard” essence of the problem). While abstract theories promote techniques of wide applicability, the elaboration of an overly general setting is often detrimental since specific structural properties are ignored and the approach becomes incapable of providing insight in directions which are relevant to the original problem. One major advantage of the global bifurcation approach is that it is accurate in all water depths and for all wave heights and periods, being thus capable of handling an otherwise intractable problem. At the same time, it offers challenges, at the cutting edge or beyond modern analysis, that should profitably occupy scientists for years to come. Let us first check the validity of the hypotheses .H1/–.H4/ of the Analytic Bifurcation Theorem presented in Sect. 4.3.1. We verified .H1/ and .H3/, at the unique solution  D 1 of (122) with n D 1, in the previous subsection. To deal with .H2/ and .H4/, we rewrite (109) in terms of the initial parameter  from (106)

22

We recall from Sect. 3.2.1 that the distinction between deep and shallow water waves has nothing to do with absolute water depth, being determined by the ratio ı of the water’s depth to the wavelength of the wave.

60

A. Constantin

as n o n o F1 .; .q; f// D 2.1  w/ C 1  2 f Cw .f0 / C  f Cw .f0 /  Cw .ff0 /  f C  ŒŒf Cw .f0 / ; while (110) takes on the form F2 .; .q; f// D

hhn

on  2 oii 1 2.1  w/ C 1  2 f .f0 /2 C  C Cw .f0 /  2; w

where DqC

1 C w : 22 w2

We now rely on the following commutator estimate (see [24] for the proof): j;s

j1;s

Lemma 1 If f 2 C2;ı .R/ and g 2 C2;ı .R/ with j  1 an integer and s 2 .0; 1/ a   j;s0 given constant, then f Cw .g/  Cw .fg/ 2 C2 .R/ for all s0 2 .0; s/, the inequality s0 < s being sharp. Consequently the continuous nonlinear mapping 2;s 1;s K1 W .0; 1/  R  C2;ı;e .R/ ! C2;ı;e .R/

given by K1 .; q; f/ D ff Cw .f0 /  Cw .ff0 /g  f C  ŒŒf Cw .f0 / ; 2;s=2

2;s .R/ into bounded sets of C2;ı;e .R/, maps bounded sets of .0; 1/  R  C2;ı;e 1;s .R/. Therefore K1 is a nonlinear and thus into relatively compact subsets of C2;ı;e compact operator. It follows (see [7]) that any of its partial derivatives is a linear compact operator. Note that

F2 .; .q; f// D K2 .; .q; f// C 2.1  w/2 q ; where K2 W .0; 1/  X ! R maps bounded and closed subsets of .0; 1/  X into bounded subsets of R, being thus a nonlinear compact operator whose partial derivatives must be linear compact operators. Consider now the open set O R  X, 1 and w 2.1  w/ C 1  2f.x/ > 0 for all x 2 Rg :

O D f.; .q; f// 2 R  X W

q > 0;  >

(131)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

61

We will seek solutions to (109)–(110) in O. If F.0 ; .q0 ; f0 // D 0 at some .0 ; f0 / 2 O, where f0 D .q0 ; f0 /, then, taking averages, we get 20 .1  0 w/ C 1 > 0, where 0 D q0 C

1 C 0 w : 220 w2

Since 0 w > 1, this means that 0 < q0
0 ;

x2R

so that the bounded linear operator from X to Y given by   .q; f/ 7! f20 .1  0 w/ C 1  20 f0 g Cw .f0 /; 2.1  0 w/2 q is invertible. Since the linear operator  .q; f/ 7! f2q.1  0 w/  2qf0  20 fg Cw .f00 / ; 0/ is clearly compact from X to Y, and the previous considerations show that the linear operator .q; f/ 7!



 @f K1 .0 ; f0 / .q; f/ ;

   @f K2 .0 ; f0 / .q; f/

is compact from X to Y, we see that @f F.0 ; f0 / is the sum of an invertible bounded linear operator and a compact linear operator. It is therefore a Fredholm operator of index zero (see [30]). The assumption .H2/ of the Analytic Bifurcation Theorem is therefore satisfied.

62

A. Constantin

To check .H4/, let us first note that O D

S

On , where

n1

On D f.; .q; f// 2 O W jj C jqj C kf kC2;s .R/  n ; 2

2.1  w/ C 1  2f.x/ 

1 for all x 2 Rg n

(133)

is a bounded and closed subset of the Banach space R  X for every integer n  1. To verify the validity of .H4/ it suffices to prove that the set An D On \ f.; f / 2 O W F.; f / D 0g is compact for any n  1. For this, note that if f.k ; .qk ; fk //gk1 is a sequence in An , then, setting k D qk C .1 C k w/=.22k w2 /, we may write F1 .k ; .qk ; fk // D 0 as f2k .1  k w/ C 1  2k fk g Cw .f0k / C K1 .k ; .qk ; fk // D 0 : Since 2k .1  k w/ C 1  2k fk .x/   1  fk D  Cw @x

1 n

for all x 2 R and all k  1, we get

 K1 .k ; .qk ; fk // ; 2k .1  k w/ C 1  2k fk

k  1:

(134)

K1 .k ; .qk ;fk // gk1 is uniThe definition of An ensures that the sequence f 2k .1 k w/C12k fk   2;s=2 formly bounded in the Banach space C2;o;e .R/. Since Cw @x is a bounded linear 3;s=2

2;s=2

operator from C2;o;e .R/ to C2;o;e .R/, from (134) we deduce that the sequence 3;s=2

f.k ; .qk ; fk //gk1 is bounded in the space R  R  C2;o;e .R/, and is therefore relatively compact in R  X. This establishes the validity of .H4/. These considerations permit us to apply the Analytic Bifurcation Theorem in order to extend the local bifurcation curve Aloc of non-trivial solutions, whose existence was established in Theorem 1, to a curve A which exhausts all possibilities of adding real-analytic arcs to Aloc in such a way that A has a real-analytic parametrization around each of its points.23 2;s Theorem 2 In the open subset O of the Banach space RRC2;o;e .R/ there exists a continuous curve A D f../; q./; f.// W  2 Rg of solutions to F.; q; f/ D 0, with the following properties:

• ..0/; q.0/; f.0// D .1 ; 0; 0/, where 1 > w1 is the unique solution to (122) with n D 1; 2;s • f.; a/ D  cos.a/ C o./ in C2;o;e .R/ for jj sufficiently small; • A has a local real-analytic parametrization around each of its points.

23 However, A is not necessarily a maximal connected subset of the solution set F.; f / D 0, since the branching out of solutions from A is not excluded.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

63

Each point on the curve A, with exception of .1 ; 0; 0/, corresponds to a Stokes wave whose symmetric profile is strictly monotone between successive crests and troughs. Proof The existence of the curve A and the properties stated in Theorem 2 follow by combining the Analytic Bifurcation Theorem with the local bifurcation result (Theorem 1), in view of the considerations preceding the statement of Theorem 2. It remains to establish the validity of the claimed pattern for the corresponding flows. Note that .1 ; 0; 0/ corresponds to a uniform flow with a flat free surface. The symmetry of the surface of the flows corresponding to the points on the curve A is ensured since by construction the functions are even. On the other hand, for points close to the bifurcation point .1 ; 0; 0/ that are different from this point, the monotonicity property of the corresponding wave profiles (and the fact that the flows are genuine waves) is already known, being a direct consequence of the second property of A, specified in Theorem 2. Away from the bifurcation point, this will be proved in what follows by means of a continuation argument (nodal analysis): we will show that all points ../; .q./; f./// 2 A with  ¤ 0 are different from .1 ; 0; 0/, and fa .; a/ < 0 for a 2 .0; / ;

faa .; 0/ < 0 < f0aa .; / ;

(135)

if  > 0 (that is, if the dimensionless wave crest is located at x D 0), with the reverse inequalities valid for  < 0. The validity of (135) near the bifurcation point .1 ; 0; 0/ was already established in our discussion of the local bifurcation result (Theorem 1). Assuming that (135) fails at some24  > 0, let 0 > 0 be the infimum of all  > 0 where this happens. If f D f.0 /, then f0 .a/ D ˇ 0 .a; 0/  0 on Œ0;  :

(136)

Let us first show that f  0 is impossible, that is, ..0 /; q.0 /; f.0 // 2 A can not correspond to a uniform flow with a flat free surface. Indeed, for f  0 Eq. (110) reads   2q.0 / 1  .0 / w D 0 ; so that q.0 / D 0, because A O ensures 1  .0 / w < 0. Since Theorem 1 specifies all the possible bifurcations from the curve f.; 0; 0/ W  > w1 g of trivial solutions, we infer that .0 / D n for some n  1. Moreover, a discussion analogous to the one made for the non-trivial solutions close to the bifurcation point .1 ; 0; 0/ shows that all non-trivial solutions in a small neighbourhood of .n ; 0; 0/

We choose  > 0 for definiteness; all the arguments can be easily adapted to deal with the case  < 0.

24

64

A. Constantin

are such that a 7! f.; a/ is periodic of period 2=n. Thus n D 1. Since 0 > 0, the only possibility to accommodate a trivial solution at 0 is that A presents a loop which starts at  D 0 and closes on itself at  D 0 > 0. But then we should have fa .; a/ > 0

for all a 2 .0; / ;

for  < 0 sufficiently close to 0 , and this is prevented by the validity of (135) for  2 .0; 0 /. Consequently, all flows corresponding to points ../; .q./; f./// 2 A with  ¤ 0 have a non-flat free surface. To show that (135) holds for all  > 0, recall the validity of (136) at 0 > 0, the infimum of all  > 0 where (135) presumedly fails. It is convenient to introduce the auxiliary function25 @.a; b/ D 

ˇa .a; b/ 2 ˇa .a; b/ C ˇb2 .a; b/

;

a 2 R ; w  b  0 ;

(137)

representing a scaled version of the vertical fluid velocity in non-dimensional variables, in view of (77). The function @ is harmonic, being the imaginary part 1 of the (complex) analytic function .a C ib/ 7! ˛a Ciˇ , where .a C ib/ 7! ˛a C iˇa is a the (complex) derivative of the analytic function .a C ib/ 7! ˛ C iˇ. Its restriction to the rectangle R D f.a; b/ W 0 < a <  ; w < b < 0g vanishes on the lower boundary and on the lateral boundaries, due to the third relation in (80) and to the fact that ˇ has to be 2-periodic and even in the avariable. On the upper boundary of R, by (136) we have @  0. If f0 .a0 / D 0 at some a0 2 .0; /, then (136) shows that a0 is a global maximum for f0 on Œ0;  and thus f00 .a0 / D 0. From (81) and (104) it follows that ˇa .a0 ; 0/ D ˇaa .a0 ; 0/ D 0. In particular, @.a0 ; 0/ D 0, so that @ restricted to R has a global minimum at .a0 ; 0/, since we already established that @  0 on the boundary of the rectangle R. The Hopf boundary-point lemma (see [34]) yields @b .a0 ; 0/ < 0. But from ˇa .a0 ; 0/ D 0 we get @b .a0 ; 0/ D 

ˇab .a0 ; 0/ ; ˇb2 .a0 ; 0/

(138)

as one can see by differentiating (137) with respect to the b-variable. But (89) reads 

1 C 2  2ˇ.a; 0/

  1 ˇa2 .a; 0/ C ˇb2 .a; 0/ D 2 ; w

a 2 R:

(139)

This function is actually also dependent on the parameter  . Since only its value at  D 0 is relevant to our considerations, we do not keep track of this parameter, in order to ease the notation.

25

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

65

Thus 1 C 2  2ˇ.a0 ; 0/ > 0 and, differentiating the above relation with respect to the a-variable and subsequently setting a D a0 , we get ˇab .a0 ; 0/ D 0. This leads by means of (138) to the contradiction @b .a0 ; 0/ D 0. Consequently the first relation in (135) holds for all  > 0. Let us now prove that the second relation in (135) also holds for all  > 0. Let 0 > 0 be the infimum of all values of  > 0 where at least one of the two inequalities fails. Setting, as above, f D f.0 /, we know that f0 .a/ < 0 for a 2 .0; /. Since @a f.; 0/ D @a f.; / D 0 for all  > 0, the function a 7! f.; a/ being even and 2-periodic, we have f00 .0/  0 and f00 ./  0, so that we only need to show that the two inequalities are strict. Suppose, on the contrary, that f00 .a / D 0 for a D 0 or a D . Then ˇa .a ; 0/ D ˇaa .a ; 0/ D 0 and thus @.a ; 0/ D @a .a ; 0/ D 0. Since the considerations in the previous paragraph show that @ has at .a ; 0/ a global extremum on R, the Serrin corner-point lemma (see [13, 34]) guarantees that not all first and second order derivatives of the harmonic function @ can vanish at .a ; 0/. By direct calculation we now check that under the present assumption all first and second order derivatives of @ will have to vanish at .a ; 0/, which constitutes a contradiction that validates (135) for all  > 0. Indeed, ˇ.a; b/ being 2-periodic and even in the a-variable, we have ˇa .0; b/ D ˇa .; b/ D 0 and thus @.0; b/ D @.; b/ D 0 for all b 2 Œw; 0, so that @b .a ; 0/ D @bb .a ; 0/ D 0. Since @ is harmonic, we must also have @aa .a ; 0/ D 0. It remains to examine the mixed derivative @ab .a ; 0/. Differentiating (137) and using the fact that ˇa .a ; 0/ D ˇab .a ; 0/ D ˇaa .a ; 0/ D ˇabb .a ; 0/ D 0, we get @ab .a ; 0/ D 

ˇaab .a ; 0/ : ˇb2 .a ; 0/

Since differentiating (139) twice with respect to the a-variable and evaluating the outcome at a D a yields ˇaab .a ; 0/ D 0, we get @ab .a ; 0/ D 0. This completes the proof.  Theorem 2 establishes the existence of Stokes waves of large amplitude. This is not automatically ensured by the mere fact that Theorem 2 represents a global bifurcation result. After all, we did not provide any quantitative estimates of the wave heights, so that both the local bifurcation curve as well as its extension to the curve A could, in principle, correspond to waves that are small perturbations of trivial flows (uniform flows with a flat free surface). To show that the solutions provided by Theorem 2 go beyond such flows, it is necessary to understand the behaviour of the solutions ../; .q./; f./// 2 A as  ! 1. The analysis performed in the proof of Theorem 2 shows that the alternative .C3 ˇ/ from the abstract Analytic Bifurcation Theorem is excluded since all solutions except .1 ; 0; 0/ inherit the monotonicity property of the eigenfunction of the linearized problem at the bifurcation point .1 ; 0; 0/. Moreover, the curve A O eventually leaves every set On , defined in (133). Analogous to the considerations made in [7] for deep-water Stokes waves (in water of infinite depth), one can show that, as  ! 1, we can extract from A a sequence of solutions which converges in

66

A. Constantin 1=2

R  R  C2;o;e .R/ to a limiting wave pattern (a Stokes wave of greatest height) that is symmetric and strictly monotone between successive crests and troughs but fails to have a tangent at the crest—instead, lateral tangents exist, at an angle of 120ı. In the neighbourhood of this extreme wave we find Stokes waves of large amplitude. Pursuing this aspect in detail would take us too far afield, since several points, which are in the main highly technical,26 are involved. Let us point out that several assumptions encompassed in our definition of a Stokes wave are not restrictive requirements: • the free surface must always be a graph, that is, overhanging wave profiles are not possible (see [55, 64]); • the symmetry of the free surface is guaranteed if the wave profile is monotone between crests and troughs (see [48]).

4.4 The Flow Beneath the Waves While the propagation of a Stokes wave occurs without change (of shape or speed), the flow beneath it presents a rich and delicate web of processes. Leonardo da Vinci captured the essence of waves beautifully back in the fifteenth century27 :    l’onda si allontani dal suo punto di creazione, mentre l’acqua non si muove; come le onde create dal vento in un campo di grano, dove vediamo le onde correre attraverso il campo mentre il grano rimane al suo posto. Leonardo da Vinci (1452–1519)

Indeed, as we shall see in this section (and as mentioned in Sect. 4.1), contrary to a possible first impression, as a regular (Stokes) wave propagates at the surface, the water does move along too, but much slower than the wave and, while the surface wave has a fixed direction of propagation, for short times the particles beneath it are displaced in the opposite direction, the net movement being however alike that of the wave. The scientific study of the behaviour of particle paths in water waves is more than 200 years old, and discussions of it are found throughout the literature. The classical attempts towards elucidating this aspect of water waves consist in performing linearizations of the governing equations (and thus deriving explicit formulas) or numerical simulations. However, starting with the exact linear solution (29), the

For example, it is convenient to derive a priori estimates in the Sobolev spaces W 1;r Œ;  of 2-periodic functions with weak derivatives in Lr Œ;  for 1 < r < 3. 27 English translation: “   the wave flees the place of its creation, while the water does not; like the waves made in a field of grain by the wind, where we see the waves running across the field while the grain remains in its place”. 26

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

67

system describing the particle motion, 

X 0 .T/ D U.X; Y; T/ ; Y 0 .T/ D V.X; Y; T/ ;

(140)

turns out to be nonlinear. Performing a linearization of the system (140), for the expressions of the velocity components provided by (29), it is possible to integrate the resulting system, obtaining elliptical particle trajectories (see the discussion in [13]). As one goes down away from the surface, the shape of these ellipses changes: they are thinner and more elongated, and on the flat bed they degenerate into a back-and-forth horizontal motion with a drift in the direction of wave propagation. There is, however, a correction of this prediction of the analysis at first order (in the small parameter "): at second order one obtains a mean (Stokes) drift in the direction of wave propagation, so that at second order the particles follow the wave; see the discussion in [32]. This raises the issue whether there is a regular pattern to the particle motion at all, or is the flow beneath a Stokes wave just too complex and chaotic to say something definite? Also, even if a general pattern can be discerned for linear waves, it might be considerable altered if a certain amplitude threshold is reached, since, after all, linearization is appropriate only for waves of small amplitude. While an approach relying on numerical simulations and/or experimental data (whether laboratory experiments or field data) would provide valuable insight, it can not be relied upon to identify clearly the important processes that are at work, due to the need to account for many different (and interdependent) wavelength, mean depth, energy, mass flux and amplitude regimes. The highly nonlinear character of the problem and the lack of explicit particular solutions (other than flows with a flat free surface) are also important factors that limit the potential of numerical investigations. It is remarkable that an interplay of techniques from harmonic analysis and dynamical systems permits one to use the specific physical structure (reflected in the way the velocity field interacts with the pressure) to ascertain a basic pattern of the particle trajectories, beneath a Stokes wave without underlying currents, that accommodates free flow parameters (wavelength, mean depth, energy, mass flux, amplitude). In particular, although details vary, the overall structure of the pattern is valid for waves of small and large amplitude. It is useful to introduce the dimensionless counterpart of the pressure P: in view of (54), (55) and (57), setting p.x; y/ D

1  jr j2 C1; 2

(141)

we can write (53) as P D Patm C Œp.x; y/  y g :

(142)

68

A. Constantin

Let us also denote by .u; v/ D 

 .U; V/ M

(143)

the dimensionless fluid velocity; in view of (51), (54), (62), (72), note that y

D c0  u ;

x

Dv:

(144)

The discrepancy between (51) and its dimensionless counterpart (144), consisting of a change of sign, comes about because M < 0 and we would like to preserve the sign of each component when passing from the physical velocity field .U; V/ to its dimensionless correspondent .u; v/. The first relation in (56) ensures that u is a harmonic function throughout the closure D of the dimensionless fluid domain D D f.x; y/ W 0 < y < 1 C .x/g. Due to the periodicity in the x-variable, from the weak maximum principle [37] we infer that the maximum and minimum values of u throughout ˝, where ˝ is the periodicity cell ˝ D f.x; y/ W x 2 .`=2; `=2/ ; 0 < y < 1 C .x/g, are attained on the top or lower boundary. At such a point, Hopf’s maximum principle [34] yields uy ¤ 0 since u can not be constant throughout ˝ for a genuine wave; indeed, u constant would force v to be constant, so that differentiation of the last relation in (56) with respect to the x-variable shows that the free surface must be flat in such a setting. Moreover, since by the third relation in (56), uy D vx D 0 on y D 0, both extrema of u in the closure ˝ of ˝ must be attained on y D 1 C .x/. We now show that the only possible locations for extrema are the wave crest and the wave trough. For this, note that the function . p  y/ is superharmonic: . p  y/xx C . p  y/yy D 

u2x C u2y C vx2 C vy2 

0

in D :

Consequently, the minimum of . p  y/ in D is attained on the lower or upper boundary of the domain [37]. Since uy D vx , the third relation in (56) yields py D 0

on y D 0 ;

so that Hopf’s maximum principle [34] prevents that the minimum of . p  y/ is attained on the bed y D 0. In view of the last relation in (56), we conclude that the minimum of . p  y/ vanishes and is attained all along the free surface y D 1 C .x/, and only there. We can therefore take advantage of Hopf’s maximum principle for . p  y/ all along the free surface. In particular, since  0 .x/ < 0 for x 2 .0; `=2/, the x-direction points outwards and we will have px .x; 1 C .x// < 0 for x 2 .0; `=2/ :

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

69

But differentiation of the second relation in (56) yields v D .u  c0 / x

on y D 1 C .x/ ;

(145)

so that we can express the previous inequality as n o2 @x u.x; 1 C .x//  c0 > 0 for x 2 .0; `=2/ :

(146)

Note that from (51), (54) and (60) we get c0  u.x; y/ D

o  n U.X ; Y/  c > 0 M

for .x; y/ 2 D ;

(147)

in agreement with (70). We infer from (146) and (147) that n o @x u.x; 1 C .x// < 0

for x 2 .0; `=2/ :

(148)

In Sect. 4.2 we showed that zero is a strict minimum of the harmonic function v D x , restricted to the closure of the region f.x; y/ W x 2 .0; `=2/; 0 < y < 1 C .x/g, with v D 0 all along the lower and lateral boundaries. Hopf’s maximum principle yields vy .x; 0/ > 0 for all x 2 .0; `=2/, while vx .0; y/ > 0 for all y 2 .0; 1 C .0// and vx .`=2; y/ < 0 for all y 2 .0; 1 C .`=2//. In view of ux D vy D  xy and uy D vx , this means that 8 < ux .x; 0/ < 0 ; u .0; y/ > 0 ; : y uy .x; 0/ < 0 ;

x 2 .0; `=2/ ; y 2 .0; 1 C .0// ; x 2 .0; 1 C .`=2// :

(149)

Since the harmonic function u, restricted to the closure of the region f.x; y/ W x 2 .0; `=2/; 0 < y < 1 C .x/g, attains its extrema on the boundary, in view of the symmetry of u and its periodicity in the x-variable, (148)–(149) ensure that the maximum and minimum of u throughout the dimensionless fluid domain are attained at .0; 1 C .0// and at .0; 1 C .`=2//, respectively. From (143) we infer that in physical variables the maximal value of the horizontal fluid velocity U is attained at the wave crest, while the minimal value of U is attained at the wave trough. The second and third relations in (56), together with (70), show that the level sets of provide a foliation of the dimensionless fluid domain D. Since M < 0, (51) and (60) show that the streamlines (the level sets of ) foliate the fluid domain in the moving frame .X ; Y/. On the other hand, the relations (66)–(69) yield that the harmonic function v D x is positive in the interior of the region f.x; y/ W 0 < x < `=2; 0 < y < 1 C .x/g, and on its upper boundary f.x; y/ W 0 < x < `=2; y D 1 C .x/g. From (51) and (143), since v.x; y/ is odd in the x-variable, we infer that every level set of above the flat bed Y D d replicates the monotonicity

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A. Constantin

Fig. 9 Depiction of the streamline pattern beneath a Stokes wave in the physical moving frame .X ; Y /: the streamlines provide a foliation of the fluid domain, and, due to (60), the motion along the streamline is backwards (from right to left)

properties of the wave profile, being the graph of a smooth function that is strictly decreasing as X runs from 0 to L=2 and strictly increasing as X runs from L=2 to 0; see Fig. 9. The previous considerations elucidate the monotonicity properties of the horizontal fluid velocity U.X ; Y/ along the free surface Y D .X / and along the flat bed Y D d, in the physical moving frame .X ; Y/. Indeed, recall that u.x; y/ is even and `-periodic in the x-variable. In view of (148) and the first relation in (149), from (54) and (143) we deduce that on the descending part of the wave profile Y D .X /, between crest and trough, U.X ; Y/ is strictly decreasing, and U.X ; d/ is also strictly decreasing along the flat bed Y D d, as X runs from 0 to L=2. We now claim that this property holds along any streamline. To prove this, it is convenient to take advantage of the conformal parametrization of the dimensionless fluid domain. Define u; v W Rw ! R by 

u.a; b/ D u.x; y/ ; v.a; b/ D v.x; y/ ;

.a; b/ 2 Rw ;

(150)

x .x; 0/

D 0 for x 2 .0; `=2/,

with .a; b/ and .x; y/ related by means of (79). Since from (76), (77) and the first relation in (149) we get

ua .a; w/ D ux .x; 0/ < 0 for a 2 .0; / : On the other hand, differentiating the relation the x-variable yields x .x; 1

C .x// C

y .x; 1

(151)

.x; 1 C .x// D 0 with respect to

C .x// x .x/ D 0 :

Consequently, for a 2 .0; / we obtain from (76)–(77) that ua .a; 0/ D D

 w.

2 x



C

2 y / yD1C.x/



y

w.

2 x

C

n



y

2 y

yD1C.x/

ux .x; 1 C .x// C uy .x; 1 C .x// x .x/

n o @x u.x; 1 C .x// < 0 ;

o

(152)

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

71

in view of (148) and (70). Moreover, ua .0; b/ D ua .; b/ D 0 ;

b 2 Œw; 0 ;

(153)

since the harmonic function u.a; b/ is 2-periodic and even in the a-variable. The harmonicity of the function ua .a; b/ in Rw and its behaviour on the boundary of the domain f.a; b/ W 0 < a <  ; w < b < 0g, reflected in (151)–(153), permit us to conclude from the strong maximum principle that ua .a; b/ < 0

for .a; b/ 2 .0; /  .w; 0/ :

Recalling (74) and the fact that ua .a; b/ is odd in the a-variable, this shows that u is strictly increasing along any streamline in f.x; y/ W `=2 < x < 0 ; 0 < y < 1 C .x/g, and is strictly decreasing along any streamline in f.x; y/ W 0 < x < `=2 ; 0 < y < 1 C .x/g. Due to (143), this proves our claim about the monotonicity of U.X ; Y/ along streamlines. The discussion in Sect. 4.2 shows that the absence of underlying currents is expressed by the relation 1 L

Z

L=2 L=2

U.X ; Y/ dX D 0 ;

which, due to (143), is equivalent to Z

`=2 `=2

u.x; 0/ dx D 0 :

Since x 7! u.x; 0/ is even and its restriction to Œ0; `=2 is strictly monotone, in view of the first relation in (149), we deduce the existence of a unique point x0 2 Œ0; `=2 where u.x0 ; 0/ D 0. From (149) we now infer that 8 ˆ u.x; 0/ > 0 for x 2 .x0 ; x0 / ; ˆ ˆ ˆ < u.0; y/ > 0 for y 2 .0; 1 C .0/ ; ˆ u.x; 0/ < 0 for x 2 Œ`=2; x0 / [ .x0 ; `=2 ; ˆ ˆ ˆ : u.0; y/ < 0 for y 2 .0; 1 C .`=2// :

(154)

The fact that u.x; y/ is strictly decreasing along all the level sets of below the descending part of the wave profile y D 1 C .x/, in combination with (154), and knowing that v.x; y/ > 0 for x 2 .0; `=2/ and y 2 .0; 1 C .x/, permit us to obtain a complete qualitative picture of the orientation of the velocity field beneath a Stokes wave (in the moving frame). More precisely, in the moving frame, we have

72

A. Constantin

Fig. 10 Velocity field beneath a Stokes wave in the physical moving frame .X ; Y /

(see Fig. 10): • the vertical velocity component V.X ; Y/ vanishes on the flat bed and on the vertical lines below the wave crest and below the wave trough, having elsewhere a strict sign, positive beneath the descending part of the wave profile and negative beneath the ascending part of the wave profile; • in a periodicity window f.X ; Y/ W L=2  X  L=2; d  Y  .X /g, the horizontal velocity component U.X ; Y/ vanishes along two smooth curves28 that connect the free surface to the flat bed, being strictly positive in the fluid region beneath the wave crest and strictly negative in the fluid region beneath the wave trough. We are now in a position to investigate the particle path pattern. Equation (140), whose integration yields the particle trajectories in a Stokes wave, corresponds in the physical moving frame to the system (

X 0 .T/ D U.X ; Y/  c ; Y 0 .T/ D V.X ; Y/ :

(155)

Note that (155) is an autonomous Hamiltonian system, the Hamiltonian being the stream function .X ; Y/, in view of (51). Consequently, any solution to (155) is confined to a level set of , specified by the initial data. Let .X .T/; Y.T// correspond to .x.T/; y.T// in the dimensionless formulation and to .a.T/; b.T// under the conformal parametrization. Since .X .T/; Y.T// is always confined to the streamline specified by the initial data .X .0/; Y.0//, we deduce from (74) that b.T/ D b.0/ for all times T. On the other hand, (74), (71), (54), (51), (143) and (155) yield d  X .T/ Y.T/ C d  d a.T/ D w  ; dT dT  

28 Each streamline ΠD constant intersects such a curve in precisely one point, and each curve lies between adjacent vertical lines below the crest and the trough, respectively, without intersecting these lines.

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

73

o w n 0 X .T/ x .x.T/; y.T// C Y 0 .T/ y .x.T/; y.T//  o w n ŒU.X .T/; Y.T//  c2 C V 2 .X .T/; Y.T// < 0 ; D M

D

due to (60) and the fact that M < 0. Using (144) and (77), we get ŒU.X ; Y/  c2 C V 2 .X ; Y/ D

M2 w2 2 2

1 ˇa2 .a; b/ C

ˇb2 .a; b/

:

Consequently, in conformal variables, the counterpart of the system (140) is 8 1 < a0 .T/ D M ; 2 2 w ˇa .a; b/ C ˇb2 .a; b/ : 0 b .T/ D 0 :

(156)

The presence of the dimensional factor M=.w2 / on the right side of (156) is due to the fact that in (156) the variable T is not dimensionless; since time T is the only such variable, there is no need to address this discrepancy—the interpretation being that the motion of the dimensionless solution .a.T/; b.T// occurs in real time. Let us take a closer look at the behaviour of the solution .X .T/; Y.T// to (155) as it moves along the level set of specified by the initial data .X .0/; Y.0//. Due to (60), along this level set (streamline), the solution .X .T/; Y.T// moves to the left as time T progresses, performing vertical oscillations if the initial data lies above the flat lower boundary Y D d of the fluid domain, and being confined to Y D d if it starts there; see the sketch in Fig. 9. Consider the case when the initial data .X .0/; Y.0// is such that Y.0/ > d. Since X .T/ moves to the left at a speed that is bounded away from zero, we may assume without loss of generality (shifting the time variable, if necessary) that X .0/ D L=2, and this location is where the streamline, to which the solution .X .T/; Y.T// is confined, reaches its lowest elevation above the flat bed Y D d (see Fig. 9). Let us mark this location by A, as in the middle sketch of Fig. 11. The detailed information that we derived previously about the sign of the components of the fluid velocity permits us to state that U < 0 and V D 0 at A. As time progresses, along the streamline traced by the solution .X .T/; Y.T// that issues from A at time T D 0, we will have U.X .T/; Y.T// < 0 and V.X .T/; Y.T// > 0 for T > 0 small enough, until we reach a point B where U D 0 and V > 0. Increasing T further, the solution .X .T/; Y.T// enters a region of the .X ; Y/-plane where U > 0, encountering first the vertical line X D 0, at the point C, where V D 0, and then reaching the boundary of this region at the point D, where U D 0. Between the points B and C, along the streamline parametrized by .X .T/; Y.T//, we have V > 0, while between C and D we have V < 0. After reaching D, if time T is further increased, then the solution .X .T/; Y.T// moves further to the left in the .X ; Y/-plane until it reaches the vertical line X D L=2,

74

A. Constantin C

B

D

A

E D

E

E

D

C

C

B

B

A

A

Fig. 11 Particle trajectories in a Stokes wave, above the flat bed: (1) in the physical .X; Y/variables (top figure); (2) in the .X ; Y /-moving frame (middle figure); (3) in conformal variables (bottom figure)

at the point E, where V D 0. Since (

X.T/ D X .T/ C cT ; Y.T/ D Y.T/ ;

important features of the trajectory traced by the particle .X.T/; Y.T// can be inferred at once from the previous considerations: starting at time T D 0 at a location above the flat bed and exactly below the wave trough, the particle moves to the left and up (U < 0; V > 0), then it moves up and to the right (U > 0; V > 0) until the wave crest is just above it (at this location U > 0 and V D 0), subsequently the particle moves down and to the right (U > 0; V < 0) and afterwards it moves down and to the left (U < 0; V < 0), until it reaches a position located exactly below the wave trough. The points where these changes of direction occur correspond to the points B; C; D (see the top sketch of Fig. 11), while the two locations just below the wave trough correspond to A and E, and the location when the particle is just beneath the wave crest corresponds to C. If T0 > 0 is the time needed for the particle path to describe the loop from A to E, going through the intermediate points B; C; D, we can express the fact that the loop is closed, self-intersecting, or open, by means

Exact Travelling Periodic Water Waves in Two-Dimensional Irrotational Flows

75

of cT0 D L; cT0 < L, and cT0 > L, respectively. Indeed, we know that Y.T0 / D Y.T0 / D Y.0/ ; while X.T0 /  X.0/ D cT0  L ; since X.0/ D X .0/ D L=2 and X.T0 / D X .T0 / C cT0 with X .T0 / D L=2. We claim that cT0 > L ;

(157)

so that the loop is always open, with a forward drift X.T0 /  X.0/ D cT0  L > 0 : This is indicative of the fact that the particle above the flat bed performs a back-andforth and up-and-down movement, with an overall displacement in the direction of wave propagation. As for a particle located on the flat bed Y D d, the boundary condition (3) ensures that it will always move horizontally. Choosing an initial reference point on the flat bed, just below the wave trough, the particle oscillates in a backward-forward-backward pattern with a net forward drift, mirroring the projection of the loop on the top of Fig. 11 to the flat bed. This follows from the sign change pattern of the horizontal fluid velocity components U on Y D d, and from the fact that (157) is valid for any particle in a Stokes wave flow, in particular also for particles confined to the flat bed. To prove (157), let us consider a particle located initially below the wave trough, say, at .X.0/; Y.0// with X.0/ D X .0/ D L=2 and Y.0/ D Y.0/ 2 Œd; .L=2/. The solution to (155) with initial data .X .0/; Y.0// is confined at all times T to the streamline D 0 , where 0 D .X .0/; Y.0//. We know that this streamline is the graph of a function X 7! Y D Y0 .X /, with (51) and (60) ensuring, by means of the implicit function theorem, that the function Y0 is real-analytic. The previous considerations show that T0 > 0 is determined by either of the following conditions: X .T0 / D L=2; x.T0 / D `=2; a.T0 / D , whereas x.0/ D `=2 and a.0/ D . Therefore (155) and (60) yield Z T0 D

0

T0

X 0 .T/ dT D U.X .T/; Y.T//  c

Z

L=2 L=2

1 dX ; c  U.X ; Y0 .X //

(158)

since X 0 .T/ < 0 for T 2 Œ0; T0  permits us to change variables X .T/ 7! X , which automatically triggers the change Y.T/ 7! Y0 .X /. On the other hand, denoting by

76

A. Constantin

x 7! y0 .x/ the dimensionless correspondent of the function Y0 , (51) and (54) yield Z L=2 n o c  U.X ; Y0 .X // dX L=2

D 

M  M 

Z Z

`=2

`=2  

y .x; y0 .x// dx D 

M 

Z



ˇb2 .a; b0 / 2 2  w Œˇa .a; b0 / C ˇb .a; b0 /

da

2M ds D ; w w

(159)

in view of (76)–(77), where we used (54) and (74) to set b0 D w .x; y0 .x// D

w w .X ; Y0 .X // D 0 : M M

From (60), (158) and (159), by means of the Cauchy-Schwarz inequality, we obtain the estimate L2 

Z

L=2

L=2



 1 dX c  U.X ; Y0 .X //

Z

L=2 L=2

2M T0 D cLT0 : w

n

o  c  U.X ; Y0 .X // dX (160)

Indeed, the only part that needs to be checked is the equality in (160). To verify it, combine (58), (73), and the fact that (62) and (72) yield c M D : c0 

(161)

The claim (157) follows now if we can show that equality in (160) is impossible for a genuine Stokes wave. A careful inspection of the way we derived (160) reveals that equality in (157) requires U.X ; Y0 .X // to be constant, and we established that in a genuine Stokes wave U changes sign along any streamline. This validates (157) and concludes our discussion of the particle trajectories beneath a Stokes wave. Further details (for example, the fact that the particle drift and the width of the loops described by a particle both increase with the minimal distance from the particle path to the flat bed so that the maximal drift and width occur for a particle on the water’s surface, and the minimal values occur for particles confined to the flat bed) are provided by the in-depth analytical studies [12, 23]. Numerical simulations that confirm the qualitative pattern depicted on the top of Fig. 11, and offer some insight into the rôle of the wavelength and the wave amplitude in shaping the form of the loops, are provided in [46]. Finally, to be acceptable in practice, theoretical predictions must stand the test of observations. For experimental data that supports the previous mathematical conclusions we refer to [62].

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5 Concluding Remarks The range of topics related to Stokes waves that were covered in this survey has necessarily been limited. The study of these wave patterns is still, after more than 100 years, in its infancy since only a few aspects are understood in any really satisfactory way. In this section we describe some directions that we expect to be among the principal themes of research in the near future. A first instance of great practical relevance is the lack of quantitative estimates for waves of large amplitude, e.g. concerning their maximal slope, as well as relations between the wave energy, the mass flux, the mean depth and the wave height. Also, while the monotonicity properties of the horizontal fluid velocity were elucidated in Sect. 4.4, the situation for the vertical fluid velocity remains to be clarified— numerical simulations (see [10, 46]) indicate that, in the moving frame, along any streamline beneath the descending part of the wave profile, the monotonicity behaviour changes precisely once (from strictly increasing to strictly decreasing). With respect to the particle trajectories, experimental evidence and field data suggest that the elongation of the loops (that is, the distance between the points marked by B and D in the top sketch of Fig. 11) increases with the distance between the curve traced by the particle and the flat bed. An important aspect that was only discussed briefly concerns the behaviour of the pressure beneath a Stokes wave. In addition to playing an important theoretical role in in the description of the particle trajectories (see Sect. 4), the pressure is essential in quantitative studies because often in practice the elevation of a surface water wave is determined from pressure data obtained at the sea bed. The measurement of water waves is a challenging engineering problem. To begin with, there is the difficulty to make sensitive electronic devices operate effectively in salt water, inherent to the fact that salt water is both a conductor of electricity and a corroder of metals. Also, floating measuring devices disturb the natural process of wave propagation, so that the gathered information may lead to erroneous conclusions. Despite all these roadblocks, remarkably accurate measuring techniques for waves at sea have been developed. They are of two types: remote sensing and in situ. Remote-sensing refers to data collected by means of electromagnetic radiation emitted from aircraft or satellites. It provides a wide synoptic view of the sea surface and represents the only practical way to obtain data from secluded regions. However, remote sensing is an indirect measurement and the desired data is retrieved by solving an inverse problem. Due to data sensitivity and the related instability issues in the retrieval process, remote-sensing outcomes must be calibrated against in situ data, provided by instruments deployed in the water at an accessible location. In this context, the widespread use of bottom-mounted pressure sensors is due to their low cost, ease of use and low liability to damage (see the discussion in [41]). Pressure sensors are often placed at fixed positions on the rigid bed, measuring and recording pressure fluctuations at these locations. They provide accurate pressure measurements, but

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the conversion of this data into wave height information29 is quite delicate. One approach—used often in open-ocean buoys—consists of assuming the applicability of the hydrostatic approximation P.X; Y; T/  Patm C g Œ.X; T/  Y ;

d  Y  .X; T/ ;

(162)

obtained by setting P D Patm on the surface Y D .X; T/ and PY  g beneath the surface. From (162) we get .X; T/ 

P.X; d; T/  Patm d: g

(163)

This is exactly the relation between the wave profile .X; T/ and the value of the pressure P.X; Y; T/ on the flat bed Y D d provided by the representation (21) of the solution to the linearized problem in the shallow-water limit. However, even for waves of small amplitude, prediction errors for the wave height in excess of 15 % frequently occur if one relies on (163); see the data in [6]. An improvement of (163) is the so-called pressure-transfer formula .X; T/  cosh

 2d n P.X; d; T/  P L

atm

g

o d ;

(164)

valid within the linear regime of water waves of small amplitude, as a consequence of (29). The hydrostatic formula (163) is recovered from (164) in the shallow-water d limit ı D ! 0. The benefit of the simplicity of the linear formula (164) is L offset by the fact that for waves of moderate amplitude it often overestimates the wave height by more than 10 %, cf. the experimental data provided in [61]. These considerations are indicative of the importance of estimates that are valid for all Stokes waves. Without providing details (for these, we refer to the discussion in [16, 17, 23]), we now list some results in this direction, formulated in the moving frame (43), in which case P.X; Y; T/ D P.X ; Y/ and .X; T/ D .X /, for wave profiles whose crest and trough are located at X D 0 and X D L=2, respectively30: .0/ C d > .L=2/ C d


P.0; d/  P.L=2; d/ g

(167)

for the wave height of a Stokes wave. Since throughout the fluid (in the moving frame) the pressure is maximal at .0; d/, while P.L=2; d/ is the minimum value of the pressure along the flat bed (see [17, 23]), we see that (167) means that the variation of the pressure on the bed, normalized by means of g, provides a lower bound for the wave height. Moreover, the estimates (165) and (166) show that the hydrostatic formula (163) underestimates the elevation of the wave crest and overestimates the wave trough level. Note that the pressure-transfer formula (164) approximates the wave height by the right-hand side of (167) times  2d  the multiplicative factor cosh > 1. As pointed out before, field data shows L that (164) overestimates the wave height. While progress was made recently on the problem of the recovery of surface waves of large amplitude from pressure measurements at the flat bed (see the discussion in [11]), much remains to be done and is probably within the reach of available technical tools. Last, but not least, is the problem of wave-current interactions. Even within the setting of irrotational flows, the presence of a (necessarily) uniform underlying current complicates the dynamics considerably. For example, new particle paths patterns can occur (see the discussion in [23]). Considerably more challenging is the problem of water waves with vorticity. Flows with constant non-zero vorticity are ubiquitous in equatorial ocean dynamics (see the discussion in [21, 58]), and, in contrast to the irrotational framework, flow reversal can lead to the existence of internal stagnation points, with streamline patterns of Kelvin cat’s eyes type (see [21, 24, 25, 65]). Two-dimensional flows with non-constant vorticity are even more complicated and very little is known about waves of large amplitude (see [13] for the state-of-the-art in this direction). Without going into further details, let us only mention that already a flow of constant non-zero vorticity appears31 to accommodate the possibility of overhanging wave profiles.

References 1. Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008) 2. Babenko, K.I.: Some remarks on the theory of surface waves of finite amplitude. Dokl. Akad. Nauk 294, 1033–1037 (1987) 3. Baquerizo, A., Losada, M.A.: Transfer function between wave height and wave pressure for progressive waves. Coast. Eng. 24, 351–353 (1995)

31

As suggested by numerical simulations and casual observations with the naked eye (see the discussions in [25, 28]).

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4. Benjamin, T.B.: Verification of the Benjamin-Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356 (1995) 5. Benjamin, T.B., Feir, J.E.: The disintegration of wavetrains in deep water. J. Fluid Mech. 27, 417–430 (1967) 6. Bishop, C.T., Donelan, M.A.: Measuring waves with pressure transducers. Coast. Eng. 11, 309–328 (1987) 7. Buffoni, B., Toland, J.F.: Analytic Theory of Global Bifurcation. An Introduction. Princeton University Press, Princeton, NJ (2003) 8. Buffoni, B., Dancer, E.N., Toland, J.F.: The regularity and local bifurcation of steady periodic water waves. Arch. Ration. Mech. Anal. 152, 207–240 (2000) 9. Buffoni, B., Dancer, E.N., Toland, J.F.: The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Anal. 152, 241–271 (2000) 10. Clamond, D.: Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves. Philos. Trans. R. Soc. Lond. A 370, 1572–1586 (2012) 11. Clamond, D., Constantin, A.: Recovery of steady periodic wave profiles from pressure measurements at the bed. J. Fluid Mech. 714, 463–475 (2013) 12. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006) 13. Constantin, A.: Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. SIAM, Philadelphia (2011) 14. Constantin, A.: Particle trajectories in extreme Stokes waves. IMA J. Appl. Math. 77, 293–307 (2012) 15. Constantin, A.: Mean velocities in a Stokes wave. Arch. Ration. Mech. Anal. 207, 907–917 (2013) 16. Constantin, A.: Estimating wave heights from pressure data at the bed. J. Fluid Mech. 743, R2 (2014) 17. Constantin, A.: The flow beneath a periodic travelling surface water wave. J. Phys. A Math. Theor. 48, 143001 (2015) 18. Constantin, A.: Fourier Analysis. Part I - Theory. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (2016) 19. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011) 20. Constantin, A., Johnson, R.S.: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 40, 175–211 (2008) 21. Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015) 22. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009) 23. Constantin, A., Strauss, W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533–557 (2010) 24. Constantin, A., Varvaruca, E.: Steady periodic water waves with constant vorticity—regularity and local bifurcation. Arch. Ration. Mech. Anal. 199, 33–67 (2011) 25. Constantin, A., Strauss, W., Varvaruca, E.: Global bifurcation of steady gravity water waves with critical layers. Acta Math. (in print, arXiv:1407.0092/2014) 26. Craik, A.D.D.: George Gabriel Stokes on water wave theory. Annu. Rev. Fluid Mech. 37, 23– 42 (2005) 27. Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321– 340 (1971) 28. da Silva, A.F.T., Peregrine, D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988) 29. Dancer, E.N.: Bifurcation theory for analytic operators. Proc. Lond. Math. Soc. 26, 359–384 (1973) 30. Davies, E.B.: Linear Operators and Their Spectra. Cambridge University Press, Cambridge (2007)

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31. Escher, J., Schlurmann, T.: On the recovery of the free surface from the pressure within periodic traveling water waves. J. Nonlinear Math. Phys. 15, 50–57 (2008) 32. Falkovich, G.: Fluid Mechanics: A Short Course for Physicists. Cambridge University Press, Cambridge (2011) 33. Fenton, J.D.: Nonlinear wave theories. In: Le Méhauté, B., Hanes, D.M. (eds.) The Sea. Wiley, New York (1990) 34. Fraenkel, L.E.: An Introduction to Maximum Principles and Symmetry in Elliptic Problems. Cambridge University Press, Cambridge (2000) 35. Garrison, T.: Essentials of Oceanography. Brooks/Cole Cengage Learning, Boston, MA (2008) 36. Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Springer, Berlin (1996) 37. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) 38. Henry, D.: Steady periodic flow induced by the Korteweg-de Vries equation. Wave Motion 46, 402–411 (2009) 39. Henry, D.: On the pressure transfer function for solitary water waves with vorticity. Math. Ann. 357, 23–30 (2013) 40. Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997) 41. Jones, N.L., Monismith, S.G.: Measuring short-period wind waves in a tidally forced environment with a subsurface pressure gauge. Limnol. Oceanogr. Methods 5, 317–327 (2007) 42. Koosis, P.: Introduction to Hp Spaces. Cambridge University Press, Cambridge (1998) 43. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895) 44. Kozlov, V., Kuznetsov, N.: The Benjamin-Lighthill conjecture for steady water waves (revisited). Arch. Ration. Mech. Anal. 201, 631–645 (2011) 45. Lighthill, J.: Waves in Fluids. Cambridge University Press, Cambridge (1978) 46. Nachbin A., Ribeiro-Junior, R.: A boundary integral formulation for particle trajectories in Stokes waves. Discrete Contin. Dyn. Syst. 34, 3135–3153 (2014) 47. Nickalls, R.W.D.: A new approach to solving the cubic: Cardan’s solution revealed. Math. Gaz. 77, 354–359 (1993) 48. Okamoto, H., Shõji, M.: The Mathematical Theory of Permanent Progressive Water-Waves. World Scientific, River Edge, NJ (2001) 49. Oliveras, K., Vasan, V., Deconinck, B., Henderson, D.: Recovering the water-wave profile from pressure measurements. SIAM J. Appl. Math. 72, 897–918 (2012) 50. Peregrine, D.H.: Water waves and their development in space and time. Proc. R. Soc. Lond. A 400, 1–18 (1985) 51. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992) 52. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987) 53. Schwartz, L.W., Fenton, J.D.: Strongly nonlinear waves. Ann. Rev. Fluid Mech. 14, 39–60 (1982) 54. Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C.-M., Pheiff, D., Socha, K.: Stabilizing the Benjamin-Feir instability. J. Fluid Mech. 539, 229–271 (2005) 55. Spielvogel, E.R.: A variational principle for waves of infinite depth. Arch. Ration. Mech. Anal. 39, 189–205 (1970) 56. Stokes, G.G.: Mathematical and Physical Papers. Cambridge University Press, Cambridge (1880) 57. Strauss, W.A.: Steady water waves. Bull. Am. Math. Soc. 47, 671–694 (2010) 58. Thomas, G.P., Klopman, G.: Wave-current interactions in the nearshore region. In: Hunt, J.N. (ed.) Gravity Waves in Water of Finite Depth. Advanced Fluid Mechanics, vol. 10, pp. 215– 319. Computational Mechanics Publications, Southampton (1997) 59. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996) 60. Toland, J.F.: On a pseudo-differential equation for Stokes waves. Arch. Ration. Mech. Anal. 162, 179–189 (2002)

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61. Tsai, C.-H., Huang, M.-C., Young, F.-J., Lin, Y.-C., Li, H.W.: On the recovery of surface wave by pressure transfer function. Ocean Eng. 32, 1247–1259 (2005) 62. Umeyama, M.: Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philos. Trans. R. Soc. Lond. A 370, 1687–1702 (2012) 63. Van Dorn, W.G.: Oceanography and Seamanship. Dodd, Mead and Co., New York (1974) 64. Varvaruca, E.: Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth. Proc. R. Soc. Edinb. Sect. A 138, 1345–1362 (2008) 65. Wahlén, E.: Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483 (2009) 66. Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (2002)

Breaking Water Waves Joachim Escher

Abstract A basic question in the theory of nonlinear partial differential equations is: when does a solution form a singularity and what is its nature? The aim of this lecture series is to offer an introduction into the analytic study of these questions for unidirectional shallow water waves models. Two particular models are investigated: the famous Korteweg–de Vries equation and the more recent Camassa–Holm equation. A rather classical approach to the Korteweg–de Vries equation is presented showing that this flow is globally well-posed for large classes of initial conditions. As a consequence wave breaking cannot be observed within the Korteweg–de Vries regime. In pronounced contrast to that a different picture may be seen in the case of the Camassa–Holm flow: while some solutions exist for ever other develop a wave breaking in finite time. Finally a geometric picture of the Camassa–Holm equation is presented as a geodesic flow on a Fréchet–Lie group consisting of smooth diffeomorphisms of the real line.

1 Introduction In the broadest sense the description of any water wave is embedded into the dynamics of Euler’s equations of motion for an inviscid perfect fluid. It is however a matter of empirical experience that the fluid dynamical behaviour may be very complicated and in fact there are huge open problems which are still beyond our reach—Euler’s equations are far from being fully understood. Against this background the long-standing goal in mathematical hydrodynamics to produce simplified models which capture as many as possible of the characteristic phenomena of water waves becomes comprehensible. The requirements for such water wave models certainly encompass a Hamiltonian structure, the existence and interaction of solitons, and the capacity of describing breaking and peaking.

J. Escher () Inst. for Applied Mathematics, Gottfried Wilhelm Leibniz University, Hannover, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2016 A. Constantin (ed.), Nonlinear Water Waves, Lecture Notes in Mathematics 2158, DOI 10.1007/978-3-319-31462-4_2

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Fig. 1 A soliton for the Korteweg–de Vries equation

c

This aim was clearly formulated by FRS G.B. Whitham1 : “Although both breaking and peaking, as well as criteria for the occurrence of each, are without doubt contained in the equations of the exact potential theory,2 it is intriguing to know what kind of simpler mathematical equation could include all these phenomena”. The famous Korteweg–de Vries equation,3 ut C uxxx  6uux D 0;

t > 0; x 2 R;

(1)

already proposed in 1895, cf. [26] to describe the height u D u.t; x/ of a shallow water wave travelling over a flat bed,4 carries a Hamiltonian structure and possesses solitary waves, whose interaction is fairly well understood, cf. [31] (Fig. 1). It is well-known (and the first goal of this lecture series to prove) that the Korteweg-de Vries equation possesses for large classes of initial conditions solutions that are defined globally (in time). So neither the phenomenon of breaking nor of peaking of waves may be observed for solutions of (1). After several attempts to modify Korteweg–de Vries’ equation artificially to produce water wave models with breaking and peaking, substantial new input was given by the introduction of the so-called Camassa–Holm5 equation ut  utxx C 3uux  2ux uxx  uuxxx D 0;

1

t > 0; x 2 R;

See [38], p. 477. The notion of “equations of the exact potential theory” is here used as a synonym for Euler’s equations, JE. 3 In fact Eq. (1) as a model for shallow water waves was first studied by Boussinesq in 1871, cf. [4]. 4 Equation (1) results as the asymptotic model from the two-dimensional Euler equations with a free surface in the so-called shallow water regime  D h2 =2  1, where h stands for the averaged height of the wave and  for its characteristic wavelength. Letting a denote a typical amplitude and accordingly " WD a=h a dimensionless amplitude parameter, the scaling in the Euler equations to derive (1) is " D O./. 5 The Camassa–Holm equation may also be derived from the two-dimensional Euler equations in p the shallow water regime  D h2 =2  1, but with the scaling " D O. /. 2

Breaking Water Waves

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where u has the same meaning as in the Korteweg–de Vries equation. Many aspects of the mathematical beauty of this equation have been exposed over the last two decades. Particularly, it has been shown, among many other properties, that the Camassa–Holm equation has a bi-Hamiltonian structure [8, 33], that there is a geometrical formulation via geodesic flows on suitable diffeomorphism groups [2, 15, 27], that it possesses interacting peaked solitons as weak solutions [6, 12], and that there are solutions, which leave their existence in finite time by producing singularities in the first spatial derivative [11, 13, 14]. In light of the forgoing comments, it becomes clear that the Camassa–Holm equation attracted some attention over the last two decades. It is in fact one of the major objects of this lecture series to offer some insight into the dynamic behaviour of the flow induced by the Camassa–Holm equation. We shall see that it possible on the one hand to describe the blow-up scenario fairly precise and on the other hand to specify initial conditions guaranteeing permanent waves. We also present a geometric picture of general geodesic flows on diffeomorphisms groups of the real line in which the Camassa–Holm dynamics naturally may be embedded into. Let E denote a suitable Banach space of functions over R, e.g. E D H k .R/

or

BUCk .R/ for some k 2 N;

i.e. the classical Sobolev space of order k 2 N, built over L2 .R/, or the Banach space of all k-times continuously differentiable functions on the real line, which are— together with all derivatives—bounded and uniformly continuous, respectively. Given ' 2 E, we shall study the following initial value problems ut C uxxx  6uux D 0; t > 0; x 2 R; u.0; x/ D '.x/; x 2 R;

(2)

for the Korteweg–de Vries equation, and ut  utxx C 3uux  2ux uxx  uuxxx D 0; t > 0; x 2 R; u.0; x/ D '.x/; x 2 R;

(3)

for the Camassa–Holm equation. We shall construct solutions to these nonlinear evolution equations belonging (among other spaces) to C.Œ0; TC /; E/; where the positive number TC WD TC .'; E/ is the maximal time of existence, depending in general on ' and E. A solution u is called global if TC D 1. If TC < 1 we say that the solution u develops a finite time blow-up.

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In addition we shall present a framework in which the following principle of continuation of solutions holds: If TC < 1 then lim sup ku.t/kE D 1: t!TC

Assume now that E C1 .R/ and that TC < 1. We call a finite time blow-up a wave breaking, if sup .t;x/2Œ0;TC /R

ju.t; x/j < 1;

lim supfsup jux .t; x/jg D 1 : t!TC

x2R

The principal theme of this course may be phrased as follows: Given ' 2 E, let u 2 C.Œ0; TC /; E/ denote a solution to (1) or (2). Specify a priori criteria on ' in order to decide whether TC .'; E/ D 1

or TC .'; E/ < 1:

In the latter case, decide whether a wave breaking occurs or not.

2 A Well-Posedness Result for the Korteweg–de Vries Equation Consider the Cauchy problem for the Korteweg–de Vries equation t > 0; x 2 R; ut C uxxx  6uux D 0; u.0; x/ D '.x/; x 2 R;

(4)

with the initial condition '. In order to construct a functional analytic framework in which we shall study (4), let S .R/ denote the Schwartz space of all rapidly decreasing smooth functions on R and by S 0 .R/ its topological dual space,6 the space of tempered distributions. The Fourier transform7 of a tempered distribution u is denoted by F .u/ or by uO . It is convenient to study this evolution equation in the following scale of Hilbert spaces. Given s 2 R, let H s WD H s .R/ WD fu 2 S 0 .R/ I Π7! .1 C jj2 /s=2 jOu./j 2 L2 .R/g

Given a Fréchet space F over R, we use F0 to denote the topological dual space of F, consisting of all continuous linear functionals on F, i.e. F0 D L .F; R/. 7 For basic facts about the Fourier transform and its applications we refer to [22]. 6

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with the norm Z kukH s WD

R

.1 C jj2 /s jOu./j2 d

1=2

:

Note that Z .ujv/H s D

R

.1 C jj2 /s uO ./v./ O d

is an inner product on H s .R/, which induces the norm in H s .R/. We collect some further useful facts of the scale fH s .R/ I s 2 Rg: • S .R/ H s .R/ H t .R/ S 0 .R/, if s  t. These embeddings are continuous and dense. • H s .R/ C0 .R/, provided s > 1=2. This a simple version of Sobolev’s embedding theorem.8 • ŒH s .R/0 D H s .R/ with respect to the duality pairing Z hu; vi D

R

u v dx;

u; v 2 S .R/:

• @ 2 L .H s .R/; H s1 .R// for each s 2 R. • Let s u WD F 1 ..1 C jj2 /s uO .//. Then for all s; t 2 R: s 2 Isom.H t .R/; H ts .R//

with

Œs 1 D s :

We next realise the Korteweg–de Vries equation as an abstract evolution equation in H 1 .R/. For this let: u 2 H 2 .R/ dom.A0 / WD H 2 .R/; A0 u WD @3x u; 2 dom.B/ WD H .R/; BŒu; u WD 6uux u 2 H 2 .R/: Observe that, given u 2 H 2 .R/, we have that ux 2 H 1 .R/ L1 .R/. Thus BŒu; u 2 L2 .R/. Using this notation, the initial value problem for the Korteweg–de Vries equation may equivalently be formulated in the form of the following abstract evolution equation ut C A0 u C BŒu; u D 0; t > 0; u.0; / D '; on R:

(5)

Note that the first equation in (5) may (and has to) be realised in C.Œ0; T/; H 1 .R//. 8

We denote by C0 the Banach space of all continuous functions on R, vanishing at infinity.

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A useful transformation. Assume that u 2 C.Œ0; T/; H 2 .R// \ C1 .Œ0; T/; H 1 .R//

(6)

is a solution to the Cauchy problem (5). Moreover let fU.t/ I t 2 Rg denote the group in L .H 1 .R// generated by A0 . Then, given t 2 R, the operator U.t/ is unitary on H 1 .R/. Particularly, we have that kU.t/kL .H 1 .R// D 1

and ŒU.t/1 D U.t/;

t 2 R:

Moreover, given v 2 H 2 .R/ and t 2 R, we have that @t U.t/v C A0 U.t/v D 0: Consider now v.t/ WD U.t/u.t/

for t 2 Œ0; T/:

Then v 2 C.Œ0; T/; H 2 .R// \ C1 .Œ0; T/; H 1 .R//; and it holds that @t u.t/ D @t .U.t/v.t// D A0 u.t/ C U.t/@t v.t/

in H 1 .R/:

But we also have that @t u.t/ D A0 u.t/ C BŒu.t/; u.t/

in H 1 .R/;

which shows that @t v D U.t/BŒU.t/v.t/; U.t/v.t/

t 2 Œ0; T/:

Note that BŒu; u D 6uux is a bilinear operator on H 2 .R/. Thus it induces in a natural way a quasi-linear operator A1 .v/ on H 2 .R/ by setting A1 .v/w WD BŒv; w

for v; w 2 H 2 .R/:

Introducing finally the following time-dependent quasilinear operator A.t; v/w WD U.t/BŒU.t/v; U.t/w

for v; w 2 H 2 .R/;

Breaking Water Waves

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we may recast the transformed Korteweg–de Vries equation derived above, i.e. the equation @t v D U.t/BŒU.t/v.t/; U.t/v.t/;

t > 0;

together with the initial condition v.0; x/ D '.x/ for x 2 R, as an abstract quasilinear evolution equation in L2 .R/ in the following way: vt C A.t; v/v D 0; t > 0; v.0; / D '; on R: There is a powerful approach to quasi-linear evolution equations of hyperbolic type due to Kato, cf. [24]. In the following we shall explain the basic construction of this approach in some detail. First we need the following notion: Let H denote a Hilbert space over R or C. An (unbounded) operator A W dom.A/ H ! H on H is called m-dissipative in the sense of Phillips [34] iff .i/ 0: Proposition 1 (Phillips [34]) A densely defined operator A is m-dissipative if and only if A and its adjoint A are dissipative. Let now Y be a closed subspace of the Hilbert space H. Given r > 0, let B D Br;Y denote the closed ball in Y with radius r and centre 0. Given .t; v/ 2 RC  B, we assume that an operator A.t; v/ W dom.A.t; v// H ! H is given, such that Y dom.A.t; v//. With the above notations, we formulate the following hypotheses on the operator family fA.t; v/ I .t; v/ 2 RC  Bg: .A1 / There is a D .r/  0 such that A.t; v/ C is m-dissipative for all .t; v/ 2 RC  B. .A2 / There is an isomorphism Q from Y onto H such that the operator QA.t; v/Q1  A.t; v/ extends for any .t; v/ 2 RC  B to a bounded operator on H. .A3 / There is a  WD .r/ such that kA.t; v/  A.t; w/kL .Y;H/  kv  wkH .A4 /

for all t 2 R; v; w 2 B:

A.; v/ W RC ! L .Y; H/ is for any v 2 B strongly continuous.

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After these preparations we may formulate the following general existence result for abstract quasilinear evolution equations of hyperbolic type: Theorem 1 (Kato [24]) If the family of operators fA.t; v/ I .t; v/ 2 RC  Bg satisfies .A1 /–.A4 /, then the initial value problem for the quasilinear evolution equation vt C A.t; v/v D 0; t > 0; v.0; / D '; on R: possesses for each ' 2 Y a unique solution v in the class C.Œ0; TC /; Y/ \ C1 .Œ0; TC /; H//: The maximal existence time TC depends on k'kY and the principle of continuation holds in Y: If TC < 1 then

lim sup ku.t/kY D 1: t!TC

In order to realise (2) within this framework, we set: H WD L2 .R/;

Y WD H 2 .R/;

Q WD 2 D id  @2x :

Then Y D dom.A.t; v// for all .t; v/ 2 RC  H 2 and Q is an isomorphism form H 2 .R/ onto L2 .R/. To verify hypothesis .A1 /, recall that A.t; v/w D U.t/fU.t/v  @x U.t/wg for t 2 RC and v; w 2 H 2 .R/. Thus integration by parts yields .A.t; v/wjw/L2 D .U.t/v  @x U.t/wjU.t/w/L2 R D  R12 R U.t/v  @x ŒU.t/w2 dx D 12 R @x U.t/v  ŒU.t/w2 dx: Let now WD

1 sup k@x U.t/vkL1 : 2 .t;v/2RC B

Observing that Sobolev’s embedding theorem yields k  kL1  k  kH 1 and using the fact that U.t/ is unitary, we obtain: 2 D sup kU.t/vx kL1  sup kU.t/vx kH 1  sup kvx kH 1  sup kvkH 2  r:

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Thus we conclude from the estimate .A.t; v/wjw/L2   kU.t/wk2L2 D  kwk2L2 that C A.t; v/ is dissipative, uniformly in .t; v/ 2 RC  Br . Similarly one shows that C A.t; v/ is dissipative as well, uniformly in .t; v/ 2 RC  Br . Invoking Proposition 1, we consequently find that the operator family fA.t; v/ I .t; v/ 2 RC  Br g satisfies the hypothesis .A1 /. In order to discuss hypothesis .A2 /, we start with the following observation. Given f 2 H 2 , the linear (multiplication) operator Mf WD Œv 7! f  @x v is not bounded on L2 .R/. However, denoting by ŒMf ; @x  the commutator of Mf and @x , we have ŒMf ; @x v D f  @x v  @x .f  v/ D @x f  v

for all v 2 H 1 .R/:

Recalling further that @x f 2 H 1 .R/ C0 .R/; we see that ŒMf ; @x  extends to a bounded linear operator on L2 .R/. Commutator estimates of this type are true in much more general situations. In fact letting Q WD 2 D 1  @2x and using ŒU.t/; @x  D 0 for all t 2 R, one can verify that   ŒQ; A.t; v/z D 6U.t/ 2U.t/vx  U.t/zxx C U.t/vxx  U.t/zx for all z 2 H 2 .R/. From this we conclude that the operator ŒQ; A.t; v/Q1 extends to a bounded operator on L2 .R/, uniformly in .t; v/ 2 RC  B. This implies in turn that the operator family fA.t; v/ I .t; v/ 2 RC  Bg satisfies the hypothesis .A2 / of Theorem 1. Let us now consider hypothesis .A3 /. Recalling that A.t; v/z D U.t/fU.t/v  @x U.t/zg for t 2 RC and v; z 2 H 2 .R/, we get kA.t; v/z  A.t; w/zkL2 D kU.t/.v  w/  @x U.t/zkL2  k@x U.t/zkL1 kv  wkL2  kzkH 2 kv  wkL2 for all v; w; z 2 H 2 .R/ and all t 2 RC . Thus hypothesis .A3 / holds true.

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Finally, since Œt 7! U.t/ is strongly continuous, it is not difficult to verify that .A4 / is true as well. Summarising we get the following result. Corollary 1 Given ' 2 H 2 .R/, there is a unique TC > 0 and unique solution u 2 C.Œ0; TC /; H 2 .R// \ C1 .Œ0; TC /; H 1 .R// of (2) and the principle of continuation in H 2 .R/ holds true: If

sup ku.t; /kH 2 < 1

t2Œ0;TC /

then TC D 1:

3 Conservation Laws and Global Existence for Korteweg–de Vries Equation We shall now identify three conservation laws for the Korteweg–de Vries equation, which allow us to apply the principle of continuation in H 2 .R/. As a consequence we get that the Korteweg–de Vries equation is globally well-posed on H 2 .R/. The argument given below is a special case of the results in [32]. To start with, assume that u is a smooth solution of the Korteweg–de Vries equation on Œ0; TC /. Multiplying ut C uxxx  6uux D 0;

.t; x/ 2 Œ0; TC /  R

by 2u, we get 2uut C 2uuxxx  12u2 ux D 0;

.t; x/ 2 Œ0; TC /  R:

But observing that @t .u2 / D 2uut ;

@x .2uuxx  u2x / D 2uuxxx;

@x .4u3 / D 12u2 ux ;

we find that .u2 /t C .2uuxx  u2x  4u3 /x D 0: Integrating this identity with respect to x yields d dt

Z R

u2 dx D 0:

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In conclusion, we get the following conservation law: Z I0 .t/ WD

u2 .t; x/ dx D I0 .0/

R

for all t 2 Œ0; TC /:

Next we claim that also Z I1 .t/ WD

 R

 u2x .t; x/ C 2u3 .t; x/ dx

is a conserved quantity under the Korteweg–de Vries flow. Indeed, we have Z d I1 .t/ D .2ux utx C 6u2 ut / dt: dt R Expanding the integrand of the latter integral by using the Korteweg–de Vries equation, we find that 2ux utx C 6u2 ut D 2ux uxxxx C 12uuxuxx C 12u3x  6u2 uxxx C 36u3 ux : We first observe that Z Z Z 2 ux uxxxx dx D 2 uxx uxxx dx D .u2xx /x dx D 0: R

R

R

Next, we have Z R

u3x

Z dx D R

ux u2x

Z dx D 2 R

uux uxx dx

and Z

Z

2

u uxxx dx D 2

R

R

uux uxx dx:

Thus we may conclude that Z R

.12uuxuxx C 12u3x  6u2 uxxx / dx D 0:

Finally, we notice that Z 36

R

u3 ux dx D 9

Z R

.u4 /x dx D 0:

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Summarising, we have established the following conservation law: Z I1 .t/ D

 R

 u2x .t; x/ C 2u3 .t; x/ dx D I1 .0/

(7)

for all t 2 Œ0; TC /. There is a further invariant of the Korteweg–de Vries flow: Z I2 .t/ WD

R

 2  uxx  10uu2x C 5u4 dx D I2 .0/ for all t 2 Œ0; TC /;

which can be verified similarly as for I1 . Having these conservation laws at hand, we are going now to derive an a priori bound for solutions of (2) in the H 2 -norm. Given t 2 Œ0; TC /, let Z M.t/ WD max ju.t; x/j;

D.t/ WD

x2R

R

u2x .t; x/ dx:

Integrating the identity @x .u2 / D 2uux with respect to x, we get Z

2

u .t; x/ D 2

Z

x 1

uux dy  2

R

juux j dx:

Applying Cauchy-Schwarz’ inequality, we therefore find: 2

Z

u .t; x/  2

1=2 Z

1=2 1=2 2 u dx  ux dx D 2I0  D1=2 .t/ 2

R

R

for all .t; x/ 2 Œ0; TC /  R. Invoking the conservation law for I1 , it follows that Z D.t/ D I1  2

R

u3 dx  I1 C 2M.t/  I0 ;

and consequently (8) implies that 1=2

M 2 .t/  2I0

 .I1 C 2M.t/  I0 /1=2 :

From the latter inequality we conclude that M 3 .t/ 

4I0 I1 C 8I02 ; M.t/

t 2 Œ0; TC /;

which shows that M0 WD sup M.t/ < 1: t2Œ0;TC /

(8)

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In particular we get from (7) the following a priori bound Z R

u2x dx  I1 C 2M0 I0 ;

t 2 Œ0; TC /:

The conservation law for I2 shows that Z Z Z u2xx D I2 C 10 uu2x dx  5 u4 dx R

But u  M.t/ and

R

R

R

t 2 Œ0; TC /:

u2x  I1 C 2M0 I0 , so that

Z R

u2xx  I2 C 5M0 .2I1 C 5M0 I0 /

t 2 Œ0; TC /;

using again the conservation law for I0 . Summarising, there is a constant C D C.I0 ; I1 ; I2 /, depending only on the conserved quantities I0 ; I1 ; and I2 such that ku.t; /kH 2  C

for all t 2 Œ0; TC /:

Combining the above estimate with Corollary 1 we obtain the following result. Corollary 2 The Korteweg–de Vries equation is globally in time well-posed on H 2 .R/. We conclude this section by adding some remarks and further references. • An inspection of the existence proof given above shows that this approach works in H s .R/, provided that s > 3=2. There is only one point that needs some attention: If s > 3=2 is not an integer then the isomorphism Q W H s .R/ ! L2 .R/ is no longer a differential but a Fourier multiplication operator. Commutators estimates for such operator with multiplication operators are more involved, but are still available, cf. [24, 36]. • Kato’s approach presented above has been refined in 1993 by Kenig et al. to the space H s with s > 3=4, cf. [25]. • Bourgain proposed in 1995 a new approach to the Korteweg–de Vries equation. A particular highlight (among many others) is that Korteweg–de Vries equation is well-posed on L2 .R/, cf. [3]. The global well-posedness follows then directly from the conservation law for I0 . • In 2003 [10] Colliander et al. extended Bourgain’s approach to H s .R/ with s > 3=4. • Also in 2003 Christ et al. [9] showed that the Korteweg–de Vries equation cannot be well-posed in H s .R/ if s < 3=4, in the sense that the corresponding solution operator, which maps an initial condition to the solution of (2) fails to be uniformly continuous with respect to the H s -norm.

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• The limit case H 3=4 .R/ was solved by Guo in 2009 by showing that (2) is wellposed in this space, cf. [23]. • The Korteweg–de Vries equation can be written in the form ut C .uxx  3u2 /x D 0; R showing that I1 WD R u dx is also a conserved quantity as well. • The conservation laws I1 and I0 are traditional and represent conservation of mass and momentum, respectively. The integral of motion I1 represents conservation of energy and it is thus related to a Hamiltonian system. It was found by Whitham in 1965, cf. [37]. In the same year Kruskal and Zabusky found the conservation law I2 , cf. [29]. Eventually, Kruskal et al. [30] showed in 1970 that these four are merely the first of an infinite sequence .In /n2N of conserved functionals. There are recursion formulas for In . • Lax showed 1975 in [32] that the sequence .In / can be used to bound k@k ukL1 for any k 2 N. We have seen that, given any initial datum ' 2 H 2 .R/, the Korteweg–de Vries equation has no breaking waves. It has also no peaked solutions, i.e. solutions which are Lipschitz continuous, but not differentiable with respect to the spatial variable. We will meet such waves in the next section.

4 The Camassa–Holm Equation and Wave Breaking In this section we study the Cauchy problem for the Camassa–Holm equation: ut  utxx C 3uux  2ux uxx  uuxxx D 0; t > 0; x 2 R; u.0; x/ D '.x/; x 2 R:

(9)

Introducing the momentum variable p D u  uxx ; the first equation in (9) can be written as: pt C 2ux p C upx D 0: Moreover, recalling the notation, Q WD 2 D id  @2x ; we have that u D Q1 p, so that the Camassa–Holm equation appears as pt C 2.Q1 p/x  p C Q1 p  px D 0:

(10)

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Finally, to formalise things further, let us introduce the following quasi-linear operator: A.r/p WD Q1 r  px C 2.Q1 r/x  p;

r 2 L2 .R/; p 2 H 1 .R/:

Then (9) reads as pt C A.p/p D 0; t > 0; p.0; / D Q'; on R:

(11)

Well-posedness of the Camassa–Holm equation can now be established analogously to the reduced Korteweg–de Vries equation, cf. Sect. 2 by applying Kato’s theory to (11). The situation here is somewhat more involved, since the operator Œp 7! A.r/p is not only nonlinear in r but depends also non-locally on the variable r. In addition, given r 2 L2 .R/, the choice of domain D of the linear operator A.r/ is not obvious, in the sense that taking D D H 1 .R/ (which seems natural at a first sight), the linear operator A.r/ W H 1 .R/ L2 .R/ ! L2 .R/ is not closed. To get a closed operator one has to choose D WD fv 2 L2 .R/ I .Q1 r/  v 2 H 1 .R/g. However these difficulties can be resolved. The result is, cf. [14]: Theorem 2 Given ' 2 H 3 .R/, there is a TC > 0 and unique solution u.t; x/ of (9) in the class9 C.Œ0; TC /; H 3 .R// \ C1 .Œ0; TC /; L2 .R//: The maximal existence time depends only on k'kH 3 , i.e. TC D TC .k'kH 3 /. Furthermore the principle of continuation holds in H 3 .R/: If

sup ku.t; /kH 3 .R/ < 1

t2Œ0;TC /

then TC D 1:

Recall that Q 2 Isom.H 2 .R/; L2 .R//. Thus the continuation principle for u D Q1 p can be formulated as If

sup kp.t; /kH 1 .R/ < 1

t2Œ0;TC /

then TC D 1:

If the flow emerges from a more regular initial condition ' 2 H 3C with  0 then the corresponding solution carries on this regularity, i.e. p 2 C.Œ0; TC /; H 1C .R// \ C1 .Œ0; TC /; H .R//;

9

Solutions of that class are called strong solutions.

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J. Escher

cf. [14]. Particularly, we shall use later on the following regularity: p 2 C1 .Œ0; TC /; H 1 .R//;

(12)

which holds if ' 2 H 4 .R/. A conserved quantity for the Camassa–Holm equation. Assume that u is a smooth solution to the Camassa–Holm flow and recall that ut  utxx D pt D upx  2pux : This can be used in the following calculation: 1d 2 dt

Z R

.u2 C u2x / dx D D

Z ZR ZR

D R

.uut C ux utx / dx u.ut  utxx / dx .u2 px  2uux p/ dx D 0;

meaning that J1 .t/ WD ku.t; /k2H 1 D J.0/ for all t 2 Œ0; TC /. Invoking Sobolev’s embedding theorem we directly notice that there are no spatially unbounded (strong) solutions of the Camassa–Holm equation. This is a necessary condition for wave breaking. Our next result shows that there are no other singularities then breaking waves and it characterises the blow-up scenario. Theorem 3 (Blow-Up Scenario) A strong solution of the Camassa–Holm equation blows-up in finite time only as a wave breaking. More precisely: If TC < 1

then

  lim inf inf ux .t; x/ D 1: t!TC

x2R

Proof Let u denote a strong solution to the Camassa–Holm flow and assume that there is a K > 0 such that ux .t; x/  K

for all .t; x/ 2 Œ0; T/  R:

By using pt D 2pux  upx , we obtain d kp.t/k2L2 D 2.pt jp/L2 dt Z Z D 4 p2 ux dx  2 upx p dx Z R Z ZR D 4 p2 ux dx  .p2 /x u dx D 3 p2 ux dx: R

R

R

(13)

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Invoking (13), we therefore find that: d kp.t/k2L2  3Kkp.t/k2L2 : dt Differentiating10 the equality pt D 2pux  upx with respect to x, we get ptx D 3px ux  2puxx  pxx u:

(14)

Thus we find that d kpx .t/k2L2 D 6 dt

Z R

p2x ux

Z 4

Z R

ppx uxx  2

R

px pxx u:

But noticing that Z

Z 2 R

px pxx u D 

R

.p2x /x u D

Z R

p2x ux

and using p D u  uxx , we get 0D

1 3

Z R

@x .p3 / dx D

Z R

p2 px D

Z

Z R

uppx 

R

uxx ppx :

Thus d kpx .t/k2L2 D 5 dt

Z R

p2x ux  4

Z R

uppx :

The conservation law for J1 in combination with Sobolev’s embedding theorem imply that ku.t/kL1  k'kH 1

for all t 2 Œ0; TC /:

Furthermore we have Z 2 ppx  2kpkL2 kpx kL2  kpk2L2 C kpx k2L2 : R

Combining these observations with the pointwise bound on ux we get from d kpx .t/k2L2 D 5 dt

10

Z R

p2x ux  4

Z R

uppx;

The regularity property of p stated in (12) allows to realise (14) in C.Œ0; TC /; L2 .R/.

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the estimate d kpx .t/k2L2  .5K C 2k'kH 1 /kpx k2L2 C 2k'kH 1 kpkL2 : dt In combination with the estimate for C D C.K; k'kH 1 / such that

d kp.t/k2L2 , we conclude that there is a constant dt

d kp.t/k2H 1  Ckp.t/kH 1 : dt Gronwall’s lemma now implies that kp.t/kH 1  eCTC kp.0/kH 1

for all t 2 Œ0; TC /:

This contradicts the continuation principle for p.t/ in H 1 .R/ and completes the proof. u t Note that Theorem 3 describes the blow-up scenario, but does not guarantee a wave breaking. Even worse: From the conservation law for J1 we know that Z sup t2Œ0;TC / R

jux .t; x/j2 dx < 1:

If we were able to derive a pointwise bound on jux .t; x/j, Theorem 3 would imply global well-posedness of the Camassa–Holm flow. Comparing this with the Korteweg–de Vries equation, where we used not only one single conservation law but three of them, one may ask whether other integrals of motion imply a pointwise bound of jux .t; x/j. In fact there exist further conservation laws for the Camassa– Holm equation, e.g. each of the functionals Z J2 D Z J0 D

R

R

4 p2x C 5=2 p pC pC u dx ;

!

Z dx ;

J1 D Z J2 D

R

R

p pC dx ;  3  u C uu2x dx

is constant along classical solutions of the Camassa–Holm equation. Furthermore there are other conserved quantities in integral form whose integrand involve uxx , but all in a nonlocal way. However, we shall see later on that—in contrast to the case of the Korteweg–de Vries flow—the strategy of using suitable conservation laws in order to establish pointwise a priori bounds for jux .t; x/j does not lead to the desired result. To understand this fact, we proceed as follows.

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A further representation of the Camassa–Holm equation. Let F denote the Fourier transform on R and recall that the fundamental solution K of Q D id @2x is given by K D F 1



1 1 C 2

;

i.e. Q1 f D K  f for f 2 L2 .R/, where  denotes convolution in L2 .R/. By the residue theorem one finds the explicit formula K.y/ D

ejyj ; 2

y 2 R;

for this fundamental solution. Let now u be a solution to the Camassa–Holm equation. A direct calculation shows that  1  Q.ut C uux / D 2uux  ux uxx D @x u2 C u2x 2 from which we may conclude that  1  ut C uux D @x K  .u2 C u2x / : 2 The above equality holds in C.Œ0; T/; H 1 .R//. Differentiating it with respect to x yields  1  utx C u2x C uuxx D @2x K  .u2 C u2x / 2  1  D .Q  id/ K  .u2 C u2x / 2 1 2 1 2 2 D u C ux  K  .u C u2x /; 2 2 and therefore 1 1 utx C uuxx D u2  u2x  K  .u2 C u2x /: 2 2 Theorem 4 (Breaking of Odd Waves) Assume that ' 2 H 3 .R/ is odd and that 'x .0/ < 0. Then the wave originating from ' breaks in finite time. In fact, TC < 1=2j'x.0/j. Proof Let u 2 C.Œ0; TC /; H 3 .R// \ C1 .Œ0; TC /; H 2 .R//

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J. Escher

denote the unique non-extendable solution to (9) with initial datum '. Define v.t; x/ WD u.t; x/

for .t; x/ 2 Œ0; TC /  R:

Then one easily verifies that v solves the Camassa–Holm equation with v.0; x/ D u.0; x/ D '.x/ D '.x/;

x 2 R:

By uniqueness we conclude that v  u on Œ0; TC /, i.e. the solution u is odd at any instant. In particular, since u.t; / and uxx .t; / are continuous, we note that u.t; 0/ D uxx .t; 0/ D 0

for all t 2 Œ0; TC /:

(15)

Let now g.t/ WD ux .t; 0/ and observe that g 2 C1 .Œ0; TC /; R/. Furthermore, we note that g.0/ is negative by hypothesis. Using (15) in combination with

1 1 utx C uuxx D u2  u2x  k  u2 C u2x 2 2 we get 1 1 g0 .t/ D  g2 .t/  2 2

Z



1 1 ejyj u2 C u2x dy   g2 .t/: 2 2

The latter inequality shows that 1 dh 1 i g0 .t/ D 2  dt g.t/ g .t/ 2 and we find after integration: 1 t 1  C : g.t/ g.0/ 2 This implies that TC < 2=g.0/, recalling that g.0/ is negative and that g is strictly decreasing. t u Wave breaking for more general profiles. We know that a wave breaking occurs if m.t/ WD min ux .t; x/; x2R

t 2 Œ0; TC /

falls below any value as t ! TC . To derive further blow up criteria we pursue the following strategy: Try to derive an ordinary differential inequality for m.t/ which

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u(t; x)

Fig. 2 The set-valued mapping Œt 7! .t/

x possesses solutions with inft2Œ0;TC / m.t/ D 1. For this assume for the moment that there is a C1 -curve .t/ such that m.t/ D ux .t; .t//;

t 2 Œ0; TC /:

Then uxx .t; .t// D 0 and thus: mt .t/ D utx .t; .t// C uxx .t; .t// 0 .t/ D utx .t; .t//; invoking the definitions of m.t/ and .t/. Recall that 1 1 utx C uuxx C u2x D u2  K  .u2 C u2x /: 2 2 Evaluating this equation in .t; .t// gives m2 mt C D u2 .t; .t//  2

Z R

K..t/  y/Œu2 .t; y/ C 12 u2x .t; y/ dy:

Before we try to estimate the right hand side of the latter equation in order to get a differential inequality for m, let us check our assumptions that we imposed so far on .t/. Recall that .t/ should be a C1 -curve such that m.t/ D min ux .t; x/ D u.t; .t//: x2R

But in general there are several points in which ux .t; / attains its minimum, cf. Fig. 2. This means that the mapping t 7! .t/ is in general multi-valued. Even if we choose a single valued selection, we cannot accept continuity of Œt 7! .t/, as indicated in Fig. 3.

104

J. Escher u(t; x)

u(t; x)

x

x

ξ(t1 )

ξ(t2 )

Fig. 3 A single-valued selection of Œt 7! .t/ is in general not continuous

Nevertheless the following result shows that the essence of our formal consideration is correct. Theorem 5 (Variation of Minima) Let T > 0 and v 2 C1 .Œ0; T/; H 2 .R// be given. Then given any t 2 Œ0; T/ there is at least one point .t/ 2 R with m.t/ WD inf Œvx .t; x/ D vx .t; .t//; x2R

and the function m is almost everywhere differentiable on .0; T/ with m0 .t/ D vtx .t; .t//

a.e. on .0; T/:

Proof Fix t 2 Œ0; T/ and define m.t/ WD infx2R Œvx .t; x/. (i) If m.t/  0 we have that v.t; / is nonincreasing on R and thus v  0 (recall that v vanishes at infinity). So we may assume that m.t/ < 0. But since vx .t; / vanishes at infinity as well there must be a .t/ 2 R with m.t/ D vx .t; .t//. (ii) Let now s 2 Œ0; T/ be given. If m.t/  m.s/ we have 0  m.s/  m.t/ D infx2R Œvx .s; x/  vx .t; .t//  vx .s; .t//  vx .t; .t//; and we conclude from Sobolev’s embedding theorem that jm.s/  m.t/j  kvx .s/  vx .t/kL1  kvx .s/  vx .t/kH 1 : Thus the mean value theorem implies that jm.s/  m.t/j  js  tj

max

0  maxfs;tg

kvtx ./kH 1 ; s; t 2 Œ0; T/:

Recalling that vtx 2 C.Œ0; T/; H 1 .R//, we see that m is locally Lipschitz on Œ0; T/ and thus almost everywhere differentiable by Rademacher’s theorem, cf. [21].

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(iii) We have that  v .t C h/  v .t/  x  x   vtx .t/ 1 ! 0 as  H h

h ! 0:

Invoking the continuous embedding H 1 .R/ C0 .R/, we get ˇ ˇ v .t C h; y/  v .t; y/ x ˇ ˇ x sup ˇ  vtx .t; y/ˇ ! 0; h y2R

(16)

as h ! 0. By the definition of m we have that m.t C h/ D vx .t C h; .t C h//  vx .t C h; .t//: Consequently, given h > 0, we obtain vx .t C h; .t//  vx .t; .t// m.t C h/  m.t/  : h h Letting h & 0 and using (16), we find that lim sup h&0

m.t C h/  m.t/  vtx .t; .t//; h

t 2 .0; T/:

A similar argument shows that lim inf h&0

m.t/  m.t  h/  vtx .t; .t//; h

t 2 .0; T/:

Since m is almost everywhere differentiable the last two observations imply that dm .t/ D vtx .t; .t// dt

a.e. on .0; T/; t u

which completes the proof. Corollary 3 Let u be a strong solution of the Camassa–Holm equation, set m.t/ WD inf Œux .t; x/; x2R

t 2 Œ0; TC /;

and choose .t/ 2 R such that m.t/ D ux .t; .t//. Then for almost all t 2 Œ0; TC / we have that mt .t/ C

m2 .t/ D u2 .t; .t//  F.t/; 2

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where Z F.t/ WD R

K..t/  y/Œu2 .t; y/ C 12 u2x .t; y/ dy:

(17)

According to our strategy, we aim the derive an ordinary differential inequality for m. In view of the above corollary, we need to investigate the term given in (17). Recall that K.z/ D 12 ejzj . Thus F.t/ D

1 2

Z R

ej .t/yj Œu2 .t; y/ C 12 u2x .t; y/ dy:

Fix t 2 Œ0; TC / and set for simplicity  D .t/ Splitting the integral into

R

1

e

R1

C

Z

and u.x/ D u.t; x/: 

 1

we first consider

ey .2u2 .y/ C u2x .y// dy:

Integration by parts shows that e

Z



ey

1

du2 dy D u2 ./  e dy

Z

 1

ey u2 .y/ dy:

But we have that du2 D 2u  uy  u2 C u2y : dy Thus we find the inequality e

Z

 1

ey .u2 .y/ C u2x .y// dy  u2 ./  e

Z

 1

ey u2 .y/ dy;

and finally u2 ./  e

Z



1

ey .2u2 .y/ C u2x .y// dy:

A similar computation yields 2

u ./  e



Z

1 

ey .2u2 .y/ C u2x .y// dy:

Breaking Water Waves

107

Fusing these two integrals gives F.t/ 

1 2 u .t; .t// 2

for all t 2 Œ0; TC /:

Recalling Corollary 5, we find that m0 .t/ C

m2 .t/ 1  u2 .t; .t// 2 2

a.e. on t 2 Œ0; TC /:

Using again the conservation of the H 1 -norm of solutions, we may estimate u2 .t; .t// against the H 1 -norm of the initial datum u.0; x/ D '.x/. Indeed: 2u2 .t; .t// D 2

Z

 .t/ 1

Z uux 

1  .t/

 uux dy

But 2

Z

 .t/ 1

Z uux 

1

 .t/

Z  uux dy
3=2 in order to still

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get strong solutions, cf. [35]. The critical value s D 3=2 is sharp, as shown by [7]: The Camassa–Holm equation is no longer well-posed in H s .R/, if s < 3=2. Furthermore, it is possible to prove well-posedness in Besov spaces, using sharp estimates on transport equations. The first relevant work in this direction is [17]. • It is well-known that the Camassa–Holm equation possesses solitary waves, which cannot be realised as strong solutions, cf. Sect. 5 below. Therefore a notion of weak solution is needed. The first construction of global weak solutions, which incorporates these solitary waves as well as their interaction was given by [12]. This approach has been refined in [16]. Later on different constructions of weak solutions, based on the H 1 .R/ conservation law were developed in [5, 6, 39]. • As for the Korteweg–de Vries equation, there is an countable family of conservation laws for the Camassa–Holm equation as well cf. [33]. However this additional information could so far not be of much help in the study of the qualitative behaviour of the Camassa–Holm flow.

5 Global Solutions for the Camssa-Holm Equation It is well-known that the Camassa–Holm equation possesses peakons, i.e. travelling wave solutions with a Lipschitz continuous, but not differentiable profile. More precisely, let '0 .x/ WD exp jxj for x 2 R. Then it can be verified that u.t; x/ WD '0 .t  x/;

t  0; x 2 R;

(23)

defines a global in time weak solution of (9). In the following we shall construct a large class of initial data for which the corresponding strong solutions to (9) exist globally in time (Fig. 4). A pointwise conserved quantity. In order to construct global in time strong solutions to (9), we first introduce Lagrangian coordinates induced by (9). Let u be a solution to the Camassa–Holm equation on Œ0; TC /. Given x 2 R, consider the flow induced by u, i.e.: 

Fig. 4 A peaked soliton for the Camassa–Holm equation

qt .t; x/ D u.t; q.t; x//; q.0; x/ D x:

t 2 .0; TC /

(24)

c

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Recall that H 2 .R/ C1 .R/ and u 2 C1 .Œ0; TC /; H 2 .R//. Thus we have that u 2 C1 .Œ0; TC /  R; R/; and therefore (24) possesses a unique solution q 2 C1 .Œ0; TC /  R; R/: Moreover, given x 2 R, we notice that qx solves the following linear initial value problem: 8
3=2 and q  r. Then we carry over these results to DiffH 1 .R/ D \q>3=2 D q .R/. (ii) Since D q .R/ is merely a topological, but not a Lie group, more regularity than continuity of the geodesic spray F defined by Eq. (29) cannot obtained from the general setting. It can however be shown that if s  1=2 then F extends to a smooth vector field Fq on TD q .R/, which is the geodesic spray of the metric, c.f. [20, Theorem 3.10]. In that case, the Picard-Lindelöf Theorem on the Banach manifold TD q .R/ ensures that, given any initial data .'0 ; v0 / 2 TD q .R/, there is a unique non-extendable solution .'; v/ of (30), defined on a maximal interval Iq .'0 ; v0 /, satisfying the initial condition .'.0/; v.0// D .'0 ; v0 /: (iii) In order to derive the result on the Fréchet–Lie group DiffH 1 .R/ from those obtained in (ii) for the scale fD q .R/ I q > 3=2g, we need to be able to rule out that Iq .'0 ; v0 / shrinks down to f0g as q tends to infinity. A remarkable observation due to Ebin and Marsden (see [18, Theorem 12.1]) states that, if the initial data .'0 ; v0 / is smooth, then the maximal time interval of existence Iq .'0 ; v0 / is independent of the parameter q. This is an essential ingredient in the proof of the local existence theorem for geodesics on DiffH 1 .R/ (see [20]). t u As a corollary, we get well-posedness for the corresponding Euler equation (31). Corollary 5 Let s  1=2 and ' 2 DiffH 1 .R/ be given and denote by Jmax the maximal interval of existence for (30) with the initial datum .idS1 ; '/. Set u WD v ı ' 1 . Then u 2 C1 .Jmax ; R/ is the unique non-extendable solution of the Euler equation ( ut D A1 Œu.Au/x C 2.Au/ux ; (32) u.0/ D ': In the special case that A D id  @2x , we get that u 2 C1 .Jmax ; R/ complies with the Cauchy problem for the Camassa–Holm equation: (

ut  utxx C 3uux  2ux uxx  uuxxx D 0; t > 0; x 2 R; u.0; x/ D '.x/; x 2 R:

15

Illustrative counterexamples against common beliefs in ordinary differential equations on Fréchet spaces can be found e.g. in [28].

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Based on Theorem 10 and using the metric introduced in (27) one can prove the following result, cf. [2]. Theorem 11 (Blow-Up Scenario of the Geodesic Flow: The Case s  1=2) Let s  1=2 and q > 3=2 be given and assume that q  s  0. Let furthermore .'; v/ 2 C1 ..t ; tC /; TD q .R// denote the non-extendable solution of the geodesic flow (30), emanating from .'0 ; v0 / 2 TD q .R/: If tC < 1 then   lim dq .'0 ; '.t// C kv.t/kH q D C1:

t"tC

A similar statement holds true if t > 1. Assume now that tC < 1. Then Theorem 11 makes it clear that there are only two possible blow-up scenarios: either the solution .'.t/; v.t// becomes large in the sense that lim .k'.t/H q k C kv.t/kH q / D 1;

t!tC

or the family of diffeomorphisms f'.t/ I t 2 .t ; tC /g becomes singular in the sense that

lim inf f'x .t; x/g D 0: t!tC

x2R

It should be emphasised that the blow-up result in Theorem 11 only represents a necessary condition. Indeed, we discussed already the case A D id  @2x , which is equivalent to the Camassa–Holm equation. The precise blow-up mechanism here has been established in Theorem 3. We know that a classical solution u blows up in finite time if and only if lim

t!tC

inf fux .t; x/g D 1;

x2R

which is somewhat weaker than blow-up in H 2 .R/. Note also that ux .t; x/ D vx ı '.t; x/ 

1 'x .t; x/

for .t; x/ 2 .t ; tC /  S1 :

(33)

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Hence in the case of a wave breaking for the Camassa–Holm equation, either jvx j becomes unbounded or vx becomes negative and 'x tends to 0 as t " tC . Finally it is worthwhile to mention that there is a natural extension of Theorem 11 to the Euler equation with respect to a general inertia operator A, provided that its order 2s exceeds the value 2. For a proof of this result we refer to [19]. Theorem 12 (Blow-Up Scenario of the Geodesic Flow: The Case s  1) Assume that s  1 and q  2s C 1 and let furthermore .'; v/ 2 C1 ..t ; tC /; TD q .R// denote the non-extendable solution of the geodesic flow (30), emanating from .'0 ; v0 / 2 TD q .R/: If the Eulerian velocity u D v ı ' 1 complies with the estimate inf

t2.t ;tC /

inf fux .t; x/g < 1

x2R

then tC D 1. A similar result holds for t . We close our consideration by showing that the geodesic flow (30) is complete, provided that the order of A is larger than 3. Corollary 6 Assume that s > 3=2. Then any solution .'; v/ of the geodesic equation (30) exists for ever as a smooth path on the bundle TDiffH 1 .R// and the corresponding Eulerian velocity u D v ı ' 1 belongs to the space C1 .R; C1 .R// and is a global solution of (32). Proof Since .'; v/ is a solution of the geodesic equation induced by the metric h; iA given in (28), the quantity Z hu.t/; u.t/iA D

Au.t/  u.t/ dx R

is constant on .t ; tC /. But hu.t/; u.t/iA D ku.t/k2H 2s and 2s > 3. Hence Sobolev’s embedding theorem implies that jux .t; x/j  Ck'kH 2s

for all .t; x/ 2 .t ; tC /  R:

In view of Theorem 12 the proof is completed.

t u

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References 1. Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319– 361 (1966) 2. Bauer, M., Escher, J., Kolev, B.: Local and global well-posedness of the fractional order EPDiff equation on Rd . J. Differ. Equ. 258, 2010–2053 (2015) 3. Bourgain, J.: On the Cauchy problem for periodic KdV-type equations. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. Special Issue, 17–86 (1995) 4. Boussinesq, J.: Théorie de l’intumescence liquide appelée onde solitaire ou de translation˙I se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris 72, 755–759 (1871) 5. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. (Singap.) 5, 1–27 (2007) 6. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007) 7. Byers, P.: Existence time for the Camassa–Holm equation and the critical Sobolev index. Indiana Univ. Math. J. 55, 941–954 (2006) 8. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) 9. Christ, M., Colliander, J., Tao, T.: Asymptotics, frequency modulation, and low regularity illposedness for canonical defocusing equations. Am. J. Math. 125, 1235–1293 (2003) 10. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on R and T. J. Am. Math. Soc. 16, 705–749 (2003) 11. Constantin, A., Escher, J.: Global existence and blow up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa XXVI, 303–328 (1998) 12. Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 47, 1527–1545 (1998) 13. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998) 14. Constantin, A., Escher, J.: Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. LI, 443–472 (1998) 15. Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003) 16. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000) 17. Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001) 18. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970) 19. Escher, J., Kolev, B.: Geodesic completeness for Sobolev H s -metrics on the diffeomorphisms group of the circle. J. Evol. Equ. 14, 949–968 (2014) 20. Escher, J., Kolev, B.: Right-invariant Sobolev metrics H s on the diffeomorphisms group of the circle. J. Geom. Mech. 6, 335–372 (2014) 21. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992) 22. Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics. Springer, Heidelberg (2008) 23. Guo, Z.: Global well-posedness of Korteweg–de Vries equation in H 3=4 .R/. J. Math. Pures Appl. 91, 583–597 (2009) 24. Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Everitt, W. (ed.) Spectral Theory and Differential Equations. Lecture Notes in Mathematics, vol. 448, pp. 25–70 (1975)

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25. Kenig, C.E., Ponce, G., Vega L.: The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993) 26. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. Ser. 5 39(240), 422–443 (1895) 27. Kouranbaeva, S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868 (1999) 28. Kriegl, A., Michor, P.W.: Regular infinite dimensional Lie groups. J. Lie Theory 7, 61–99 (1997) 29. Kruskal, M.D., Zabusky, N.J.: Exact invariants for a class of nonlinear wave equations. J. Math. Phys. 7, 1256–1267 (1966) 30. Kruskal, M.D., Miura, R.M., Gardner, C.S., Zabusky, N.J.: Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws. J. Math. Phys. 11, 952–960 (1970) 31. Lax, P.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968) 32. Lax, P.: Periodic solutions of the KdV equation. Commun. Pure Appl. Math. 28, 141–188 (1975) 33. Lenells, J.: Conservation laws of the Camassa–Holm equation. J. Phys. A 38, 869–880 (2005) 34. Phillips, R.S.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc. 90, 193–254 (1959) 35. Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46, 309–327 (2001) 36. Taylor, M.: Commutator estimates. Proc. Am. Math. Soc. 131, 1501–1507 (2003) 37. Whitham, G.B.: Non-linear dispersive waves. Proc. R. Soc. Ser. A 283, 238–261 (1965) 38. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974) 39. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)

Asymptotic Methods for Weakly Nonlinear and Other Water Waves Robin Stanley Johnson

Abstract The model and equations that describe the classical (inviscid) waterwave problem are introduced and some standard applications listed. The particular interests here are outlined, based on the relevant non-dimensionalisation and scalings; the resulting form of the equations is then interpreted in terms of the various parameters. The main development in this contribution makes use of the ideas and techniques of asymptotic (parameter) expansions, methods which are briefly described. This approach is then used to develop some important, approximate equations that are generated by the water-wave problem, many of which are of completely integrable type i.e. they are ‘soliton’ equations, for which an inverse scattering transform (IST) exists; these important ideas are briefly described. The corresponding problems of modulated waves are also mentioned. Further, the various models and approximations introduced here can be improved in order to represent, more accurately, physically realistic flows (but always in the absence of viscosity). To this end we provide, as an example, the problem of variable depth for weakly nonlinear, dispersive waves. A class of problems, of considerable practical and current mathematical interest, involves the inclusion of vorticity (i.e. the prescription of some background flow on which a wave propagates). In particular, we will present the corresponding problems associated with some of the classical examples introduced earlier, such as the Korteweg-de Vries and Camassa-Holm equations. We conclude this type of problem by discussing the propagation of ring waves on a flow with some prescribe flow (i.e. non-zero vorticity) in a given direction. We finish with an introduction to the way in which we can use asymptotic expansions to extract some details of solutions that exist only in a formal sense. As examples of this, we look at two rather special wave-propagation problems: periodic waves with vorticity and a novel approach to the problem of edge waves.

R.S. Johnson () School of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne, UK e-mail: [email protected] © Springer International Publishing Switzerland 2016 A. Constantin (ed.), Nonlinear Water Waves, Lecture Notes in Mathematics 2158, DOI 10.1007/978-3-319-31462-4_3

121

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R.S. Johnson

1 Introduction and Outline We first introduce the model and equations that describe the classical water-wave problem, and then list some of the applications that are possible. The particular scenario of interest here will be outlined, and the associated non-dimensionalisation and scalings developed. The resulting form of the equations will then be interpreted, and two classical problems mentioned: fully nonlinear long waves and the solitary wave. As an introduction to the main discussion, we first describe, in general terms, the ideas and techniques of asymptotic (parameter) expansions. We then use this approach to develop some important, approximate equations that are generated by the water-wave problem: Korteweg-de Vries (KdV), 2DKdV (or KadomtsevPetviashvili (KP)), concentric KdV (cKdV), a nearly-concentric KdV (ncKdV), Boussinesq, 2D Boussinesq and, finally, Camassa-Holm (CH). (The 2DKdV, 2D Boussinesq and ncKdV equations will be presented only in outline.) Many of these equations are of completely integrable type i.e. they are ‘soliton’ equations, for which an inverse scattering transform (IST) exists. These important ideas will be briefly described, by presenting some results for our standard soliton equations. (This is a very large and deep topic, the development of which is outside the scope of the material presented here, so the background and technical details will be omitted, although appropriate references will be provided.) We will then briefly describe the problem of modulated waves as they appear in water-wave propagation; this leads to the Nonlinear Schrödinger (NLS) equation, and a two-dimensional variant of this: the Davey-Stewartson (DS) equation. The IST theory for the NLS equation will be mentioned. With this basic material as a background, we then describe how these models can be improved in order to represent, more accurately, physically realistic flows (but always in the absence of viscosity, turbulence and mixing!). To this end, we outline the problem of variable depth (associated with the KdV equation), and indicate that this results in a quite complicated technical problem (which we will not pursue here). A more accessible class of problems, of considerable practical and current mathematical interest, involves the inclusion of vorticity (which can be thought of as the prescription of some background flow on which a wave propagates). In particular, we will present the corresponding problems associated with the KdV, CH and Boussinesq equations. We conclude this type of problem by discussing the propagation of ring waves on a flow with some prescribe flow (i.e. non-zero vorticity) in a given direction. To complete this introduction to the way in which we can use (formal) asymptotic expansions to extract some details of the flow, we look at two very different and rather special wave-propagation problems: periodic waves with vorticity (which has been of considerable interest, over the last decade or so, from the rigorous viewpoint) and a novel approach to the problem of edge waves. Much of this material appears in various papers and texts; we have therefore provided an extensive list of references for the interested reader.

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123

2 Governing Equations The problem, typically referred to as the classical problem of water waves, takes as its model an incompressible, inviscid fluid with zero surface tension. Further, the water moves over an impermeable bed (which we take here to be stationary), with a constant pressure—atmospheric pressure—at the free surface. The fundamental governing equations are then Euler’s equation and the equation of mass conservation: 1 Du D  rp C F; Dt 

ruD0

(1)

with p D pa D constant & w D

Dh Dt

on z D h.x? ; t/,

(2)

and w D .u?  r? /b

on z D b.x? /,

(3)

where F D .0; 0; g/; for constant g, and  is the constant density of water; D=Dt is the familiar material derivative. The velocity in the fluid has been written as u D .u? ; w/, with h D h.x? ; t/ (the free surface), where x? is the 2-vector perpendicular to the z-coordinate, with the associated velocity vector u? ; w is the component of the fluid velocity in the z-direction. It is useful, at this early stage, to retain the two-dimensionality of the surface (even though much of our analysis will be for one-dimensional plane waves) because we need this extra freedom in a couple of calculations that we present here. We shall seek a solution, constructed from a suitable function-set, in the domain z 2 Œb.x? /; h.x? ; t/

and

t  0;

with, for example, 1 < x < 1 (and no ydependence) or 1 < y < 1 and r. y; t/  x < 1 (e.g. for edge waves propagating in the ydirection). The domain in z describes a region whose upper boundary is the free surface, which we do not know a priori. This exemplifies one of the essential difficulties of the water-wave problem: the determination of the domain where the solution exists is part of the solution process. Although it would be natural to prescribe some general initial state, and then examine how this evolves in time (a very difficult problem), we take a different route here. Our main interest is in the form of governing equation which, at leading order (in a sense to be defined), describes a particular type of water wave. Thus our approach is to emphasise this equation, and how we derive it (and, of course, the

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assumptions underpinning it); then we may address the question of what type of initial data is required to generate solutions of this equation. The fundamental assumptions that we have described above reflect the physical reality of these types of fluid flow. The incompressibility of water is well known [66]: e.g. at 150 atmospheres, water is compressed by considerably less than 1 %. The effects of laminar viscosity are important only on length scales much greater than the wave lengths and propagation distances that we envisage; [22]. Further, for ocean waves that are not close to breaking, the effects of turbulent-mixing (and associated viscosity) are similarly unimportant [4]; thus we may reasonably ignore the rôle of viscosity altogether in our model. We also exclude from our model any wind action on the waves, either by the direct transfer of momentum by pressure changes, or by the shearing action of the air. On the other hand, surface tension is certainly important in the class of waves normally described as ‘capillary’, but these waves have wave lengths of only a few centimetres and our waves (in the ocean, for example) are typically of wave lengths 100–150 m; [4, 48, 66]. Our focus of attention will be on the generation and propagation of gravity waves, which arise by virtue of the balance between gravity and the inertia of the system: capillary waves will be absent from our model. At this point, we can begin to appreciate the various types of problem that could now be examined, each being some appropriate solution of this system but satisfying suitable boundary and initial conditions. Some of the classical problems that we will not address here include: ship-wave pattern, sloshing in tanks, hydraulic jump and bore, short-crested waves, tsunamis, storm surges, Kelvin-Helmholtz instability, Rayleigh-Taylor instability, propagation of capillary waves, general energy and momentum flux in wave propagation and, of course, the effects of viscosity on suitably long scales. Far more background to the theory of many types of waterwave propagation can be found in [21, 24, 25, 48, 72, 78]. The next stage in any problem of this type—or, indeed, in a systematic approach to problem solving in applied mathematics—is to non-dimensionalise the set (1)– (3) by introducing suitable general scales that describe the class of problems under consideration. Let h0 be the depth of the water in the absence of waves for some fixed (perhaps average) level bottom, and  an average or p typical wavelength of the wave (see Fig.p1); an associated speed scale is then gh0 , with a corresponding time scale = gh0 . This speed scale, however, is used only for the horizontal velocity components (u? D .u; v/); in order to be consistent with the equation of mass conservation (and so, equivalently, consistent with the existence of a stream function), p the vertical component of the velocity (w) is non-dimensionalised by using h0 gh0 =. Thus we non-dimensionalise according to the transformation x? ! x? ; u? !

p gh0 u? ;

z ! h0 z;

p t ! .= gh0 /t;

p w ! .h0 gh0 =/w;

(4) b ! h0 b;

Asymptotic Methods for Weakly Nonlinear and Other Water Waves Fig. 1 The coordinates and typical scales

z

125

λ

ho a g

y b

x

where ‘!’ is to be read as ‘replace by’ (so that, for convenience, the current notation is retained, but now all the variables are non-dimensional versions of those introduced earlier). Further, we also introduce h D h0 C a

p ! pa C g.h0  z/ C gh0 p;

and

(5)

where a is a measure of the wave amplitude and the new p is a non-dimensional pressure that measures the deviation away from the hydrostatic pressure distribution. (Note that, in the transformation for p; the original, dimensional z is used.) We now use (4), (5) in Eqs. (1)–(3) to give Du D r? p; Dt

ı2

@p Dw D ; Dt @z

ruD0

(6)

on z D 1 C ".x?; t/,

(7)

with p D " &

wD"

n @ @t

o C .r?  u? /

and w D .u?  r? /b

on z D b.x? /.

(8)

Here, we have introduced the two familiar, and fundamental, parameters that characterise the classical water-wave problem: " D a=h0 , the amplitude parameter, and ı D h0 =, the long wavelength (or shallowness) parameter. The final stage in the construction of an appropriate form of these equations is to scale with respect to ". The case " D 0 recovers undisturbed conditions in the absence of any waves; indeed, as " ! 0, so the disturbance associated with the propagating wave vanishes. Although there are many problems for which we may not wish to take this limit— we could elect to examine the “fully nonlinear’ problem—the equations must be

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consistent with this choice. Thus we further redefine our variables according to the additional transformation .u? ; w; p/ ! ".u? ; w; p/

(9)

when the underlying flow-configuration is that of stationary water; we will allow for an existing background vorticity later. The final form of our governing equations, at this stage, therefore becomes Du? D r? p; Dt

ı2

@p Dw D ; Dt @z

ruD0

(10)

with pD &

wD

@ C ".r?  u? / @t

on z D 1 C ",

(11)

and w D .u?  r? /b

on z D b,

(12)

where @ D D C ".u  r/: Dt @t This version of the problem allows for the identification of various problems of practical interest, accessed by making suitable choices of the two parameters: neglecting either one or the other, for example, or choosing some special relation between them. Thus, to be specific, we have two standard approximations to these equations: 1. 2.

" ! 0 (ı fixed): the linearised problem; ı ! 0 (" fixed): the long-wave or shallow-water problem.

The first case clearly recovers the most general linear problem, whereas the second is fully nonlinear, but the pressure correction (due to the passage of the wave) is missing: there is no dispersion in this case. We see how these simple approximations can be used, and expanded, in some of what we do later. In the case that the flow is strictly irrotational (so ! D r  u D 0 i.e. u D r), then an alternative formulation of this problem is available (and sometimes this is more natural than the version above, which is based on the Euler equation). However, much of the work here will be developed from the Euler equation, even if the flow is irrotational, so that direct comparison with flows with vorticity can readily be made. For irrotational flow, we have 2 zz C ı 2 r? D0

(13)

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with 8 2 ˆ < z D ı ft C "..r? /  r? /g; "n 1 o ˆ : t C  C .r? /2 C z2 D 0 2 ı2

on z D 1 C ",

(14)

and z D ı 2 ..r? /  r? /b

on z D b.

(15)

These equations have been non-dimensionalised and scaled exactly in accordance with Eqs. (10)–(12); the second equation in (14) is the pressure equation (sometimes referred to as the ‘unsteady Bernoulli’s equation’) evaluated at the free surface, with constant atmospheric pressure imposed. In these equations, we have introduced subscripts to denote partial derivatives. For some applications, it is relevant (and useful) to remove the explicit dependence on ı; this is accomplished by performing one more transformation: ı x ? D p X? ; "

ı t D p T; "

ı wD p W "

(16)

which produces the set (10)–(12) with ı 2 replaced by ", for arbitrary ı. Note that, because the terms in ı 2 (now ") are associated with the dispersive property of the wave, " ! 0 will describe weakly nonlinear, weakly dispersive waves. (This choice of variables is equivalent to using the single scale length h0 , rather than both h0 and .) In this case, we consider only " ! 0, for any ı, and so we can no longer access the problem of ı ! 0 (" fixed) for which we must revert to the earlier form of the governing equations. Nevertheless, this alternative version of the equations is useful in a description of, for example, the classical derivation of the Korteweg-de Vries equation; see Sect. 5.1. This, for many examples, will be coupled with the restriction to one-dimensional (plane wave) propagation.

3 Two Classical Problems in the Theory of Water Waves There are many problems, of both practical and mathematical interest, that are generated by our governing equations (even though we have chosen to use the simplest model for the fluid and its surface boundary conditions). Although the main thrust here is to present the techniques and results associated with special (e.g. ‘soliton’) equations, we start with two different types of problem that, perhaps, may exemplify the broad sweep of the subject.

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3.1 Nonlinear, Long Waves This problem is based on the long-wave (ı ! 0) version of the equations, for waves propagating in only one dimension; thus Eqs. (10)–(12) become ut C ".uux C wuz / D px ;

pz D 0;

ux C wz D 0

(17)

with pD &

w D t C "ux

on z D 1 C ",

(18)

and wD0

on z D 0.

(19)

Here, we have written u? D .u; 0/, suppressed any dependence on y (so the wave is propagating in the x-direction:  D .x; t/) and we have taken the bottom to be simply z D 0 (i.e. b.x/  0). The essential character of the long wave is now evident: the second equation in (17) gives, directly, that p D p.x; t/ and then the relevant surface boundary condition in (18) requires p D .x; t/ (0  z  1 C "). Thus the variation of pressure with depth in the fluid is provided solely by the hydrostatic pressure distribution. The equations then reduce to ut C ".uux C wuz / D x ;

ux C wz D 0;

(20)

with w D t C "ux

on z D 1 C "

and

w D 0 on z D 0.

(21)

These equations admit a solution for which u D u.x; t/ (which is the only solution if, somewhere, u is independent of z, for then it will remain so); this is simply stating that the flow is irrotational, which is the additional simplification that we now invoke. (The vorticity is—here in the y-direction—equal to .uz  wx /, which is zero if, somewhere, there is the uniform flow u D u.x; t/ and w D 0.) Thus, from (20), we see that wz .D ux / is not a function of z, and so the solution for w which satisfies the boundary conditions is wD

t C "ux zI 1 C "

the two equations in (20) therefore become ut C "uux C x D 0;

.1 C "/ux C t C "ux D 0:

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There is no assumption here about the size of ": we have retained ‘full’ nonlinearity, so we could set " D 1 (and then we are using simply h0 as the length scale i.e. no ‘a’); indeed, we may also simply write the surface as 1 C ".x; t/ D h.x; t/ to obtain ut C "uux C hx D 0;

ht C .uh/x D 0:

(22)

This pair of equations, (22), can be solved by p the method of characteristics; to accomplish this, we first introduce c.x; t/ D h (and we certainly have h  0, because this is the total depth of the water). Then c is the speed of propagation (in nondimensional variables) of infinitesimal waves in water of depth h. The two equations now become ut C"uux C2ccx D 0 and 2cct Cc2 ux C2uccx D 0

or .2c/t Cu.2c/x Ccux D 0I (23)

these are added to give .u C 2c/t C u.u C 2c/x C 2ccx C cux D 0; and subtracted to give .u  2c/t C u.u  2c/x C 2ccx  cux D 0: These, in turn, can be written as n@ @t

C .u C c/

n@ @o @o .u C 2c/ D 0 and C .u  c/ .u  2c/ D 0; @x @t @x

and so u C 2c D constant on lines

dx D u C c; dt

(24)

u  2c D constant on lines

dx D u  c: dt

(25)

and

This provides the basis for the construction of the solution using the characteristic lines and Riemann invariants; see Stoker [20, 35, 78].

3.2 The Solitary Wave The solitary wave, as it appears in water waves, has a long and illustrious history, going back to J. Scott Russell’s report [75]; see, for example, Johnson [48]. We will

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be presenting some aspects of the solitary wave of small amplitude later (because it plays an important role in soliton theory and the KdV equation). Here, we make a few observations about some more general properties of this wave, not restricted to small amplitude. This wave, in its simplest manifestation, propagates in irrotational flow (and we will consider the case of the wave propagating into stationary water), so we will start with Eqs. (13)–(15): 2 zz C ı 2 r? D0

with z D ı 2 ft C ".u?  r? /g;

t C  C

"n 1 o .r? /2 C 2 z2 D 0 on z D 1 C ", 2 ı

and z D 0

on z D 0,

where b.x/  0. This set admits solutions that represent (one-dimensional) waves of elevation that propagate with unchanging form (shape)—usually called a travellingwave solution—which decay exponentially away from the peak. There is also a limit on the amplitude of the wave; see Stokes [80], Varvaruca [82], Varvaruca and Weiss [83]. The small-amplitude approximation is accessed by imposing " ! 0, but we will not restrict ourselves to this choice, so we set " D 1; ı will be arbitrary. A number of general properties of the solitary wave, and its associated flow field, can be defined; we consider a wave of permanent form, propagating at a speed c, and so introduce  D x  ct, with  D .; z/ and  D ./, then zz C ı 2   D 0

(26)

with z D ı 2 .  c/

&

 c C  C

1 n 2 1 2o  C  D 0 on z D 1 C , 2  ı2 z

(27)

and z D 0

on z D 0.

(28)

Then we define: Z total mass

1

M WD 1

 d;

total momentum (or impulse)

Z

1

Z

I WD 1

0

1C

 dz d;

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

T WD

total kinetic energy

1 2

Z

1

C WD

circulation

1

Z

1

1C

0

V WD

potential energy of the wave Z

1



Z

1 2

2 C

1 1

131

1 2  dz dI ı2 z

2 dI

u  ds D Œ1 1 I

where this last integral is taken along any streamline. As an example of the identities that exist between these various properties of the solitary wave, it is convenient to start from the equation of mass conservation (see (10)), written in the moving frame: .u  c/ C wz D 0: Then integration in z, with the relevant boundary conditions imposed, gives d n d

Z

1C 0

o .u  c/ dz D 0

and so Z

1C 0

Z .u  c/ dz D constant D

1 0

.c/ dz D c

since both u D  and  tend to zero as jxj ! 1. Thus Z

1C

 Z u dz D

1C

0

0

1C

Z

  dz D c

and then Z

1 1

Z 0

 dz d D c

1 1

 d

i.e. I D cM.

This identity was first obtained by Starr [77]; another identity is 2T D c.I  C/; which was first derived by McCowan [71]. In 1974, Longuet-Higgins found 3V D .c2  1/M: Other results that apply to the solitary wave, up to the largest wave, together with applications to numerical and other approximations, can be found in [67–70].

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4 Asymptotic Expansions (With a Parameter) Our primary interest here is in the representation, as a suitable expansion, of functions that involve a parameter ("), as that parameter approaches zero. (This will prepare the way for our application of these ideas to the construction of asymptotic solutions of various water-wave problems.) The function will otherwise depend on a variable (which can be a vector); in this introduction to the ideas, we consider the real, scalar function f .xI "/, where x is also a real scalar. The parameter here is " (real) and we shall always define it so that " > 0 W the limit process is always therefore " ! 0C i.e. " & 0 (even if we often write simply " ! 0). The function is defined on some domain, x 2 D, and then the critical question is this: is an expansion of f .xI "/, as " ! 0 at fixed x, appropriate (valid) for 8x 2 D? First, the expansion of a function, in the asymptotic sense, is constructed with respect to an asymptotic sequence, usually written fn ."/g; n D 0; 1; 2; : : :. The elements of this sequence are defined so that lim ŒnC1 ."/=n ."/ D 0

"!0

for every n D 0; 1; 2; : : : ;

usually expressed as nC1 ."/ D oŒn ."/

for " ! 0

where we have used the Landau symbol. Then, for a sequence appropriate to a given function, we write f .xI "/ D

N X

an .x/n ."/ C OŒNC1 ."/

nD0

as " ! 0, at fixed x; for suitable coefficients an .x/. (We have used the other Landau symbol here.) This statement is often expressed as f .xI "/

N X

an .x/n ."/

for " ! 0;

nD0

and read as ‘asymptotically equal to’ or ‘varies as’; we must allow, at least in principle, that N ! 1 (although we are rarely able to construct all the terms explicitly; nevertheless, their asymptotic behaviours are usually available, confirming a breakdown or otherwise; see below). As implied above, fn ."/g cannot be arbitrarily assigned; for example, the p sequence f"n g is appropriate for the function 1 C x C " but not p for exp.x="/. On the other hand, fn ."/g is not unique to a given function: for 1 C x C " we could also use fsinn ."/g or ftann ."/g or even fln.1 C "n /g, but we would normally choose the simplest of various (essentially equivalent) sequences.

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To proceed, we suppose that f .xI "/ is defined for x 2 D; we construct the asymptotic expansion of f with respect to the asymptotic sequencefn."/g at fixed x; so that, for every N  0, f .xI "/

N X

an .x/n ."/:

nD0

Question: is this asymptotic representation, for 8N, also valid for 8x 2 D‹ First, we address the notion of validity in this context. The asymptotic expansion is said to be valid—we normally refer to this as ‘uniformly’ valid—if anC1 .x/nC1 ."/ D oŒan .x/n ."/ for every n D 0; 1; 2; : : : and 8x 2 D, as " ! 0. Thus the function p 1 C x C ";

f .xI "/ D

x0

has an asymptotic expansion which starts f .xI "/

p " 1CxC p I 2 1Cx

it is easily checked—here we can construct the terms to all orders—that this asymptotic expansion is uniformly valid on the given domain. Note, however, that this would not be the case if the domain were given as x  1I indeed, the example f .xI "/ D

p x2 C x C ";

x0

has the asymptotic expansion which begins f .xI "/

p " x2 C x C p 2 x2 C x

(29)

which is not uniformly valid as x ! 0C (which is in the given domain). An asymptotic expansion which is not uniformly valid is said to ‘break down’, this occurring for a particular size (or, possibly, sizes) of x (in the domain); this breakdown is determined by any xs that satisfy anC1 .x/nC1 ."/ D OŒan .x/n ."/ for any n, as " ! 0. In the preceding example, (29), these first two terms in the expansion show a breakdown where x D O."/ (and it is a straightforward exercise to confirm that this is the only breakdown of the complete asymptotic expansion on the given domain). The procedure is to define a new x, of the given size, and repeat

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R.S. Johnson

the construction of an asymptotic expansion for the new x held fixed as " ! 0. So here, let us set x D "X (a ‘scaled’ x), to give f .xI "/ D f ."XI "/  F.XI "/ D

p "2 X 2 C ".1 C X/;

and then we have the corresponding asymptotic expansion which starts F.XI "/

p np "X 2 o " 1CXC p 2 1CX

(30)

as " ! 0 for X fixed. (It is readily checked that this new asymptotic expansion is valid as X ! 0C , but not as X ! 1—and then the associated breakdown simply takes us back to the requirement to use the x variable.) One final comment: the two expansions that we have generated, of the same function, namely (29) and (30), are said to ‘match’. That is, an appropriate further expansion of (29), and correspondingly of (30), produce two expressions that are identically the same. We will not spell out the rules here—further details can be found in the references mentioned below—but we will present the relevant calculation for this example. We have p p " " D "2 X 2 C "X C p x2 C x C p 2 2 2 x Cx 2 " X 2 C "X n 1 p p 1 1 p o " X C p C " X 3=2  X 2 4 2 X

(31)

p p retaining terms O. "/ and O." "/; and o p np "X 2 o p np x2 =" " 1CXC p D " 1 C x=" C p 2 1CX 2 1 C x="    p 1 " 1 1" C x3=2 1  ; x 1C 2x 2 2x

(32)

retaining terms O.1/ and O."/. The final expressions in (31) and (32) are identical, when written in the same variable; this is called matching (and this property, which also involves the careful designation of the terms to be retained, is called the matching principle). A problem that requires more than one (matched) asymptotic expansion, to cover the given domain, is called a singular perturbation problem. Before we proceed to use these ideas in our studies of the water-wave problem, we make an important observation (which is at the heart of modern singular perturbation theory): we normally do not have a function to expand! Rather, we aim to seek a function which is the solution of a partial (or ordinary) differential equation, or a set of such equations. Thus we need to adjust the approach slightly: we first assume a suitable asymptotic sequence and associated expansion, consistent with the form of the governing equation(s) and boundary conditions. This generates

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a sequence of problems, obtained at each order, which are then solved sequentially. Once we have an asymptotic solution constructed in this way, valid in some domain, we may examine its validity: is it valid throughout the domain of interest? Is there a breakdown requiring a rescaling of the variable(s)? (The interested reader may wish to explore these ideas further; a selection of relevant texts is: [19, 41, 53, 60, 76, 81].) Of course, this approach is based on a ‘formal’ procedure: it simply follows a well-defined path, guided by clear principles. It does not incorporate any element of rigour, in the sense of proving uniqueness—typically, rather unimportant in this context because solutions are usually known to be unique—and existence. This latter point can be viewed on two levels. On the one hand, we need to be satisfied that a solution exists within a certain class of functions. This is not quite so critical here, because the asymptotic development almost always leads to the specification of the functions that are allowed, and readily identifies failures (which may then, if appropriate, be examined via scaling and matching). On the other hand, there is a far more significant issue relating to the nature of the asymptotic expansion itself: does it exist in the sense of constituting a convergent series with a non-zero radius of convergence? Sadly, the answer is very often ‘no’ in the context of classical singular perturbation theory. Asymptotic expansions, as described above, are, in general, strictly only asymptotic: they may well diverge when viewed conventionally. One of the aims of a rigorous approach is to prove that, for some "0 > 0; the expansion converges for 0  " < "0 (at least for a certain class of functions). This can sometimes be accomplished, but typically with other constraints e.g. in wave-propagation problems—particularly relevant here—convergence often requires  < 0 for some 0 fixed as " ! 0, where  is a scaled time (e.g.  D "), and for suitably smooth initial data. (That such a solution is not valid for  ! 1 is not normally regarded as a significant difficulty in any use of these ideas to model physical reality. It is understood that such a problem, in all likelihood, has been generated from physical considerations, and has already ignored some physical properties, e.g. viscosity, which become important, typically on timescales far longer than O."1 / W we do not expect our asymptotic solution to be appropriate for  ! 1.) Without the bonus of convergence, our asymptotic expansions are, therefore, no more than formal representations of the solution, expressed as divergent series—but this is not quite as worrying as it might appear. The fundamental difference between divergent and convergent series is simply stated: the error in using a convergent series can be made vanishingly small by increasing the number of terms indefinitely. In the case of a divergent series, for a given x and " > 0 (where we use our notation introduced above), there is a choice of N (the number of terms) which minimises the error; this is the best that we can accomplish. Nevertheless, even if only numerical estimates are required, this still gives a very useful (and mathematically robust) result; the real strength of this approach is that the detailed mathematical (i.e. algebraic/functional) structure of the solution is obtained by these techniques. (For those who wish to explore these ideas further, a very illuminating discussion of the relevance and use of divergent series can be found in [37]; see also Ford [32].)

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R.S. Johnson

We submit, therefore, that the asymptotic approach can certainly contribute to a clearer understanding of complicated phenomena—but, of course, it can never compete with rigorous theories. It may be the spring-board for a rigorous development, or it may attempt to fill-in some of the detail not accessible from the rigorous approach. A broad view will hold that the rigorous and asymptotic approaches should go forward hand-in-hand. With these observations in mind, we now use the method of asymptotic expansions to investigate a number of important and illuminating problems that are generated by the classical water-wave equations.

5 Weakly Nonlinear, Weakly Dispersive Waves As an introduction to our discussion, and to lay the foundations for much that we shall present here, we will begin by deriving the Korteweg-de Vries (KdV) equation as it appears in water waves. This will provide the background to the derivation of related equations, the majority of which turn out to be integrable equations (in the soliton or inverse scattering sense; see Chap. 6). These equations represent asymptotic approximations to the water-wave problem which exhibit some appropriate balance between nonlinearity and dispersion. Although many of these equations are useful in the description of important properties of water waves, the emphasis here will be on the method of derivation (and we should also note that we encounter a surprising number of interesting equations that are generated by this one problem).

5.1 The Korteweg-de Vries (KdV) Equation This is arguably the most important equation of this type in this context. It was the first to be obtained (originally by Korteweg and de Vries [65])—although we will give a more modern derivation of it—and it can be regarded as the archetype for the study of soliton problems. We choose to start with Eqs. (10)–(12), with b.x/  0 and the transformation (16) incorporated, and we consider only plane waves propagating in the X-direction, where x? D .X; Y/ and u? D .u; 0/I there will be no dependence on Y. (We use the Euler equation even though we will assume irrotationality here, because we will then be able to compare this with the calculations with vorticity (discussed later).) So we introduce  D X  T;  D "T

(33)

which describes a far field for right-running waves. [The corresponding near field arises with the choice  D X  T and T [i.e. T D O.1/; not T D O."1 / as used in (33)]; this generates a perturbed linear problem, which produces an asymptotic

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

137

solution not uniformly valid for T D O."1 /: typically, the second term includes a term "T; we have therefore gone directly to the far field. In these types of wave propagation problems, it is usual to find that the far field is uniformly valid as  ! 0. The development of the wave as it moves into the far field can be interpreted as the growth of weak nonlinearity and weak dispersion, so that it eventually becomes important at leading order. The return back to the near field merely reduces this effect, making uniformity in this direction possible.] With this choice of variables, the governing equations become u C ".ut C uu C Wuz / D p I "fW C ".Wt C uW C WWz /g D pz I

(34)

u C Wz D 0; with pD

& W D  C ".t C u /

on z D 1 C "

(35)

and W D0

on z D 0:

(36)

In these equations, we have again introduced subscripts to denote partial derivatives. We seek a solution of this set for z 2 Œ0; 1 C "; 1 <  < 1 and   0; and for suitable initial data. The procedure now is to assume a formal asymptotic solution of the system (34)– (36), expressed as .; I "/

1 X

"n n .; /

and

q.; ; zI "/

nD0

1 X

"n qn .; ; z/;

(37)

nD0

where q (and correspondingly qn ) represents each of u; W and p. (The structure of the equations, and the result of a simple iterative procedure, both suggest this form of asymptotic expansion.) These asymptotic expansions, defined for " ! 0 are, typically, uniformly valid for 0   < 0 ; for some fixed 0 ; provided that the initial data decays sufficiently rapidly as jj ! 1. Indeed, matching to the near field (where T D O.1/; and containing T D 0/ shows that the problem in the far field must satisfy the initial data prescribed at T D 0I this will take the form of some given .X; 0/; and then this function will have to decay sufficiently rapidly at infinity in order to ensure the existence of solutions of the KdV equation. Thus this far-field solution will be uniformly valid back into the near field. (Periodic solutions require a more careful treatment.) The procedure is altogether routine; at leading order we obtain p0 D 0 ;

u0 D 0 ;

W0 D z0

(all for 0  z  1/;

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R.S. Johnson

where we have imposed the condition that the perturbation in u is caused only by the passage of the wave i.e. u0 D 0 whenever 0 D 0. Because the boundary conditions are automatically satisfied, the leading-order contribution to the surface wave .0 / is arbitrary—we must go to the next order to determine it. Evaluation on the free surface is, at this order, simply z D 1I to proceed, we define all the functions on z 2 Œ0; 1 and so we invoke Taylor expansions (of the surface boundary conditions) about z D 1 (which is equivalent to mapping the domain z 2 Œ0; 1C" to z 2 Œ0; 1/. This is a valid manoeuvre here because the solution, at all orders, turns out to be polynomial in z; which ensures that the asymptotic solution is certainly uniformly valid in z. (Indeed, the construction of a power-series solution in z was the basic structure invoked by Korteweg and de Vries in 1895 [65].) Thus the two surface boundary conditions, (35), are written as )

p0 C "0 p0z C "p1 D 0 C "1 C O."2 / W0 C "0 W0z C "W1 D 0  "1 C ".0 C u0 0 / C O."2 /

on z D 1

together with, at O."/; u1 C u0 C u0 u0 C W0 u0z D p1 I

p1z D W0 I

u1 C W1z D 0:

These equations and boundary conditions then give, for example, o n 1 p 0 C " 1 C .1  z2 /0  2 and o n   1 1 W z0 C "  1 C 0 C 0 0 C 0   z C z3 0   ; 2 6 both defined for 0  z  1. In order to satisfy the surface kinematic boundary condition, the function 0 .; / satisfies the Korteweg-de Vries (KdV) equation 1 20 C 30  C 0   D 0; 3

(38)

leaving the next approximation to the free surface, 1 , undetermined at this order; the equation for this function is found by going to the next order. A corresponding result holds for left-going waves; this involves using the characteristic XCT with the sign of the  derivative reversed. In both cases, the higher-order terms are described by an equation of the form 1 2n C 3.0 n / C n   D Fn ; 3

n  1;

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

139

Table 1 Table showing the distances for the KdV balance, for various depths and amplitudes Amplitude Depth 5 m Depth 10 m Depth 100 m Depth 1 km Depth 4 km

0.1 m 1.77 km 10 km 3162 km 106 km 3.2  107 km

0.2 m 0.16 km 0.89 km 283 km 8.9  104 km 2.8  107 km

1m 56 m 0.32 km 100 km 3.2  104 km 106 km

5m N/A 28 m 8.9 km 2828 km 9  104 km

10 m N/A N/A 3.2 km 1000 km 3.2 104 km

for right-running waves, for example, where Fn is a forcing term containing, in general, contributions from 0 ; 1 ; : : : ; n1 . The KdV equation provides the basis for a discussion of a class of water waves, which includes the solitary wave and soliton interactions; the evidence (see, for example, Constantin and Johnson [14]) is that, on suitable distance and time scales, this is a good model for nonlinear, dispersive water waves. Indeed, the scales on which the KdV equation might be relevant is of some interest (if only because there has been some debate over the relevance of KdV, and its soliton solutions, to the formation and development of tsunami waves). The distance scale (and corresponding time scale) which dictates where the KdV equation is appropriate is given by X D O."1 / i.e.  D O.ı="3=2 / [which is obtained from (16) and (33)]; this produces a length scale measured by 

ı "3=2

D h0

 a 3=2 h0

.a < h0 /;

which depends only on the (average) amplitude and (average) depth. This can be used to estimate the distance at which the KdV balance occurs, for various depths and amplitudes (Table 1). We observe that, for moderate amplitudes in a depth typical of rivers, the balance occurs over a few kilometres which makes it possible (as J. Scott Russell did in 1834) to follow a solitary wave on horseback; see Russell [75]. On the other hand, the 2004 Boxing Day tsunami (for which the initial a was barely 1 m and the local ocean depth 4 km) requires distances of about one million kilometres! This tsunami was not of soliton/KdV origin; see Constantin and Johnson [15].

5.2 The Two-Dimensional Korteweg-de Vries (2D KdV) Equation The KdV equation obtained in the previous section describes nonlinear plane waves propagating in the X-direction (on stationary water). We now pose the question: how might this equation be modified if the propagation is on the two-dimensional surface? A simple example would be the oblique interaction of two waves that, at

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R.S. Johnson

infinity, were plane but not parallel. However, we aim to derive an equation that is essentially a KdV-type equation and, hopefully, also completely integrable; this requires the additional dependence (on Y) to be special i.e. appropriately ‘weak’. One way to see what is involved is to formulate, initially, the near-field, linear, longwave equation for this problem: TT  .XX C YY / D 0;

(39)

to leading order as " ! 0I this is the classical, two-dimensional wave equation. A solution of this equation is  D ei.kXClY!T/

where ! 2 D k2 C l2 ;

and we consider l=k small:  1 l2  ! k 1C 2 k2 as l=k ! 0; this then gives the dispersion relation for waves that are propagating predominantly in the X-direction. In order that the contribution from the behaviour in the Y-direction is the same size as the nonlinearity p and dispersion already this is more present in the KdV equation, we must choose l D O. "/. However, p conveniently accommodated by rescaling according to  D "Y and then, for consistency with the equation of mass conservation p (i.e. for the existence of a suitable stream function) we also require v D "V (where u? D .u; v//. The governing equations, again written in the far-field variables  and , then become u C ".u C uu C "Vu C Wuz / D p I V C ".V C uV C "VV C WVz / D p I "fW C ".W C uW C "VW C WWz /g D pz I u C "V C Wz D 0; with pD

& W D  C ". C u C "V /

and WD0

on z D 0:

on z D 1 C "

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

141

The asymptotic expansion, and general procedure, follow exactly as for the KdV derivation, producing now, for example, o n   1 1 W z0 C "  1 C 0 C 0 0 C 0   C V0 z C z3 0   2 6 and then finally the equation for the leading-order term representing the wave is given as:   1 20 C 30  C 0   C 0  D 0:  3

(40)

This equation, (40), is the two-dimensional KdV equation (sometimes referred to as the Kadomtsev-Petviashvili (KP) equation; see Kadomtsev and Petviashvili [59]), and it is another completely integrable equation (see Chap. 6). It turns out that there are two variants of this equation, both of some interest, defined by ˙0  I the equation with our sign—the case applicable to water waves—is usually designated KPII. It is instructive to give an interpretation of the scalings that we have used in the derivation of the 2D KdV equation, (40). This equation is an appropriate leadingorder approximation, valid at times  D O.1/ (so the original non-dimensional, scaled time, T, is large: O."1 // and where  D O.1/ (which is a measure of p the width of the wave). However, the solution is also expressed in terms of  D "Y; which can be regardedp as a ‘weak’ dependence on Y. Thus, along any wave front, we have dY=dX D O. "/ W in the original, non-dimensional p variables, .X; Y/ for some  D O.1/; the wave front deviates only a little .O. "// away from a plane wave (defined by lines  D constant ). So, for example, in the case of two obliquely crossing waves—an exact solution of the 2D KdV equation (see Chap. 6)—the angle between the waves in .X; Y/ or .; Y/ coordinates is small: the two waves are nearly parallel in the original physical frame (and they propagate in essentially the same direction: head-on collisions are not allowed; see Sect. 5.5). Furthermore, the construction of the asymptotic solution for oblique interactions of two waves at arbitrary angles, with weak nonlinearity and weak dispersion, exhibits a nonuniformity as the waves become nearly parallel; see Miles [73]. Incorporating the scaling associated with this breakdown recovers, at leading order, precisely the 2D KdV equation.

5.3 The Concentric (or Cylindrical) Korteweg-de Vries (cKdV) Equation We have derived the versions of the Korteweg-de Vries, in both one and two dimensions, as described in Cartesian geometry; we now demonstrate that a corresponding equation exists in cylindrical coordinates. The same general approach

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is possible, but some elements of the formulation are different in important details. To see one of the main differences, it is useful first to consider the underlying linear problem for axi-symmetric long waves in this geometry:  1  tt  rr C r D 0; r

(41)

cf. Eq. (39). Here, r is the radial coordinate, non-dimensionalised with respect to the wave length ./ and, at this stage, without the additional scaling described in (16). The solution of (41), for outward propagation and at large radius, is 1  p f .r  t/ r

(as r ! 1 , for r  t D O.1/ )I

this shows the expected geometrical decay of the amplitude of the wave, as the radius increases, and this will need to be incorporated in any scaling that we adopt. The relevant governing equations, written in cylindrical coordinates, but with circular symmetry, are ut C ".uur C wuz / D pr I

ı 2 fwt C ".uwr C wwz /g D pz I

1 ur C u C wz D 0; r

with pD

&

w D t C "ur

on z D 1 C "

and wD0

on z D 0

cf. (10)–(12). There is a scaling, equivalent to (16), that replaces the two parameters, " and ı; by a single parameter; we introduce D

"2 .r  t/ ı2

and

RD

"6 r ı4

and define D

"3 H; ı2

pD

"3 P; ı2

uD

"3 U; ı2

wD

"5 W; ı4

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143

which describes outward propagation at a suitable radius. The governing equations then become U C .UU C WUz C UUR / D .P C PR /I

fW C .UW C WWz C UWR /g D Pz I  1  U C Wz C UR C U D 0; R with PDH

& W D H C D.UH C UHR /

on z D 1 C H

and WD0

on z D 0,

where D e4 =ı 2 is the single parameter that appears in this system. Thus the problem associated with ı ! 0 can be interpreted as, for example, " ! 0 at fixed ı (and the amplitude parameter is now defined as that appropriate to the region where r D O.1/ and t D O.1/); the amplitude in the far field is . Indeed, with  D O.1/ (the neighbourhood of the wave front) and R D O.1/; where R D ."6 =ı 3 /.r=ı/ D D3=2 .r=ı/; then ! 0 implies r=ı ! 1 W this scaling corresponds, for fixed ı, to large radius. The procedure is exactly as in the two previous derivations; an asymptotic solution is sought, as presented in (37), but now the expansion parameter is . At leading order, we obtain P 0 D H0 ;

U0 D H0 ;

W0 D zH0

.0  z  1/;

for arbitrary H0 .; R/I at the next order, we find, for example,  1  W1z D U1  U0R C U0 R

(42)

which leads to the equation for H0 .; R/ W 2H0R C

1 1 H0 C 3H0 H0 C H0   D 0; R 3

(43)

which should be compared with the classical KdV equation, (38). This is the concentric KdV equation—another completely integrable equation (even if the variable coefficient might suggest otherwise!).

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5.4 The Nearly-Concentric Korteweg-de Vries (ncKdV) Equation We have seen how the classical one-dimensional KdV equation (Sect. 5.1) has a twodimensional counterpart (with weak y-dependence): the 2D KdV equation (Sect. 5.2). The same general property arises with the cKdV equation; a large radius wave—the far field—can have, in addition, a weak dependence in the -direction (when written in cylindrical coordinates). The wave equation for linear, long waves, in cylindrical geometry, is   1 1 tt  rr C r C 2   D 0I r r the far field, with a suitable weak -dependence, requires the introduction of the additional scaled variable defined by  D ."2 =ı/, together with v D ."5 =ı 3 /V (where u? D .u; v/ in cylindrical polar coordinates). The procedure is exactly as described for the cKdV equation (and the leading order is identical) but now we have, for example, W1z D U1  U0R 

1 .U0 C V0 / R

cf. Eq. (42). The ncKdV equation (sometimes called the ‘Johnson equation’) appears at this order:   1 1 1 2H0R C H0 C 3H0 H0 C H0   C 2 H0 D 0;  R 3 R

(44)

but this is not a completely integrable equation for general initial data; see Sect. 5.8.

5.5 The Boussinesq Equation The examples discussed so far describe propagation in one direction only—either to the right (or left) or outwards. We now consider the possibility of a model that allows waves to propagate to the left and to the right, both being solutions of the same equation (which, of course, is the situation pertaining to the original, governing equations). This will allow waves to collide head-on. We shall find, however, that allowing both together—so we do not follow either one or the other (cf. Sect. 5.1)— implies that the equation we require must allow propagation predominantly in either direction, which is possible only with the weak nonlinearity and weak dispersion appearing as small corrections.

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We start with our standard equations for one-dimensional propagation, (10)–(12) with (16) incorporated: uT C ".uuX C Wuz / D pX I

"fWT C ".uWX C WWz /g D pz I

uX C Wz D 0;

with pD &

W D T C "uX

on z D 1 C "

and W D0

on z D 0:

We seek a solution in the standard form, (37), and then, at leading order, we find that p0 D 0 I

u0T D 0X I

W0 D zu0X I

u0X D 0T

(all for 0  z  1);

and so 0TT  0XX D 0:

(45)

As before, we expand the surface boundary conditions about z D 1I at the next order we then obtain o n 1 1 1 p1 D  .1  z2 /u0XT C 1 I W1T D .u0 u0X /X C 1XX  u0XXXT z C z3 u0XXXT ; 2 2 6 and eventually 1 .u0 u0X /X C 1XX  u0XXXT  .0 u0X /T D 1TT C .u0 0X /T : 3

(46)

Finally, we invoke the leading order (for example, in the form Z

1

u0 D

0T dX; X

on the assumption that decay conditions exist ahead of the wave i.e. any rightrunning wave is entering undisturbed conditions; indeed, u0 ! 0 as jxj ! 1 if 0 is the sum of two waves that each decay at infinity). Then we construct the equation for  D 0 C "1 C O."2 / W TT  XX  "

n1 2

2

 C

Z

1

T dX X

2 o

1  "XXXX D O."2 /: 3

(47)

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This is the Boussinesq equation [7] and, in this context, it is valid for X D O.1/ and T D O.1/, combining the O.1/ and O."/ (and ignoring the error term, of course). It admits solutions that describe waves that propagate to the left and to the right, together, but with a weak nonlinear interaction between them and for weak dispersion. However, in suitable far fields, Eq. (47) recovers the KdV equations for, separately, right- or left-propagating waves. We comment that this type of equation is the least satisfactory outcome when we use this asymptotic structure: there are terms of different asymptotic order (here O.1/ and O."/) retained together in the same equation and truncated thereafter; cf. the KdV equation, (38). This equation, (47), is another of our completely integrable equations, although the form in which it is written does not allow immediate solution by this method. To do this, we must transform the equation (and take advantage of the overall error: O."2 / here). We introduce H D   "2 I

Z and

1

DX

.X 0 ; TI "/ dX 0 ;

(48)

X

which is equivalent to writing the equation in a Lagrangian rather than an Eulerian frame; this gives 3 1 HTT  H  ".H 2 /  "H D O."2 /I 2 3

(49)

with zero on the right-hand side, this is the Boussinesq equation written in completely integrable form. Complete integrability holds for any " > 0I solutions describing collisions, however, are relevant only in the region defined by T D O.1/ and  D O.1/; with H D O.1/. With this understanding, we may scale 2 H !  HI "

r .; T/ !

" .; T/ 3

to recover the Boussinesq equation written in the standard form: HTT  HXX C .H 2 /  H D 0:

(50)

This equation possesses solutions, and most particularly soliton solutions, that may propagate either all in the same direction (much as for the KdV equation) or in opposite directions, producing head-on collisions between the waves; see Sect. 6.4.

5.6 The Two-Dimensional Boussinesq Equation The KdV equation, as we have seen, can be generalised to accommodate a weak dependence in the y-direction, and so admit solutions representing obliquely inter-

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

147

acting waves; see Sect. 5.2. The same approach can be adopted for the Boussinesq equation, by allowing a weak dependence in y and so this will model the oblique, head-on collision of waves. The procedure follows the method of derivation used for the 2D KdV equation, by extending the development that leads to the Boussinesq equation (Sect. 5.5) with the inclusion of a suitable p Y-dependence. So we define a scaled Y, by writing  D "Y p and then, for consistency with the equation of mass conservation, introducing v D "V (where u? D .u; v/); see Sect. 5.2. We obtain—perhaps not surprisingly—the 2D Boussinesq equation 3 1 HTT  H  ".H 2 /  H  "H  D O."2 /I 2 3

(51)

written in the same Lagrangian frame as for Eq. (49). This, in turn, can be expressed in the standard form for an integrable system, by transforming 2 H !  HI "

r .; T/ !

" .; T/I 3

 !

p 3"

to produce HTT  H C 3.H 2 /  H  H  D 0:

(52)

As we shall mention later, this equation has some interesting properties that relate to integrability, but it falls short of being completely integrable. More details about this equation, its solutions and relevance, can be found in [47].

5.7 The Camassa-Holm (CH) Equation This equation is the last that we shall discuss under the heading of weakly nonlinear, long waves i.e. weakly dispersive waves, but it is by the far most involved in terms of its derivation. Furthermore, it has the same asymptotic character as the Boussinesq equation: we retain terms of different orders in the one equation. Indeed, this is further complicated by the need to use both our fundamental small parameters (" and ı) independently and in the same equation. Some relevant background information can be found in [9]. We start with Eqs. (10)–(12), with both " and ı retained as separate and independent parameters (with the understanding that " ! 0; ı ! 0); we consider one-dimensional wave propagation (in the x-direction). The method that we describe here is explained more fully, particularly in the context of other model equations for water waves, in [49]. First, we introduce D

p ".x  t/;

p  D " "t;

wD

p "w;

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for right-running waves (and the geometry is restricted to two dimensions, .x; z/); we form the equation for .; I "; ı/ 00 C "10 C ı 2 01 C "ı 2 11 , which gives 1 3 1 2 C3 C ı 2   "2  D  "ı 2 .23  C10 /CO."2 ; ı 4 /; 3 4 12

(53)

but this is not a CH equation—although it does take the form of higher-order corrections to the KdV equation. The details underpinning this derivation are quite straightforward, following the general approach adopted for the KdV equation, but working to higher order. Here, we can interpret the procedure as: expand first in " (so each term in this asymptotic expansion also depends on ı), and then expand each of these terms in ı. We elect to retain terms as far as O."/ in the first expansion, and then each of the O.1/ and O."/ terms are themselves expanded, to retain terms as far as O.ı 2 /. (A comment on a rigorous justification of this approach will be made later.) To proceed, we find that, to this same order of approximation, we have 1 1  1 u   "2 C "ı 2  z2  4 3 2

(for 0  z  1),

(54)

which describes the horizontal velocity component in the flow at various depths. We now select a specific depth, denoted by z D z0 .0  z0  1/ and introduce

D

1 1 2  z I 3 2 0

writing uO D u.; ; z0 I "; ı/; we invert (54), evaluated at z D z0 ; to give 1  uO C "Ou2  "ı 2 uO  I 4

(55)

we assume that the functions uO and , appropriate to this problem, permit this manoeuvre. This is used in (53) to obtain the corresponding equation in uO : n o 1 5 29  2Ou C 3OuuO  C ı 2 uO  D "ı 2 6 C uO  uO  C uO uO  C O."2 ; ı 4 /; 3 12 6 or n o 29  1 5 2.OuOt C uO xO / C 3"OuuO xO C "ı 2 uO xO xO xO D "2 ı 2 6 C uO xO uO xO xO C uO uO xO xO xO C O."3 ; "ı 4 /; 3 12 6 (56) p p when expressed in (essentially) original variables: xO D "x; Ot D "t. Finally, we add the term "ı 2 .OuxOxOOt  uO xOxOOt / to the left-hand side of (56), and use 3 uO Ot .OuxO C "OuuO xO / 2

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

in the first of these; finally, we choose  D give

1 2

C 4 with  D

149 5 6

(so that D

1 12 /;

to

1 5 5 2 2 " ı f2OuxO uO xO xO C uO uO xO xO xO g C O."3 ; "ı 4 /: 2.OuOt C uO xO / C 3"OuuO xO  "ı 2 uO xO xO xO  "ı 2 uO xO xOOt D 2 6 12 (57) This is a CH equation (because a frame shift allows the term uO xOxO xO to be subsumed into uO xO xOOt ), and then the reversion to qthe standard form qrequires no more than a simple 5 3O O scaling transformation; so  D 2 .Ox  t/; uO D 5 uO (and Ot unchanged) gives 3

5

3

n o uO Ot C 2 uO O C 3"OuuO O  "ı 2 uO OOOt D "2 ı 2 2OuO uO OO C uO uO OOO ;

(58)

p at this order, where D .2=5/ 3=5: We have demonstrated that the CamassaHolm equation does indeed describe a class of water waves, but this description applies only p to the horizontal velocity component in the flow, at a specific depth .z0 D .1= 2//, and then, most importantly, if we retain [when expressed in .; / variables; see (53)] only terms O.1/, O."/; O.ı 2 / and O."ı 2 /. The equation for the surface, (53), as we have previously observed, is not a CH equation; the behaviour of the surface (at this order) is obtained from (55) with uO determined by (58). It should be noted that any number of different equations, all valid at this same level of approximation, are clearly possible, by using the lower-order terms in various higher-order terms (and incorporating additional terms, suitably approximated, as we did with uO xOxOOt ). In no sense is the CH equation unique, but it is certainly one of the possibilities, and to show that this was the case was the main aim of the exercise. Of course, we prefer to generate equations like CH because they are completely integrable and, in this context, they incorporate appropriate higher-order effects associated with water-wave propagation than does, say, just the KdV equation. (The complete recovery of CH requires that " and ı also be scaled out from (58), but the caveats that we mentioned about the Boussinesq equation apply equally here.) This analysis and derivation, and the existence of the CH equation as a model for water waves, has been put on a rigorous basis by Constantin and Lannes [16]. There, it is shown that the CH equation for the horizontal velocity component, at the depth that we have found, is a proper approximation of the water-wave problem provided that "  Mı, for some M > 0 (independent of " and ı) and ı < ı0 for some ı0 > 0; with an overall error O."ı 4 / in Eq. (58). This, it is shown, constitutes a well-posed problem, such p that there is some fixed T0 ensuring that the solution exists for times Ot 2 Œ0; T0 =" ". We conclude by observing that there is a second equation, of CH type, that is also completely integrable: the Degasperis-Procesi equation [26, 27] n o 2 2 uO Ot C 2 uO O C 4"OuuO O  "ı 2 uO OO 3O u D " ı u O C u O u O Ot O OO OOO ;

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which we have written in the same format as our CH equation, (58). This arises in the water-wave problem q q in exactly the same way that CH does, but at a depth

21 3 z0 D 13 11 2 with 2 D 40 10 ; see Johnson [52]. Furthermore, it is also possible to obtain a 2D CH equation (Oy D "y; vO D v="); the resulting equation that corresponds to (58) is then

    uO Ot C 2 uO O C 3"OuuO O  "ı 2 uO OO C "2 uO yOyO D "2 ı 2 2OuO uO OO C uO uO OOO I Ot O

O

see Johnson [49]. There is no information, currently, as to the complete integrability, or otherwise, of this equation.

5.8 Transformations Between These Equations Before we move on, we make one further general observation about some of the equations described above. The four equations: KdV (38), 2D KdV (40), cKdV (43) and the ncKdV (44), are different representations of the same type of propagation problem—weakly nonlinear, weakly dispersive gravity waves on the surface of water—but written in different coordinate systems. Because we may interpret the two equations in cylindrical geometry as being appropriate for large radius, which presumably corresponds to nearly plane waves locally, we might expect some transformations to exist between these equations. O /; Thus we see that the 2D KdV equation (40), expressed in the form 0 D H.; 2 with O D  C =2, gives (after one integration in O and invoking decay conditions at infinity) 1 1 2H C H C 3HHO C HOOO D 0:  3 This is precisely the cKdV equation, (43), expressed here in terms of  rather than O R; O /, R. Similarly, in the case of the ncKdV equation, (44), we write H0 D .; 2 O O where  D   R =2 and  D R; to give 

 1 2R C 3O C OOO C O O D 0; O 3

which is the 2D KdV equation, (40), expressed in terms of R rather than . Thus, although the ncKdV equation is not completely integrable (for arbitrary initial data), this equation can be solved by soliton methods for initial data consistent with the above transformation (because, of course, the 2D KdV equation is completely integrable).

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151

6 The Inverse Scattering Transform (‘Soliton’ Theory) The development of what we now call the inverse scattering transform method came out of some observations of the numerical solution of a related problem [31] and then of the KdV equation [85]. The results were a considerable surprise: waves that interacted nonlinearly but retained their identities. This led a group at Princeton to investigate the properties of the equation, which produced [36] a method of solution that treats the unknown function (a solution of KdV) as the time-dependent potential in a one-dimensional, linear scattering problem. This linear problem, with the associated inverse scattering problem, coupled with the time evolution of the potential (consistent with the KdV equation), produce a solution-method that maps the nonlinear PDE into a linear integral equation. This constitutes the inverse scattering transform method (and is somewhat analogous to the classical transform methods for solving linear PDEs). From this relatively small—and apparently very special—beginning has sprung a whole range of methods which are applicable to an extremely large family of important equations. In particular, we now have a good understanding of the underlying properties that lead to complete integrability, and many different techniques for constructing solutions. Here, we will do no more than give the simplest statement of the formulation of the solution of a few equations via an appropriate integral equation. (This will not include any development of the method itself, and certainly not the description in terms of a quantum scattering problem.) The topic has expanded to include Hamiltonian structure, conservation laws, Bäcklund transforms, Hirota’s bilinear form, prolongation structures, Painlevé equations and much more; the interested reader is directed to e.g. [1, 2, 28].

6.1 The Korteweg-de Vries Equation The solution, u.x; t/; of the KdV equation ut  6uux C uxxx D 0; which has been written in the standard form (by simply applying a suitable scaling transformation to any other equivalent KdV equation), is related to the function K.x; zI t/ by u.x; t/ D 2

d K.x; xI t/: dx

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(The wave of elevation then corresponds to u.) Here, K satisfies the integral equation Z

1

K.x; yI t/F. y; z; t/ dy D 0;

K.x; zI t/ C F.x; z; t/ C x

which is usually called the Marchenko equation (or, sometimes, the Gel’fandLevitan equation). In this equation, F.x; z; t/ satisfies the pair of linear equations Fxx  Fzz D 0;

Ft C 4.Fxxx C Fzzz / D 0I

however, because we require evaluation on z D x (in order ˇto recover u), the relevant solution for F depends on .x C z/ (because F D f .x  z/ˇzDx D f .0/ Dconstant is redundant) and so this pair reduces to the single equation Ft C 8F   D 0

where  D x C z:

With this choice of argument, the integral equation becomes Z

1

K.x; zI t/ C F.x C z; t/ C

K.x; yI t/F. y C z; t/ dy D 0: x

This formulation is appropriate for solving the initial-value (Cauchy) problem for the KdV equation, provided that certain existence conditions are satisfied: Z

Z

1

1

ju.x; t/j dx < 1

1

and 1

.1 C jxj/ju.x; t/j dx < 1

8t:

These state that the solution, for all time, must satisfy absolute integrability and, indeed, it must decay sufficiently rapidly at infinity (and exponential decay is what we usually encounter). The simplest solution of the KdV equation is obtained by choosing the solution for F as F D ek.xCz/C!tC˛ ; where k.> 0/ is a parameter and ˛ an arbitrary constant (equivalent to writing F D Aek.xCz/C!t ); !.k/—the dispersion relation—is to be determined. We find that ! D 8k3 , and then the solution for K follows directly (the integral equation being separable): K.x; zI t/ D

ekz I .ekx =2k/ C ekx8k3 t˛

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153

thus we obtain a solution of the KdV equation as u.x; t/ D 2k2 sech2 Œk.x  x0 /  8k3 t

p . 2ke˛=2 D ekx0 /:

This is the solitary-wave solution of the KdV equation, and corresponds to the smallamplitude solitary wave mentioned in Sect. 3.2. The two-soliton solution is generated by the choice F D e 1 C e2

where i D ki .x C z/ C 8ki3 t C ˛i ;

k1 ¤ k2 ;

which eventually leads to the solution u.x; t/ D 8

k12 E1 C k22 E2 C 2.k1  k2 /E1 E2 C A.k22 E1 C k12 E2 /E1 E2 ; .1 C E1 C E2 C AE1 E2 /2

where 3

Ei D e2ki .xx0i /8ki t ;

i D 1; 2;

A D .k1  k2 /2 =.k1 C k2 /2 I

here, x0i are two arbitrary phase shifts. An example of a two-soliton solution is shown in Fig. 2.

1

–u

1

0.75 0.5 0.25 0

5

0 –10 0

–5 10 X=x–t

Fig. 2 Perspective view of a 2-soliton solution of the Korteweg-de Vries equation

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R.S. Johnson

6.2 The Two-Dimensional Korteweg-de Vries Equation The solution of the 2D KdV equation, written in the normalised form .ut  6uux C uxxx /x C 3uyy D 0; can be expressed as u.x; t; y/ D 2

d K.x; xI t; y/; dx

where Z

1

K.x; zI t; y/ C F.x; z; t; y/ C

K.x; y0 I t; y/F. y0 ; z; t; y/ dy0 D 0:

x

The function F satisfies the pair of equations Fxx  Fzz  Fy D 0;

Ft C 4.Fxxx C Fzzz/ D 0;

and then the solitary-wave solution is obtained with the choice F D expŒ.kx C lz/ C .k2  l2 /y C 4.k3 C l3 /t C ˛I cf. the solution of KdV, Sect. 6.1. The solution based on the sum of two exponentials for F produces the two-soliton solution (i.e. an oblique collision), an example of which is shown in Fig. 3.

6.3 The Concentric Korteweg-de Vries Equation We write this equation in its standard form ut C

1 u  6uux C uxxx D 0; 2t

with the Marchenko equation which is unchanged: Z

1

K.x; yI t/F. y; z; t/ dy D 0;

K.x; zI t/ C F.x; z; t/ C x

but where F now satisfies the pair of equations Fxx  Fzz D .x  z/F;

3tFt  F C Fxxx C Fzzz D xF C zFz :

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

2

155

–u 20 y

1.5 1

10

0.5 0 –20

0 –10 0

–10 10 20

x

–20

Fig. 3 A 2-soliton solution of the 2D Korteweg-de Vries equation

In this case, the transformation that recovers the solution is u.x; t/ D 2.12t/2=3

dK ; d

where K D K.; I t/ with  D x=.12t/1=3 ;

and it is this appearance of a similarity variable which complicates the solution somewhat! The complication is immediately evident in that the solitary-wave solution of the cKdV equation involves not only the introduction of a similarity variable, but the solution itself comprises the integral of the Airy function squared.

6.4 The Boussinesq Equation For our final two KdV-type problems, we simplify the presentation by invoking Hirota’s version of the soliton-solution problem [40]. This replaces the original partial differential equation by another equation which is also nonlinear but which contains some amazing structure. So the soliton solutions of the Boussinesq equation, written as utt  uxx C 3.u2 /xx  uxxxx D 0; can be recovered from u D 2@2 =@x2 Œln f .x; t/; for suitable f .x; t/. The solitarywave solution is then given by the choice f D 1Cexp.kx!tC˛/ with ! 2 D k2 Ck4 I

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R.S. Johnson

4 2 0 1.5 1 –4 0.5 t

–2 0

0 2 4

–0.5

Fig. 4 Perspective view of a 2-soliton solution of the Boussinesq equation, depicting a head-on collision

the two-soliton solution is generated by f D 1 C E1 C E2 C AE1 E2 ; q where Ei D e2ki x"!i tC˛i with !i D 2ki 1 C 4ki2 ; AD

.!1  !2 /2  .k1  k2 /2  .k1  k2 /4 .!1 C !2 /2  .k1 C k2 /2  .k1 C k2 /4

and " D ˙1 (describing propagation either to the left or to the right). In Fig. 4 is shown a two-soliton solution of the Boussinesq equation.

6.5 The Two-Dimensional Boussinesq Equation The soliton solutions of the 2D Boussinesq equation follow exactly the same pattern as shown for the standard Boussinesq equation (in Sect. 6.4). The equation written in the form utt  uxx C 3.u2 /xx  uxxxx  uyy D 0;

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has the solution u D 2@2 =@x2 Œln f .x; y; t/; where f .x; y; t/ can be written f D 1 C e1 for the solitary-wave solution, and f D 1 C e1 C e2 C Ae1 e2 ; for the 2-soliton solution, where AD

. p1  p2 /.q1  q2 / : . p1 C p2 /.q1 C q2 /

Here, we have introduced parameters pi and qi , so that i D . pi C qi /x C

p 3. p2i  q2i /y  !i t C ˛i ;

with !i D ki3 C 3

l2i ki

(where ki D pi C qi I li D p2i  q2i :)

The 2-soliton solution is a quite general one of this type (because it contains the appropriate number of free parameters); however, a corresponding 3-soliton solution does not exist. The 2D Boussinesq equation is not completely integrable in the accepted sense (because it does not satisfy the 3-soliton test originally introduced by Hirota, and developed by Hietarinta [39]; also see Johnson [47]). But, as also happens in the case of the 2D KdV equation, we do have a 2-soliton resonant interaction, corresponding to a parameter choice that gives A D 0, as well as special—not general—N-soliton solutions (for N > 2); two different soliton solutions of the 2D Boussinesq equation are shown in Figs. 5 and 6.

6.6 The Camassa-Holm Equation The construction of the solution to the CH equation, written in the form ut C 2!ux C 3uux  uxxt D 2ux uxx C uuxxxx ;

! > 0;

is altogether much more complicated than any of the preceding methods, essentially because of the nature of the eigenvalue problem in this case. The description that we give here, based on [9, 10, 12], follows the implementation that can be found in [50].

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R.S. Johnson

0.6 0.4

–u

0.2 0 5 –10 0

–5

y

0 x 5

–5 10

Fig. 5 Head-on-oblique-collision solution of the 2D Boussinesq equation

2.0 1.5

4

–u 1.0 2

0.5 0

0 –5

y –2

0 x

5

–4 10

Fig. 6 A resonant-wave solution of the 2D Boussinesq equation

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159

The soliton solutions, which may or may not be peakons, are generated by the choice F.X; t/ D where ki D equation

  ki ki exp  p X  t C ˛i ; i ! iD1

N X

p 1 C 4!i =2; i 2 .1=.4!/; 0/; this is then used in the Marchenko Z

1

K. y; zI t/F.z C x; t/ dz D 0;

K. y; xI t/ C F. y C x; t/ C y

to give K.y; xI t/. We now construct Q. y; t/ D 2d=dyŒK. y; yI t/; which provides the coefficients in the ordinary differential equation for ˚.yI t/ W  1  0 ˚ 000  4 Q C ˚  Q0 ˚ D 0; 4! the solution of which gives q.yI t/ D ˚ 2 : Finally, the solution that we require, u.yI t/; is expressed in parametric form (parameter y) as the solution of qu00 C q0 u0  u D !  q

with

p dy D q. yI t/: dx

The solitary-wave solution, obtained by using just one exponential term in F; can be written uD

.c  2!/sech2 ./ sech2 ./ C .2!=c/ tanh2 ./

.c > 2! > 0/;

with r x  ct  x0 D 

 cosh.   /  4c 0 C ln ; c  2! cosh. C 0 /

tanh 0 D

p 1  2!=cI

here,  is the parameter (which is more convenient than the y introduced above). A corresponding 2-soliton solution is shown in Fig. 7.

7 Modulation of Waves The analysis that leads to the Nonlinear Schrödinger (NLS) equation is certainly important in any general discussion of the theory of water waves, but it sits rather outside the type of problem that is of most interest in the development here. Thus

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R.S. Johnson

u/w

8 6 4 2 0 –3

–20 –10

–2 –1

0

0 T

1

x

10 2 3

20

Fig. 7 Perspective view of a 2-soliton solution of the Camassa-Holm equation .T D 2!t/

we will only briefly outline the method of derivation, and mention some of the results; more details can be found in the literature that we cite. The problem that we discuss retains arbitrary (linear) dispersion, so the underlying asymptotic limit will be: " ! 0; ı fixed; this is based on the original work of Hasimoto and Ono [38]. We seek a solution for which the initial wave-profile takes the form A."x/ exp.ikx/ C c:c: where ‘c.c.’ denotes the complex conjugate and k is a general wave number; the governing equations are those that retain the parameter ı, so we use the set (10)– (12) or (13)–(15). The appropriate ‘slow’ variables that we need here to describe the evolution of the wave are  D x  cp t;

 D ".x  cg t/;

 D "2 t;

where cp .k/ and cg .k/ are the phase and group speeds, respectively. The asymptotic expansion, at fixed ı for " ! 0; for the surface wave, takes the form .; ; I "/

nC1 X X

Anm .; /Em C c:c:;

nD01 mD0

where E D exp.ik/ and A00 D 0I all the various functions are expanded, following this same pattern (with the z-dependence included, of course); we take b.x/  0. The calculation is very lengthy and rather cumbersome, but fairly routine; by

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

161

working as far as the terms that arise at O."2 /; and balancing all the terms of the type Em ; we find that c2p D

tanh.ık/ ık

and

c2g D

1 cp Œ1 C 2ıkcosech.2ık/; 2

for which the familiar identity cg D d!=dk .! D kcp / holds. Finally, we obtain the classical Nonlinear Schrödinger (NLS) equation for the leading term that describes the amplitude modulation of the surface wave:  2ikcp A01  kcp

d2 ! A01 C ˇA01 jA01 j2 D 0; dk2

(59)

where ˇD

 k2 h 1  1 C 9 coth2 .ık/  13sech2 .ık/  2 tanh4 .ık/ c2p 2 i  2  2cp C cg sech2 .ık/ .1  c2g /1 :

The relevance and application of this equation to our understanding of water waves is considerable, but outside our main interest here; for more information, see [48, 72]. The extension to two dimensions, with a suitable weak dependence in the ydirection, is accomplished by allowing the initial profile to take the form A."x; "y/ exp.ikx/ C c:c:; so the scale of the (slow) evolution in x and y is the same in this model. We follow exactly the same procedure as we did for the derivation of the standard NLS equation, but now we include Y D "yI the resulting equation [cf. (59)] is  2ikcp A01  kcp

 d2 !  2 k2  A  c c A C ˇ C A01 jA01 j2 01 p g 01YY dk2 c2p .1  c2g /

C  k2 A01 f D 0; where .1  c2g /f C fYY D 

 .jA01 j2 / ; c2p

and  D 2cp Ccg sech2 .ık/. This coupled pair is usually called the Davey-Stewartson (DS) equations (see Davey and Stewartson [23]), and this system is completely integrable for long waves .ı ! 0/I see Anker and Freeman [3]. Further, there is

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a correspondence between the KdV equation and the NLS equation for long waves [48], and similarly between the 2D KdV equation and the DS equations [33].

7.1 The Inverse Scattering Transform for the NLS Equation The Nonlinear Schrödinger equation, written in the form iut C uxx C ujuj2 D 0;

(60)

possesses a rather different structure when compared with all our previous IST calculations: it requires a matrix formulation; see Zakharov and Shabat [86, 87]; also Drazin and Johnson [28]. (The NLS equation written here—appropriate for water waves—is to be compared with the alternative version with ujuj2 I this is relevant to self-defocussing and dark solitons in optics.) First, we introduce F.x; z; t/; which here is a 2  2 matrix, and which satisfies the pair of equations



l 0 l 0 D0 Fx C Fz 0m 0m



and

Fxx  Fzz  i˛Ft D 0;

where l; m and ˛ are arbitrary, real constants. Then the 2  2 function K.x; zI t/ is a solution of the matrix Marchenko equation Z

1

K.x; yI t/F. y; z; t/ dy D 0I

K.x; zI t/ C F.x; z; t/ C x

it is convenient to set

ab KD ; cd and then we find that u.x; t/ D b.x; xI t/

and

u.x; t/ D c.x; xI t/;

where the over-bar denotes the complex conjugate, gives a solution of the NLS equation i˛.l  m/ut C .l C m/uxx C

2 .l  m/.l2  m2 /ujuj2 D 0: lm

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163

As an example, we outline the calculation that generates the solitary-wave solution of the NLS equation. The choice for F is FD

0 f1 f2 0



with f1 Da1 expŒ1 .mx  lz/ C i21 .l2  m2 /t=˛; f2 Da2 expŒ2 .lx  mz/ C i22 .m2  l2 /t=˛;

where ai and i .i D 1; 2/ are arbitrary, real constants; solving for K (which is altogether routine), and then extracting b.x; xI t/ (and suitably redefining the various arbitrary constants), gives a solution of our NLS equation, (60), as oi h n1 p u D a exp i c.x  ct/ C !t sechŒa.x  ct/= 2: 2 This solution—the solitary wave—expressed in terms of two arbitrary (real) constants a and c, represents an oscillatory wave packet that propagates at the speed c with a maximum amplitude of a; the dispersion relation is !D

1  2 1 2 a C c : 2 2

8 Variable Depth Although our main interest hereafter will be on the important class of problems that generalise the basic description of flows (as given earlier) to include vorticity, we should mention another type of problem—and one that is technically quite involved. This arises when we allow the wave to propagate over variable depth; in this situation, the leading-order governing equation is almost never of completelyintegrable type. Indeed, to analyse this problem in any detail requires a careful discussion of many different elements (often, in this context, working to quite high order in the asymptotic expansion). As an example of what we encounter, we will outline the problem associated with the KdV equation in the presence of variable depth.

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R.S. Johnson

8.1 The Korteweg-de Vries Equation for Variable Depth We consider (see Sect. 5.1) one-dimensional propagation, using the equations that have been scaled to remove the parameter ı; so we start with uT C ".uuX C Wuz / D pX I

"fWT C ".uWX C WWz /g D pz I

uX C Wz D 0; (61)

with pD &

W D T C "uX

on z D 1 C "

(62)

and W D ub0 .X/

on z D b.X/:

(63)

One of the crucial questions in this problem is: on what scale should we allow the depth to vary? Let us write b D B.˛X/; then we have, within a conventional asymptotic formulation, three distinctly different cases: ˛ D o."/; ˛ D O."/I " D o.˛/I alternatively, we could treat ˛ as fixed (equivalently ˛ D O.1// as " ! 0. This last possibility is by far the most difficult, so we will present some of the details for the mathematically most interesting problem that is also reasonably accessible: ˛ D O."/. Thus we choose b D B."X/ and then introduce  D "1 k./  T

and

 D "X;

(64)

where k./ accommodates the expected variable speed of the wave as it moves over variable depth. The characteristic variable, , has been written to ensure that d=dX D O.1/ i.e. the constant-depth case is recovered from the choice k./ D  W  D ="  T D X  T. (Notice that this formulation uses a characteristic variable (for right-running waves) together with a large distance variable, rather than large time; cf. (33).) With this transformation to these far-field coordinates, Eqs. (61)–(63) become u C "Œu.k0 u C "u / C Wuz  D .k0 p C "p /I 0

"fW C "Œu.k W C "W / C WWz g D pz I

0

k u C "u C Wz D 0;

(65) (66)

with pD &

W D  C "u.k0  C " /

on z D 1 C "

(67)

and W D "uB0 ./

on z D B./:

(68)

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

165

We seek a solution of this set, following the standard form of asymptotic expansion, (37), and although, in principle, we could also expand k; that is unnecessary here; at leading order we find that p0 D 0 I

u0 D 0 k0 ./I

W0 D .B  z/.k0 /2 0

(all for B  z  1),

with Z



k./ D 0

d0 p D.0 /

where D./ D 1  B./.> 0/ is the local depth. Here, we have selected the positive square root to correspond to rightward propagation. At the next order, we obtain the KdV-type equation appropriate to this problem: p 1 D0 ./ 3 1 2 D0 C p 0 C 0 0 C D0   D 0: 2 D D 3

(69)

This equation immediately recovers our familiar KdV equation, (38), when we take D D 1; but otherwise, for general depths, this equation is not completely integrable. Often, the equation is rewritten with the change of variable H.; / D D1=4 0 ; which gives 1 2H C C3D7=4 HH C D1=2 H   D 0 3 —more evidently a KdV-type equation. The depth-dependent factor in 0 D D1=4 H is usually called Green’s law. One final transformation is worthy of mention; with Z p  D d 0 D D H ; 2b



and the choice D D .a C b/9=4 ; b is the cKdV equation (which is for arbitrary constants a; b; the equation for H completely integrable; see Sects. 5.3 and 6.3). Most of the work done on this equation, linked to a discussion of its role in wave propagation, has been for the case ˛ D o."/; in this situation, the leading term is the standard KdV equation, but with variable coefficients that depend on a new, third, variable. However, the interesting aspects are not restricted to the analysis of this equation in isolation; rather, it is the discussion of all the various wave-propagation phenomena that occur: primary wave, wave reflections and rereflections. This problem is explored in [46, 61, 62, 74].

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R.S. Johnson

9 Weakly Nonlinear, Weakly Dispersive Waves with Vorticity We now turn to a discussion of a model that will more accurately represent another aspect of realistic flows. To this end, we consider the propagation of weakly nonlinear, (usually) long waves, of various types, in a flow that is moving according to some prescribed vorticity. (Because such a flow can be regarded as a model for a real flow, which can exhibit viscous and turbulent properties, this is often called a ‘shear’ flow.) The aim is to derive the leading-order approximation, based on a suitable asymptotic expansion, of the nonlinear surface wave in the presence of vorticity. This will then make accessible the possibility of describing the effects of a background flow on the propagation process (and the choice of this flow can certainly model realistic, observed flows). The starting point is the set of governing equations, (10)–(12), which will be restricted so that they are appropriate to the discussion of plane waves propagating in the x-direction. Now we need to introduce the given background vorticity—and we take this to be O.1/ relative to the scales previously defined—which we do in the form (for plane waves) "u? D ".u; 0/ is replaced by .U.z/ C "u; 0/; where U.z/ is given. The equations therefore become ut C Uux C U 0 w C ".uux C wuz / D px ; ı 2 Œwt C Uwx C ".uwx C wwz / D pz I

(70) ux C wz D 0;

(71)

on z D 1 C ",

(72)

with pD

& w D t C Ux C "ux

and wD0

on z D 0,

(73)

where U 0 D dU=dzI we shall consider only constant depth (so b.x/  0/. Thus if, somewhere, the flow is an undisturbed parallel flow, .U.z/; 0/; the vorticity is simply U 0 .z/. The set of equations currently contains both our fundamental parameters, but we shall, for some of the following calculations, invoke the scaling that removes ı 2 in favour of "I see (16). In this case, we obtain the equivalent set uT C UuX C U 0 W C ".uuX C Wuz / D pX ; "ŒWT C UWX C ".uWX C WWz / D pz I

uX C Wz D 0;

(74) (75)

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167

with pD &

W D T C UX C "uX

on z D 1 C ",

(76)

and WD0

on z D 0.

(77)

9.1 The Korteweg-de Vries Equation with Vorticity This problem is described by Eqs. (74)–(77) and recast—as we did for the classical KdV equation (Sect. 5.1)—in suitable far-field variables, although in this case we do not know, a priori, the speed of propagation, c, of (linear) waves; thus we introduce the far-field variables  D X  cT;

 D "T:

Equations (74)–(77) then become .U  c/u C U 0 W C ".u C uu C Wuz / D p I "f.U  c/W C ".W C uW C WWz /g D pz I

u C Wz D 0;

with p D  & W D .U  c/ C ". C u /

on z D 1 C ",

and WD0

on z D 0,

and we seek an asymptotic solution exactly as before; see (37). At leading order, we obtain the problem described by .U  c/u0 C U 0 W0 D p0 I

p0z D 0I

u0 C W0z D 0;

with p0 D 0

& W0 D .U  c/0

and W0 D 0

on z D 0.

on z D 1,

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R.S. Johnson

The appropriate solution of this set, defined for z 2 Œ0; 1; is p0 D 0 I

W0 D .U  c/I2 0x I

u0 D 0

d Œ.U  c/I2 ; dz

where Z I2 .z/ D

z 0

dz0 ŒU.z0 /  c2

and then c is determined from Z I2 .1/ D 1

1

or 0

dz0 D 1: ŒU.z0 /  c2

(78)

(In the light of the notation developed below, we will write this in the shorthand I21 D 1.) This last result, (78), is the famous Burns condition [8], which, for some choices of U.z/; can lead to the presence of critical layers (i.e. where U.z/ D c for some z 2 .0; 1/), even for infinitesimally small waves; for more on these ideas see Benney and Bergeron [6] and Johnson [43, 45]. We will consider, here, only situations where critical layers do not appear in the linearised problem. The procedure at the next order follows the pattern laid down for the classical KdV equation (Sect. 5.1), although a little more involved; this eventually produces  2I31 0 C 3I41 0 0 C J1 0   D 0

(79)

where Z In1 D In .1/ D

1 0

dz0 ; ŒU.z0 /  cn

Z J1 D

0

1

Z

1 z

Z

 0

ŒU./  c2 dZd dz: ŒU.z/  c2 ŒU.Z/  c2

(More details can be found in [34], which describes a development, and generalisation, of the seminal work presented in [5].) If we set aside the complications that arise when critical layers are present, the results represented by (79) are most encouraging. This is a KdV equation, with appropriate constant coefficients, which has, for example, solitary-wave and soliton solutions—and this holds for any background vorticity. Thus we can expect that these solutions will be relevant to wave propagation in (almost) any realistic onedimensional flow of water. As a simple example, let us examine the special case of constant vorticity. We suppose that the original equations (expressed in X; T; u variables) have been written in the Galilean frame in which U.0/ D 0I indeed, this will be the physical frame if U.z/ models a ‘shear’ flow with a no-slip condition on the bed. Now we write U.z/ D  z, for  D constant, and then the speeds of small-amplitude waves

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169

are given [from (78)] by cD

p 1 . ˙ 4 C  2 /; 2

(80)

and no critical layers exist for this choice; then, correspondingly, the KdV equation (79) becomes ˙

p  1 1 0   D 0: 4 C  2 0 C .3 C  2 /0 0 C 3 c

(81)

(We see immediately that the problem of zero vorticity,  D 0; recovers our classical KdV equation, (38), for propagation to the right i.e. the upper sign.) It is of some interest, and possible relevance to more general theories of waves with vorticity, to investigate how the surface-wave profile is affected by the underlying shear flow, according to this model. To this end, we might consider two cases:  D ˙!.D constant), where the upper sign corresponds to positive vorticity in much of the rigorous work that has be done in recent years. (This mismatch has arisen because here we have elected to use the .x; z/ coordinate system, whereas the other work uses the .x; y/ system, both being interpreted in the conventional .x; y; z/ righthanded, rectangular Cartesian system, and then the two vorticities differ by a sign.) We should note, however, that here the choice of sign for the constant vorticity is immaterial: the underlying flow is then either to the left or the right—both generate the same mathematical problem—and then the crucial identification, in either case, is whether the surface wave is propagating upstream or downstream. To exemplify the essential character of the effect of constant vorticity, Fig. 8 shows three solitary waves, all of the same amplitude, and all exact solutions of Eq. (81). The middle profile is for zero vorticity—the irrotational case  D 0I the outer (‘wider’) solitary wave is an example ( D 1) of the profile obtained for upstream propagation (the sign of  is immaterial), and the narrower profile is that obtained for downstream propagation. This observation—equivalently a longer wave upstream and shorter wave downstream—is precisely that made by Benjamin [5]. Similar observations apply to other solutions of this KdV equation, such as soliton interactions and cnoidal waves.

Fig. 8 The solitary wave of the KdV equation, for constant vorticity: middle profile is  D 0I outer (broader) profile is  D 1; upstream propagation; inner (narrower) profile is  D 1; downstream propagation

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9.2 The Boussinesq Equation with Vorticity We now turn to a discussion of a relatively new problem (for which the investigation is still ongoing). The derivation of the Boussinesq equation, Sect. 5.5, as we have seen, is considerably more involved than the corresponding problem for, say, the KdV equation. In the light of this, and also to make the details more transparent, we will examine the problem of obtaining a Boussinesq equation appropriate for propagation in the presence of constant vorticity (U.z/ D  z; for constant  ). The governing equations are therefore (74)–(77), with this choice of U; which give uT C  zuX C  W C ".uuX C Wuz / D pX ; "ŒWT C  zWX C ".uWX C WWz / D pz I

(82) uX C Wz D 0;

(83)

on z D 1 C ",

(84)

with p D  & W D T C .1 C "/X C "uX and WD0

on z D 0.

(85)

The form of the asymptotic expansion is exactly as used throughout our discussions in this work (and specifically in the case for  D 0I Sect. 5.5). In this approach, as we have seen, it is necessary to expand the surface boundary conditions in Taylor expansions about z D 1; which we repeat here. Thus, at leading order, we find that, for z 2 Œ0; 1; p0 D 0 ;

u0X D .0T C  0X /;

u0T D 0X ;

W0 D .0T C  0X /z; (86)

which gives 0TT C  0TX  0XX D 0;

(87)

and u0 .X; T/ satisfies this same equation (but related to 0 .X; T/ as described in the appropriate equations in (86)). At the next order, we find (for example) that p1 D

1 1 QT .1  z2 / C QX .1  z3 / C 1 ; 2 3

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171

where Q.X; T/ D 0T C  0X ; and W1T D

n1

o 1 QXXT C  QXXX C 1XX C .u0 u0X /X .z  z3 / 2 3 ˚ C 1TT C  1XT C .0 0X /T C .u0 0X /T  .Q0 /T z3 :

The equation for 1 .X; T/ is 1TT C  1TX  1XX D

i h  1 1 1 0   u0 XXXX C u20 C 20 C  0 u0 C  2 20 ; XX 3 2 2 (88)

with Z

1

u0 D  0 C

0T .X 0 ; T/ dX 0 ;

(89)

X

assuming decay conditions ahead .X ! 1/ of any right-running wave. The equation for  0 C "1 , constructed from (87)–(89), becomes TT C TX XX D

h1 i  1  "    u0 XXXX C" .1 C  2 /2 C u20 C  u0  CO."2 /; XX 3 2 (90)

where (if we need to use it) u0 satisfies (87) with u0X .T C  X / (or (89) with 0 ); this is therefore a generalisation of Eq. (47) which, in that case, can be transformed into the completely integrable version of the Boussinesq equation. The new equation, (90), cannot, however, be transformed into anything equivalent to the classical Boussinesq equation, (50), by any transformation that corresponds to the Lagrangian form used in Sect. 5.5 i.e. X ! X using a transform linear in integrals of  and u0 ; together with  ! H using all terms of degree two in  and u0 . (The difficulties are readily seen by writing (90), first, in a frame moving at speed 12  (which removes the term  XT ), introducing  D    u0 and noting that Z 1  1 2 1 1 C  u0    .   u0 / dT; 2 2  with  D X  12  TI all this should be compared with (47)–(49).) Nevertheless, the equation does admit (ignoring the O."2 / error term) a sech2 solitary-wave solution:

r p  1 a 2 2 2 .3 C  /f.2 C  / ˙ .4 C  /g.X  cT/ ;  "asech 2 2 2

p where c D 12 . ˙ 4 C  2 / [from (80)]. This solution exhibits the same properties as observed for the solitary-wave solution of the KdV equation, (81),

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R.S. Johnson

for upstream/downstream propagation when  ¤ 0. This is to be expected because the farfield approximation of (90), following either left—or right—going waves, recovers precisely the KdV equation (81), to leading order as " ! 0. Now the fact that we obtain the two variants of the KdV equation from the one Boussinesq equation, and each of these admits scaling transformations that convert them into any desired, standard version of the KdV equation for any  (a property of the KdV equation over any shear) explains the difficulties mentioned above. The scaling transformation depends on c and therefore the sign of c (i.e. the direction of travel), and we cannot define a single transformation that will accommodate both directions of travel at the same time: either one or the other (as in the KdV equation), but not both. Thus we cannot transform our new Boussinesq-type equation into a standard Boussinesq equation. So we have a new equation, (90): Boussinesq with constant vorticity; we can expect, therefore, that this equation may describe new phenomena (but this is for future study).

9.3 The Camassa-Holm Equation with Vorticity We now examine the problem posed by considering the possible existence of a CH equation describing a class of water waves with some background vorticity. Although the problem for general vorticity can be formulated—this procedure is outlined in Johnson [51]—we will present the details for the case of constant vorticity .U D  z/. The method follows that for the case of zero vorticity (Sect. 5.7), except that we must use  D ".x  ct/;

p  D " "t;

wD

p "w;

where c is given by the Burns condition Z

1 0

dz D1 . z  c/2

i.e. c D

p 1 . ˙ 4 C  2 /I 2

see Sect. 9.1 and Eq. (80). It turns out that, in the presence of an underlying vorticity, the equation for the horizontal component of the velocity in the flow at a specific depth .u/ must be replaced by vO D uO C "ˇ.c/Ou2 C O."2 ; ı 4 /;

(91)

for some constant ˇ (which is zero for  D 0 i.e. for c D ˙1). (It is more convenient, and simpler, to express the various constants in terms of c rather than  .) The procedure is exactly that described in Sect. 5.7, with only fairly minor adjustments (in addition to the use of vO above), resulting in a CH equation for v. O

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173

With the choice ˇD

c5 .c4 C c2  2/ ; 2.c4 C c2 C 1/.1 C c2 /2

which we note is indeed zero for c D ˙1; and selecting, this time, D

c4 C 6c2 C 3 ; 2.c4 C c2 C 1/.1 C c2 /

we obtain .1 C c2 /.vOOt C vOxO / C ".c4 C c2 C 1/vO vO xO C "ı 2 f..1=3c/  /vO xO xOxO  vO xO xOOt g D "2 ı 2

c4 C 6c2 C 3 .2vO xO vO xO xO C vO vO xO xO xO / C O."3 ; "ı 4 /: 6.1 C c2 /2

(92)

Then, with the transformation s O D

1 C c2 .Ox  .1=3c/Ot/; 

p .1 C c2 / vO ! 3 4 v; O c C c2 C 1

we obtain 2 2 vOOt C 2 vO O C 3"vO vO O  "ı 2 vO OO O O vO OO C vO vO OOO / Ot D " ı .2v

(to this order) exactly as in (58), where p D

 c4 C c2 C 1 3=2 2 .1 C c2 /2 4 : 3c c C 6c2 C 3

The depth at which the horizontal component of the velocity is defined for (91) (to produce v) O is now p 2c12 C 16c10 C 33c8 C 31z6 C 32c4  3c2  3 z0 D p : .c4 C c2 C 1/.c2 C 1/ 6 In conclusion, we have demonstrated that, just as for the corresponding KdV problem—but not the Boussinesq case—the inclusion of vorticity (albeit constant here) does not fundamentally affect the underlying propagation structure. We still obtain a standard (constant coefficient) variant of the CH equation. The details of this result, and its relevance to this water-wave problem, are discussed in [51]; here, we will simply note how the constant vorticity distorts the solitary wave, based on Eq. (92), for various  . (The wave profile at the surface is recovered by using uO deduced from (91) in (55), but the dominant behaviour is simply given by  v.) O

174

R.S. Johnson 5 4 3 2 1

–8

–6

–4

–2

0

2

4

6

8

Fig. 9 The solitary wave of the CH equation, for constant vorticity: middle profile is  D 0I outer (broader) profile is  D 1; upstream propagation; inner (narrower) profile is  D 1; downstream propagation

In Fig. 9 we show three examples—just as in Fig. 8—of exact solutions of (92); the previous observation (longer waves upstream, shorter waves downstream) is repeated here. Comment: The corresponding analysis for the propagation of a modulated wave in the presence of vorticity (i.e. NLS over a shear flow) is a very considerable undertaking; this will not be pursued here. A discussion of this problem can be found in [42].

9.4 Ring Waves with Vorticity A very different type of problem which involves vorticity is provided by combining two types of geometry: a concentric wave expanding over a shear flow which moves in one direction. So we consider how a ring wave (Sect. 5.3)—and it may not be circular—propagates on the surface of water that is moving (in the x-direction, say) with some prescribed vorticity. In order to formulate this problem, we first take Eqs. (10)–(12), with the parameter ı scaled out in favour of " [via (16)], retain the dependence on a second horizontal dimension . y/, and replace "u by U C "u; for general U.z/I thus we get uT C UuX C U 0 W C ".uuX C vuY C Wuz / D pX I

(93)

vT C UvX C ".uvX C vvY C Wvz / D pY I

(94)

"ŒWT C UWX C ".uWX C vWY C WWz / D pz I

uX C vY C Wz D 0; (95)

with pD &

W D T C UX C ".uX C vY /

on z D 1 C ",

(96)

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175

and WD0

on z D 0.

(97)

We have also taken the bed to be horizontal (b.X/  0). Our aim here is to provide an overview of this problem, highlighting some of the difficulties that we encounter, and producing some details for the case of constant vorticity. (This work is based on [44], although the specific choice of constant vorticity is not discussed in any detail there.) At the outset, we observe that the geometry of this problem is a complication (requiring a mix of rectangular Cartesian and polar representations). The full details, which will not be developed here, can be found in the reference cited above. The first stage is to transform to a plane-polar coordinate system that is moving (in the x-direction) at a constant speed c; this prescribes a moving frame, the speed of which may be assigned as we wish. Thus we transform according to X D cT C r cos ;

Y D r sin 

(98)

with u ! u cos   v sin ;

v ! u sin  C v cos ;

(99)

so that .u; v/ now represents the horizontal velocity vector written in the polar system that is moving in the x-direction. Further, it is convenient to consider a large radius; it will then be possible to extend the analysis to a far field that accommodates the growth of initially weak nonlinearity and dispersion, and so correspond to our cKdV equation (Sect. 5.3), for example. The leading order, with this choice, will still generate the underlying linear problem (which is appropriate for weak nonlinearity and dispersion in the near field). Thus we further transform according to  D rk./  T;

R D "rk./;

(100)

where k./ is to be determined; the wave front represented by  Dconstant, for given T; is a circle if k./ Dconstant, which recovers the concentric wave. Thus we incorporate (98) and (99), with (100), into Eqs. (93)–(97); then we seek an asymptotic solution in the usual form [see (37)], to give, at leading order, u0 C .U  c/.k cos   k0 sin /u0 C U 0 W0 cos  D kp0 I v0 C .U  c/.k cos   k0 sin /v0  U 0 W0 sin  D k0 p0 I p0z D 0I

ku0 C k0 v0 C W0z D 0;

with p0 D 0

&

W0 D 0 C .U  c/.k cos   k0 sin /0

on z D 1,

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R.S. Johnson

and W0 D 0

on z D 0,

where k0 D dk=d (and U 0 D dU=dz; as used earlier). It is altogether routine to find that a solution exists of this set (giving 0 arbitrary at this order, which is therefore to be determined at the next order; see below) provided that .k2 C k02 /

Z

1 0

dz D 1; Œ1  fU.z/  cg.k cos   k0 sin /2

(101)

which is a generalisation of the Burns condition; cf. (78). At this stage, and in the context of our previous calculations, we choose to consider a constant vorticity flow: U.z/ D  z (other choices are discussed in [44]); in this case, (101) becomes .k2 C k02 / D 1: 1 C .k cos   k0 sin /

(102)

Here, we have made the obvious choice: c D  D U.1/; and so we are in a frame that moves at the surface speed of the water. The first order, nonlinear ordinary differential equation, (102), has the general solution p k./ D a cos  ˙ 1 C a  a2 sin ; (103) which is real if the (real) parameter a satisfies a2  1 C a; for a given  . However, even when this is the case, this solution has the property that k./ D 0 where a tan  D p ; 1 C a  a2 and this implies that r ! 1 on the wave front: a ring wave does not exist. Further, (103) does not recover the circular ring-wave solution, corresponding to k./ D constant , on a stationary flow i.e. when  D 0I we must conclude that (103) is not the solution that we seek. However, the equation also possesses a singular solution r 1 1 k./ D  cos  ˙ 1 C  2 ; (104) 2 4 and here we elect to use the upper sign, so that k > 0 (and we see that k D 1 for  D 0). This solution accommodates all the properties that we seek and expect, most particularly the recovery of the circular ring-wave; the wave front,  Dconstant, for

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

177

a given T; is then expressed as rD

1  2

q

1 C 14  2 ; q 1 1 2  cos  C 1 C  2 4 C

(105)

which has been normalised so that r D 1 at  D 0. Three examples of the wave front, based on this description and including the circular case ( D 0), are shown in Fig. 10. The distorting effect of the underlying flow, which is moving from left to right, is quite evident—the wave front is an ellipse—involving downstream propagation on the front edge of the wave front . D 0/; and upstream at the back . D ˙/. There is no critical level at any point below this ring wave (although more general vorticity distributions can exhibit this complication; see Johnson [44]). The O."/ problem generates the equation for 0 (with 1 now arbitrary at this order), because we have chosen to construct the problem in an appropriate far field; cf. Sect. 5.3. This produces, after extensive manipulation, an equation of the form A1 0R C

A2 A3 0 C 0 C A4 0 0 C A5 0   D 0; R R

(106)

where each Ai D Ai ./ is an involved function containing many integrals over z; although careful analysis shows that A3 D 0 for all shear flows. In the case of a ring wave over stationary water (so the wave front is a circle), Eq. (106) becomes 20R C

1 1 0 C 30 0 C 0   D 0; R 3

(107)

the concentric or cylindrical KdV (cKdV) equation (which is completely integrable); see (43) and Sect. 6.3. Fortunately, the absence of the -derivative term in (106) means that this equation can be treated as a version of (107) under a scaling

2 1.5 1 0.5

–5

–4

–3

–2

–1

1

Fig. 10 Shape of the ring-wave wave-fronts (half the profile is plotted) for propagation over constant vorticity. The semi-circular profile is for  D 0; the middle curve:  D 1; outer (largest) curve:  D 2. For each profile, the region near 1 . D 0/ is propagating downstream, and at the other end . D / is propagating upstream

178

R.S. Johnson

transformation (because  now appears only as a parameter), although the equation is only completely integrable if A1 =A2 D 2. In detail, with U.z/ D  z; we find that A5 D 1=3, and 1 A1 D 2 C  2 C  cos I 2 i  h     C 12  cos  .1 C  2 / cos  C  12 C cos2  C 14  3 .1 C cos2 / 1   A2 D I  C  cos  1 C 12  2 C  cos  2 A4 D 3 C 

  1  C  cos  3 C  2 C  cos  ; 2 2

1

p where  D 1 C  2 =4. (The check for the case  D 0 follows directly.) Then the scaling transformation 1=3 O

 D A2

;

 2=3 0 D 3A2 =A4 /O 0 ;

converts (106) into ˛ O 0R C

1 1 O 0 C 3O 0 O 0O C 0O OO D 0; R 3

(108)

where ˛./ D A1 =A2 ; and this is equal to 2 only for  D 0; but otherwise the equation takes the standard form, (107). Equation (108) is not completely integrable, for general  , nor does the usual form of solitary-wave solution exist (based on the integral of the square of the Airy function, Ai ), although a similarity solution— which is not physically realistic—does. This is not the place to begin an extensive discussion and analysis of this equation; suffice it to note that this considerably more complicated flow problem—a ring wave over a shear flow—can be tackled by our techniques. The wave front is readily described (and evident from observations in the field), but we are left with an interesting mathematical problem posed by Eq. (108), for general ˛./.

10 Two Examples: Periodic Waves with Vorticity and Edge Waves We conclude this discussion of nonlinear water waves by examining two rather different types of problem. They both involve a small parameter, so we may invoke our techniques of asymptotic expansions, but they relate only loosely to our previous examples. They do, however, show how broad are the problems that are encompassed by the classical water-wave equations. First, we will examine an application of considerable current interest, from the rigorous viewpoint: the

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

179

propagation of periodic waves in the presence of constant vorticity, and then a rather unusual mode of propagation: edge waves along a beach. Some additional information relating to this approach for these two problems can be found in [58].

10.1 Periodic Waves over Constant Vorticity In recent years, there have been some important developments that have confirmed the existence of periodic waves—not necessarily of small amplitude—with arbitrary vorticity; see, for example, Constantin and Strauss [17, 18] and Constantin and Escher [13]. These wave profiles, and the underlying flow field, can exhibit some quite unusual and extreme properties, not least the appearance of stagnation points (where the speed of propagation of the wave is equal to the speed in the flow at some level below the wave i.e. a critical level exists; see Sect. 9.1). In addition, there has been some illuminating numerical work, for both constant and variable vorticity, by Ko and Strauss [63, 64]. This work has proved the existence of such flows—and Ko and Strauss provide some graphical examples of them—but there is no analytical detail of the structure of the solutions; this is a gap that an asymptotic (parameter) approach might fill. This is not the place to provide all the background information that underpins the formulation—this is available elsewhere—but we will briefly outline how we obtain the relevant equations. In the context of the work presented earlier, it is convenient to start from Eqs. (10)–(12), but we allow propagation only in the x-direction (and there is no dependence on y). Further, we simplify the description by using just one scale length for the non-dimensionalisation, so we set " D ı D 1. The waves that we shall describe are steady, moving at a constant speed c, so we introduce  D x  ct; and then write U.; z/ D u  c. Thus for a wave without stagnation in the flow field, we have U D u  c < 0 everywhere: the wave is moving faster than any point in the body of the fluid, or any point of the surface. Stagnation occurs wherever U D u  c D 0 i.e. the horizontal component of the velocity vector, in this frame, is zero. However, the usual formulation of the problem as adopted by many authors—and we will follow this approach—means that U D 0 cannot be attained (although U ! 0 can still be examined). Equations (10)–(12) now become UU C wUz D p I

Uw C wwz D pz I

U C wz D 0;

with pDh

& w D Uh

on z D h./

and wD0

on z D H;

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R.S. Johnson

where the bottom is given by z D H.< 0/ D constant and h D 0 corresponds to the undisturbed free surface in this description of the problem. Equivalently, the equations imply Bernoulli’s equation which, evaluated on the surface, is U 2 C w2 C 2. C H/ D Q D constant

on z D h./;

where Q is the conserved energy (often called the ‘total head’); we shall find that this is a useful equation in this discussion. The vorticity is Uz  w D  D constant; where the sign . / here is a convenience (chosen to correspond to the sign used in much of the earlier work). It is appropriate, in the light of the transformation that we shall introduce shortly, also to define the stream function, .; z/ W UD

and

z

wD



and to write the total mass flux, in this moving frame, as Z

h./ H

Z .u  c/ dz D

h./ H

U dz D p0 .< 0/ D constant;

and then to use 0   p0 .> 0/; corresponding to h./  z  H. At this stage, we seek a solution in H  z  h./

and

 1 <  < 1:

To proceed, we replace the unknown free surface by a known, fixed, boundary. The transformation used in much of the rigorous work is the Dubreil-Jacotin transformation [29], which uses the variables  and (rather than  and z). We set D.; / D z C H; and then with @=@z  .1=D /.@=@ / and @=@ replaced by @=@  .D =D /.@=@ / (so that U D 1=D and w D D =D ), we find that .1 C D2 /D

 2D D D C D2 D  D  D3 ;

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

181

with 1 C .2D  Q/D2 C D2 D 0

on

D 0;

and DD0

D p0 :

on

The implementation of this approach shows that stagnation cannot be reached, because this corresponds to D ! 1. (This difficulty is avoided in the alternative descriptions given by Ehrnström and Villari [30] and Wahlén [84].) A convenient reformulation, with an associated asymptotic approach in mind, is provided by the rescaling p D D d= !

Q D !q

and

where  D ˙! .! > 0/ is the constant vorticity; this then produces d

d3 D ! 1 .2d d d  d2 d   d2 d

/

(109)

with 1  .q  2! 3=2 d/d2 C ! 1 d2 D 0 on

D 0;

and

d D 0 on

D p0 : (110)

(In the terminology of much of the work cited above, positive vorticity is associated with the upper sign.) This restatement of the problem allows, directly, the analysis of, for example, the case of large !, although we will not follow that route here. Finally, we note that we require a solution (for d.; /) in 0

 p0

     ;

and

this latter being the choice for a periodic solution (and we take the crest to be at  D 0); this is equivalent to regarding the original (single) length scale, , as the physical wavelength. The problem that we address here is one of the examples discussed in [57], where more details can be found. We examine what is, essentially, a rather routine problem: the small-amplitude perturbation of a uniform state, but—as we shall see—this does provide a good example of expansion breakdown, rescaling and matching; see Chap. 4. We first need the uniform-flow solution, for positive vorticity (which produces the most interesting problem in this context), of Eqs. (109)–(110); this is easily seen to be d. / D

p b2



p b  2p0 ;

(111)

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R.S. Johnson

with p p q D b C 2! 3=2 . b  b  2p0 /;

(112)

where we require (for a real solution) b  2p0 . The arbitrary (real) constant, b, is associated with the speed of the uniform flow on the bottom (say) because, from p p U D 1=D D  ! b  2 ; we obtain p p  ! b  2p0

on the bottom.

Let " measure the (non-dimensional) amplitude of the wave that perturbs the uniform state. The main aim here is to find an explicit representation of the structure of the flow field, and to examine the nature of near-stagnation, and so produce details not immediately evident in the rigorous approach (although some of this can be seen in the numerical results). We seek a solution in the form d.; I "/ d0 . / C

1 X

"n dn .; /;

(113)

nD1

where d0 . / is the uniform state defined by (111); we consider " ! 0; at fixed b; p0 ; q and !. In this exercise, we note that the asymptotic parameter does not appear in the governing equations: we have introduced " as the amplitude of the wave-solution that we seek. The asymptotic sequence, f"n g; is the natural one to choose, because a term "1 ; appearing in the equations and surface boundary condition by virtue of (113), automatically generates a term "2 ; and so on. The problem that defines d1 .; / is d1

 3d002 d1 D ! 1 d002 d1 

(114)

with ˚ 2qd00 d1 C 2! 3=2 2d0 d00 d1 C d002 d1 D 0

on

D 0;

and d1 D 0

on

D p0 ;

where d00 D d.d0 /=d . All higher-order terms each satisfy a similar partial differential equation, with solutions that are readily constructed. Equation (114) possesses an exact solution which is periodic (period 2), with a peak at  D 0,

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

and which satisfies the bottom boundary condition (on d1 .; / D p

cos  b2

h 1 p sinh p . b  2 !



183

D p0 ): i p b  2p0 / ;

(115)

with h 1 p i p b tanh p . b  b  2p0 / D p ; 1 ! ! C b!

(116)

which can be interpreted as defining b, for given p0 and !, and then q is given by (112). (Note that Eq. (116) has real solutions for b only if 0 < !=p0  ; where  7:009; which corresponds to b D 2p0 .) Higher-order terms merely add small corrections to the results just presented; indeed, these terms generate the expected higher-harmonics that are driven by cos . The resulting asymptotic expansion is uniformly valid for      and 0   p0 , for any .b  2p0 / fixed as " ! 0I then the first two terms of a uniformly valid expansion give d

p b2



p cos  b  2p0 C " p b2

h 1 p sinh p . b  2 !



i p b  2p0 / ;

(117) subject to (116) and (112). Solutions exist (the rigorous approach tells us this) for b ! 2p0 I this is the case that we now examine. With this choice of parameters in the problem, we see now that the asymptotic expansion given in (117) is not uniformly valid—it ‘breaks down’—if b ! 2p0 ; and then where ! p0 (remember that 2 Œ0; p0 ). That is, the asymptotic ordering of the terms (O.1/; O."/; etc.) no longer holds; this heralds the breakdown of the asymptotic expansion. The precise identification of this breakdown leads to a new scaling and a resulting reformulation of the original problem. The breakdown evident in (117) occurs with the particular choice b  2p0 D O."2 /; and then where  p0 D O."2 /; which corresponds to a region near the bottom boundary of the flow. (Note: the higher-order terms in the p asymptotic expansion are easily confirmed to be no more singular than ."= 2p0  2 /n , as ! p0 ; for each "n dn .) To proceed, we first define b  2p0 D "2 2 . > 0/ and then from (117), for  p0 D O.1/; we obtain d

p 2p0  2

cos   " C " p 2p0  2

i h 1 p sinh p . 2p0  2 ; !

(118)

correct at O."/. As we have just observed, this asymptotic expansion is not uniformly valid for p0 D O."2 /; and so we introduce D p0 "2 .  0/I we then see that this implies d D O."/; and hence we also introduce d D "d. ; I "/.

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R.S. Johnson

The problem for d then becomes "2 2 2 d C d D .2d d d  d d   d d ! with d D 0 on D 0; 3

9 > /; = > ;

(119)

the condition at the surface being accommodated through solution (118) to which d must be matched. We seek a solution of (119) in the form d. ; I "/ d0 . ; / as " ! 0; which gives directly d0 D

p p 2 C A./  A./

where A./  0 is an arbitrary function. We now invoke the matching principle, and apply it to the two-term asymptotic expansion (118) and the one-term expansion d d0 I matching occurs only if 1 A./ D   p cos  !

p so   1= !;

which gives the leading-order solution d

q

2 C .  ! 1=2 cos /2  .  ! 1=2 cos /

(120)

for " ! 0 and D O.1/. (It is straightforward to confirm that (120) is the first term of an asymptotic expansion valid as ! 0. Thus no further asymptotic (scaled) regions are necessary for the description of this problem.) The solution represented by (118) and (120) is appropriate to the parameter ! close to p0 (because b is close to 2p0 ), and so we must have, in this case, p q D 2p0 C 2! 3=2 . 2p0  "/ C O."2 /

as " ! 0.

It is instructive to examine the predictions implied by (120), which is the (asymptotic) solution for d valid near the bed (i.e. on D p0 or D 0). For example, we can investigate if this admits a stagnation point. By virtue of the Dubreil-Jacotin transformation, we obtain p p p q 1 ! " ! UD D  D " ! 2 C .  ! 1=2 cos /2 D "d d0

(121)

to leading order, which shows that indeed a stagnation point is approached on D 0 p (the bed), and then only below the crest . D 0/, if  D 1= ! (where, we note,  is a measure of the difference between the constants b and 2p0 ). An example of

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

185

Fig. 11 An example of the streamline pattern for small-amplitude waves over constant vorticity U –2.5

–2

–1.5

–1

–0.5 0

–0.1

–0.2

–0.3 - psi –0.4

–0.5

–0.6

Fig. 12 The horizontal velocity component, U, in the flow below the crest (corresponding to the solution depicted in Fig. 11) for p0 D 0:6389

a solution with a stagnation point, based on our asymptotic results, is represented by the streamlines (Fig. 11) and the associated behaviour of U on the vertical line below the crest (Fig. 12). We must emphasise that these graphical representations are not expected to correspond to solutions of some specific problem with known (numerical) error. To do this would require the construction of more terms in the asymptotic expansions and then the identification of the number of terms that produce minimum error overall. Rather, we wish to demonstrate that our asymptotic results appear to have captured, in all essential details, the character of this problem.

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R.S. Johnson

10.2 Edge Waves over a Slowly Varying Depth Edge waves are a rather unusual phenomenon: they are waves that propagate along a sloping beach, with an amplitude that decays exponentially away from the beach; they were first reported by Stokes in 1846 [79]. A general introduction to the various theories of edge waves can be found in [55], and the work described here is based on [54]. An example of a typical edge wave is shown in Fig. 13. Our approach is prompted by the work of Constantin [11], where a particularly simple parametric representation of the run-up pattern on the beach was described (for the classical problem of a beach with constant slope). (This is an example of an exact solution of the classical water-wave problem, based on Gerstner’s famous exact solution.) We will employ, in this discussion, a variant of a parameter expansion called the method of multiple scales. The method of formulation here requires a different starting point as compared with all our previous analyses. We introduce a non-dimensionalisation based on just one scale length, so we use Eqs. (6)–(8) with " D ı D 1. The edge waves, which we will assume in this model are simply travelling waves, propagate in the y-direction, which is along the beach, 1 < y < 1; the ocean will exist from the edge of the beach (the boundary being described by the run-up pattern) out to infinity .x ! 1/. The bottom topography, which includes the beach profile, is given by a suitable b D b.x/. Thus we have, initially, the set Du? D r? p; Dt

@p Dw D ; Dt @z

r uD0

(122)

on z D h.x? ; t/,

(123)

with pDh

& wD

@h C .r?  u? /h @t

beach Sea beach dunes run-up pattern

RSJ

Fig. 13 Sketch of an example of edge waves along a gently-shelving beach

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

187

and wDu

db dx

on z D b.x/.

(124)

Note that, in this statement of the problem, the undisturbed surface h.x? ; t/  0; is now at z D 0. Guided by the exact solution written down in [11], we introduce suitable scaled variables consistent with the appearance of the slope in that solution. So we express this in terms of the small parameter "; the magnitude of the slope b0 .0/ and, further, we assume that the depth varies slowly on this scale; thus we shall define the bottom by z D b.x/ D B.X/; X D "x; with B0 .0/ D 1. (The change of sign is merely a convenience.) The appropriate choice of characteristic variable, describing propagation along the beach (in the y-direction), and a ‘slow’ variable used to measure the behaviour of the solution seawards, are p

 D `y  ! "t;

 D"

Z

1

X 0

˛.X 0 I "/dX 0 ;

(125)

respectively, where `.> 0/ is a given wave number in the y-direction, and !.D constant) is to be determined, as is the function ˛.XI "/. In order to formulate the problem of edge waves, we note that, in [11], velocity components, .u; v; w/; are p the p p proportional, correspondingly, to .cos ˇ sin ˇ; sin ˇ; sin ˇ sin ˇ/; where tan ˇ is the uniform slope of the bottom; similarly, p and h are proportional to sin ˇ. With this in mind, u; p and h are transformed (rescaled) according to .u; v; w/ !

p ".u; v; "w/I

. p; h/ ! ". p; h/I

(126)

the resulting non-dimensional, scaled equations [from (122)–(124)], written in the variables (125), are Du D .˛p C "pX /; Dt

Dv D `p ; Dt

"

Dw D pz ; Dt

˛u C `v C "uX C "wz D 0;

(127) (128)

with pDh

& w D !h C ˛uh C `vh C "uhX

on z D "h,

(129)

and w D uB0 .X/

on z D B.X/,

(130)

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R.S. Johnson

where @ @ @ @ @ D  ! C ˛u C `v C "u C "w : Dt @ @ @ @X @z These equations, (127)–(130), provide the basis for our discussion of the problem of edge waves (along a straight beach), interpreted in the limit " ! 0. We seek an asymptotic solution in the usual form, (37), and construct the problem at each order, imposing appropriate uniformity conditions: the behaviour of the solution as jj ! 1; jj ! 1 and X ! 1 (and/or X ! 0). These will be necessary in order to determine completely the earlier terms in the expansions. (This technique of introducing various versions of essentially the same variable, but scaled differently (as with  and X here), with uniformity conditions invoked, is at the heart of the method of multiple scales.) In addition to the expansion of the dependent variables, we must also allow here ˛.XI "/ ˛.XI 0/ C

1 X

"n ˛n .X/;

nD2

where the term "˛1 .X/ is omitted because it can be subsumed into any amplitude function (which, in general, depends on X/I in this particular problem—and exceptionally—we do not also expand the constant ! (but this could be included). The leading-order problem from (127)–(130) then generates the nonlinear set !u0 C ˛0 u0 u0 C `v0 u0 D ˛0 p0 I p0z D 0

and

!v0 C ˛0 u0 v0 C `v0 v0 D `p0 I ˛0 u0 C `v0 D 0;

with p0 D h0

on z D 0I

note that w0 is absent at leading order. This set of shallow-water-type equations has the particular exact solution (selected to correspond to the (linearised) solution found by Stokes, and others) 1 `2 2 2 A e I 2 !2 0 l v0 D A0 e cos ; !

p0 D h0 D A0 .X/e cos  

(131)

l u0 D  A0 e sin I !

(132)

for arbitrary A0 .X/ and !I we have selected ˛0 D l so that  `X=" (i.e. e ! 0 as X ! 1/. This solution is then consistent with the initial surface profile h A0 ."x/e`x cos.`y/ 

`2 2 2`x A e ; 2! 2 0

Asymptotic Methods for Weakly Nonlinear and Other Water Waves

189

for some A0 ."x/. At the next order, we obtain the set of equations  !u1 C ˛0 .u0 u1 / C `.v0 u1 C v1 u0 / C u0 u0X D .˛0 p1 C p0X /I  !v1 C ˛0 .u0 v1 C u1 v0 / C `.v0 v1 / C u0 v0X D `p1 I  !w0 C ˛0 u0 w0 C `v0 w0 D p1z I

˛0 u1 C `v1 C u0X C w0z

D 0;

with p1 D h1

and

w0 D !h0 C ˛0 u0 h0 C `v0 h0

on z D 0,

and w0 D u0 B0 .X/

on z D B.X/.

(The boundary conditions on the surface, z D "h; are rewritten to all orders as evaluations on z D 0 by assuming the existence of Taylor expansions about z D 0; a procedure adopted in our earlier problems.) This set has an appropriate solution on z 2 ŒB.X/; 0; although its form is quite involved; see Johnson [54]. It turns out that the asymptotic expansions are uniformly valid as  ! 1 and jj ! 1 only if A0 B0 C 2BA00 D

!2 A0 : `

(133)

With this condition imposed, the solution for h1 can be written h1 D

 `2 A30 3 `  `3 A30 2 0 2 e e cos  C  4A 0 A0 e cos.2/ 8! 2 B 8! 2 ! 2 B `  0 `3 A30 2  2 2A e A0 e : C  0 4! 2 !2 B

(134)

The uniformity condition, (133), is solved to give 1

n !2 Z

A0 .X/ D p exp 2`2 B.X/

X

dX 0 o ; B.X 0 /

(135)

and the non-uniformity that is evident in h0 C "h1 ; as B ! 0 for general !; is discussed in [54]; it turns out that this imposes a restriction on the modes that can be accessed by this approach.

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To the order at which we have obtained the details, the surface wave is  2 3 `2 2 2 ` A0 3 A e C " e cos  2! 2 0 8! 2 B  `  `3 A30 2 0 2 e  4A C 0 A0 e cos.2/ 8! 2 ! 2 B  `  0 `3 A30 2  2 C 2 2A0  2 e A0 e : 4! ! B

h.; X; I "/ A0 e cos  

We take the shore (beach) to be described by B.X/ X as X ! 0; and so (135) gives A0 .X/ kX ˇ ;

ˇD

 1  !2 1 2 `

as X ! 0;

(136)

where k is a constant which is determined by the amplitude of the wave for some X > 0. If A0 .X/; and all its derivatives, exist as X ! 0; then we require ˇD

 1  !2 1 Dn 2 `

n D 0; 1; 2; : : :;

which recovers the classical result for the modes of linear edge waves (conventionally obtained via the properties of the Laguerre equation). We see that the exponential decay as X (and x/ ! 1 ensures the validity seawards, and is the property that shows that we have a trapped wave here. Our main interest in this brief discussion of this problem is the run-up pattern on the beach; this is given by the intersection of the surface wave with the bottom profile there i.e. z D B.X/ D "h and with B.X/ X as X ! 0; we will take this to be x A0 e cos  

`2 2 2 A e 2! 2 0

(137)

at leading order. Equation (137) therefore describes the run-up pattern at the shoreline, with A0 .X/ given by (136) (for ˇ D n) and  D `X=e D `x. The result expressed by Eq. (137) has been generated by a multiple-scale technique, and so we should treat X;  and  each as O.1/ and independent; however, for the purposes of producing graphical results, we must revert to the original coordinates. Then it is useful to consider a suitable normalised version of this equation: 1 C Z n1 eZ cos  

2 Z 2n1 e2Z D 0; 2.1 C 2n/

(138)

Asymptotic Methods for Weakly Nonlinear and Other Water Waves Fig. 14 A cycloid-like run-up pattern

191

14

12

10

8 longshore coordinate 6

4

2

0

1

seawards --> where Z D `x;  D k"n =`n1 (for n D 1; 2; : : :) and the root Z D 0 has been eliminated. It is quite straightforward to show that solutions exist of this equation that are continuous, bounded and periodic for   n > 0 (for appropriate n ). The solutions for  < n comprise closed curves, spaced periodically, which coalesce for  D n to form two near-cycloids that meet at their cusps; for  > n ; these curves separate to become a pair of curves that correspond to trochoids. These profiles are analogous to the cycloid and trochoid given in [11], although here we have a pair in each case, and either is an acceptable solution— pointing either towards the shore or seawards. An example of a cycloid-like profile .n D 2; 2 D 21  54) is presented in Fig. 14 (this being one of a pair), and a pair of trochoid-like profiles .n D 2;  D 25/; together with closed-curve solutions which are not physically realisable, is shown in Fig. 15. Solutions for  < n do not describe a physically realistic phenomenon.

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14

12

10

8 y 6

4

2

0

1

2

3

4

5

x Fig. 15 A pair of possible solutions, together with closed-contour solutions

0 –2 0 5 10

longshore

15 14

12

10

8

6

4

2

0

seawards

Fig. 16 A three-dimensional plot of an edge wave

This approach can be extended to include a long-wave current or sand bars beyond the beach [54] and, if a background vorticity is added [56], just one of the pair of solutions mentioned above is selected. A typical edge wave, presented as a surface plot which depicts the run-up pattern, is shown in Fig. 16.

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This concludes our examination of two examples in classical water-wave theory, each of which demonstrates the broad spectrum of problems that can be accessed by the asymptotic approach.

References 1. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Notes, vol. 149. Cambridge University Press, Cambridge (1991) 2. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1981) 3. Anker, D., Freeman, N.C.: On the soliton solutions of the Davey-Stewartson equations for long waves. Proc. R. Soc. Lond. A 360, 529–540 (1978) 4. Barnett, T.P., Kenyon, K.E.: Recent advances in the study of wind waves. Rep. Prog. Phys. 38, 667–729 (1975) 5. Benjamin, T.B.: The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97–116 (1962) 6. Benney, D.J., Bergeron, R.F.: A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48, 181–204 (1969) 7. Boussinesq, J.: Théorie de l’intumescence liquid appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus Acad. Sci. (Paris) 72, 755–759 (1871) 8. Burns, J.C.: Long waves on running water. Proc. Camb. Philos. Soc. 49, 695–706 (1953) 9. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) 10. Camassa, R., Holm, D.D., Hyman, J.: An integrable shallow water equation with peaked solitons. Adv. Appl. Mech. 31, 1–33 (1994) 11. Constantin, A.: Edge waves along a sloping beach. J. Phys. A Math. Gen. 34, 9723–9731 (2001) 12. Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001) 13. Constantin, A., Escher, J.: Analyticity of periodic travelling free surface waves with vorticity. Ann. Math. 173, 559–568 (2011) 14. Constantin, A., Johnson, R.S.: On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations of water waves. J. Nonlinear Math. Phys. 15(2), 58–73 (2008) 15. Constantin, A., Johnson, R.S.: Propagation of very long waves, with vorticity, over variable depth, with applications to tsunamis. Fluid Dyn. Res. 40, 175–211 (2008) 16. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Rat. Mech. Anal. 192, 165–186 (2009) 17. Constantin, A., Strauss, W.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527 (2004) 18. Constantin, A., Strauss, W.: Rotational steady water waves near stagnation. Philos. Trans. R. Soc. Lond. A 365, 2227–2239 (2007) 19. Copson, E.T.: Asymptotic Expansions. Cambridge University Press, Cambridge (1967) 20. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1967) 21. Crapper, G.D.: Introduction to Water Waves. Ellis Horwood, Chichester (1984) 22. da Silva, A.F.T., Peregrine, D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)

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23. Davey, A., Stewartson, K.: On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110 (1974) 24. Dean, R.G., Dalrymple, R.A.: Water Wave Mechanics for Engineers and Scientists. World Scientific, Singapore (1984) 25. Debnath, L.: Nonlinear Water Waves. Academic, London (1994) 26. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999) 27. Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1461–1472 (2002) 28. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction, 2nd edn. Cambridge University Press, Cambridge (1992) 29. Dubreil-Jacotin, M.-L.: Sur le détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pure Appl. 13, 217–291 (1934) 30. Ehrnström, M., Villari, G.: Linear water waves with vorticity: rotational features and particle paths. J. Differ. Equ. 244, 1888–1909 (2008) 31. Fermi, E, Pasta, J., Ulam, S.M.: Studies in nonlinear problems. Technical Report, LA-1940, Los Alamos Science Laboratory, New Mexico (1955) (Also in Newell, A.C. (ed.) (1979), Nonlinear wave motion, American Mathematical Society, Providence, RI 2008) 32. Ford, W.B.: Divergent Series, Summability and Asymptotics. Chelsea, New York (1960) 33. Freeman, N.C., Davey, A.: On the evolution of packets of long waves. Proc. R. Soc. Lond. A 344, 427–433 (1975) 34. Freeman, N.C., Johnson, R.S.: Shallow water waves on shear flows. J. Fluid Mech. 42, 401–409 (1970) 35. Garabedian, P.R.: Partial Differential Equations. Wiley, New York (1964) 36. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967) 37. Hardy, G.H.: Divergent Series. Clarendon, Oxford (1949) 38. Hasimoto, H., Ono, H.: Nonlinear modulation of gravity waves. J. Phys. Soc. Jpn. 33, 805–811 (1972) 39. Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. I KdVtype bilinear equations. J. Math. Phys. 28, 1732–1742 (1987) 40. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971) 41. Holmes, M.H.: Introduction to Perturbation Methods. Springer, New York (1955) 42. Johnson, R.S.: On the modulation of water waves on shear flows. Proc. R. Soc. Lond. A 347, 537–546 (1976) 43. Johnson, R.S.: On the nonlinear critical layer below a nonlinear unsteady surface wave. J. Fluid Mech. 167, 327–351 (1986) 44. Johnson, R.S.: Ring waves on the surface of shear flows: a linear and nonlinear theory. J. Fluid Mech. 215, 145–160 (1990) 45. Johnson, R.S.: On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow, with or without a critical layer). Geophys. Astrophys. Fluid Dyn. 57, 115–133 (1991) 46. Johnson, R.S.: Solitary wave, soliton and shelf evolution over variable depth. J. Fluid Mech. 276, 125–138 (1994) 47. Johnson, R.S.: A two-dimensional Boussinesq equation for water waves and some of its solutions. J. Fluid Mech. 323, 65–78 (1996) 48. Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997) 49. Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 457, 63–82 (2002) 50. Johnson, R.S.: On solutions of the Camassa-Holm equation, Proc. R. Soc. Lond. A459, 1687– 1708 (2003)

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51. Johnson, R.S.: The Camassa-Holm equation for water waves moving over a shear flow. Fluid Dyn. Res. 33, 97–1111 (2003) 52. Johnson, R.S.: The classical problem of water waves: a reservoir of integrable and nearlyintegrable equations. J. Nonlinear Math. Phys. 10, 72–92 (2003) 53. Johnson, R.S.: Singular Perturbation Theory. Springer, New York (2004) 54. Johnson, R.S.: Some contributions to the theory of edge waves. J. Fluid Mech. 524, 81–97 (2005) 55. Johnson, R.S.: Edge waves: theories past and present. Philos. Trans. R. Soc. Lond. A365, 2359– 2376 (2007) 56. Johnson, R.S.: Water waves near a shoreline in a flow with vorticity: two classical examples. J. Nonlinear Math. Phys. 15, 133–156 (2008) 57. Johnson, R.S.: Periodic waves over constant vorticity: some asymptotic results generated by parameter expansions. Wave Motion 46, 339–349 (2009) 58. Johnson, R.S.: A selection of nonlinear problems in water waves, analysed by perturbationparameter techniques. Commun. Pure Appl. Math. 11, 1497–1522 (2012) 59. Kadomtsev, B.P., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970) 60. Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer, New York (1996) 61. Knickerbocker, C.J., Newell, A.C.: Shelves and the Korteweg-de Vries equation. J. Fluid Mech. 98, 803–818 (1980) 62. Knickerbocker, C.J., Newell, A.C.: Reflections from solitary waves in channels of decreasing depth. J. Fluid Mech. 153, 1–16 (1985) 63. Ko, J., Strauss, W.: Large-amplitude steady rotational water waves. Eur. J. Mech. B Fluids 27, 96–109 (2008) 64. Ko, J., Strauss, W.: Effect of vorticity on steady water waves. J. Fluid Mech. 608, 197–215 (2008) 65. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 5, 422–443 (1895) 66. Lighthill, M.J.: Waves in Fluids. Cambridge University Press, Cambridge (1978) 67. Longuet-Higgins, M.S.: On mass, momentum, energy and circulation of a solitary wave. Proc. R. Soc. A 337, 1–13 (1974) 68. Longuet-Higgins, M.S.: Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. A 342, 157–174 (1975) 69. Longuet-Higgins, M.S., Cokelet, E.D.: The deformation of steep surface waves on water I. A numerical method of computation. Proc. R. Soc. 350, 1–26 (1976) 70. Longuet-Higgins, M.S., Fenton, J.: Mass, momentum, energy and circulation of a solitary wave II. Proc. R. Soc. 340, 471–493 (1974) 71. McCowan, J.: On the solitary wave. Philos. Mag. 5, 45–58 (1891) 72. Mei, C.C.: The Applied Dynamics of Ocean Surface Waves. World Scientific, Singapore (1989) 73. Miles, J.W.: Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169 (1977) 74. Miles, J.W.: On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190 (1979) 75. Russell, J.S.: Report on waves. Report of the 14th Meeting of the British Association for the Advancement of Science, New York, pp. 311–390. John Murray, London (1844) 76. Smith, D.R.: Singular-Perturbation Theory: An Introduction with Applications. Cambridge University Press, Cambridge (1985) 77. Starr, V.T.: Momentum and energy integrals for gravity waves of finite height. J. Mar. Res. 6, 175–193 (1947) 78. Stoker, J.J.: Water Waves. Interscience, New York (1957) 79. Stokes, G.G.: Report on recent researches in hydrodynamics. Report of the 14th Meeting of the British Association for the Advancement of Science, pp. 1–20 (1846) [Also Maths. Phys. Papers I, pp. 157–187. Cambridge University Press, Cambridge (1880)]

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80. Stokes, G.G.: Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. Phys. Papers I 1, 225–228 (1880) 81. van Dyke, M.D.: Perturbation Methods in Fluid Mechanics. Academic, New York (1964) (Also, with annotations, The Parabolic Press, 1975) 82. Varvaruca, E.: On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differ. Equ. 246, 4043–4076 (2009) 83. Varvaruca, E., Weiss, G.S.: A geometric approach to generalized Stokes conjectures. Acta Math. 206, 363–403 (2011) 84. Wahlén, E.: Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483 (2009) 85. Zabusky, N.J., Kruskal, M.D.: Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965) 86. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional selffocussing and onedimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972) 87. Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem I. Funct. Anal. Appl. 8, 226–235 (1974)

A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems Gabriele Villari

Abstract The aim of this note is to show how phase plane analysis is a strong tool for the study of a mathematical model, in view of its application in water waves theory. This because only in recent work such method was actually used in water waves theory and people working in this field area might be interested in a discussion of the basic ideas of phase plane analysis, which we call “a survival kit”. In this light, at first we review some classical results in Dynamics of Population and Epidemiology, and then we investigate more carefully the phase portrait of the classical Liénard equation. In particular, starting from the Van Der Pol equation, the problem of existence and uniqueness of limit cycles will be treated and the methods used to attack this problem will be presented. Finally we come back to water waves theory and present in details the results of a joint paper with Constantin (Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, Wiley, New York, 1977) in which, as far as we know, for the first time phase plane analysis was used in this kind of problems.

1 Dynamics of Population Starting from the well known logistic equation, we will briefly discuss the competition between two species in an ecological niche and examine in more details the classical predator-prey model.

2 The Logistic Equation The simplest equation which models the growth of an isolated population is xP D "x

(1)

G. Villari () Department of Mathematics and Informatics “Ulisse Dini” University of Florence, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2016 A. Constantin (ed.), Nonlinear Water Waves, Lecture Notes in Mathematics 2158, DOI 10.1007/978-3-319-31462-4_4

197

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G. Villari

Clearly in this case the rate of growth is proportional to the size of the population, and actually the growth rate is the positive coefficient ". But this model it is not realistic, because gives an exponential growth. We must take in account limited resources as well as limited space. Hence, it is necessary an inhibitory effect when the population is large, and this leads to the so called logistic equation xP D "x  x2

(2)

which was proposed by the Belgian mathematician Pierre Francois Verhulst, who in 1838 introduced the solution of such equation in the theory of dynamics of population [33]. Here the inhibitory effect is given by the negative coefficient  . We can easily solve this equation, being actually a Bernoulli equation, but we prefer a simpler and qualitative approach. We just study the sign of the right member, namely x."  x/; it is trivial to see that x D 0 and x D "= are constant solutions, while for 0 < x < "= solutions are monotone increasing and monotone decreasing for x > "= . Hence we proved that the logistic equation has an asymptotically stable solution at "= , and we know the qualitative behavior of all its solutions, without actually having solved the equation. The asymptotically stable solution x D "= is called saturation level or total carrying capacity. For small values of x the curve solution is S-shaped and it is called sigmoid (Fig. 1). There are numerous experimental data for the growth of protozoa and bacteria that fit a sigmoid curve (see for instance the data concerning the Glaucoma Scintillans or the Paramecium Aurelia, cfr. Gause [12]) (Fig. 2). Fig. 1 The logistic equation

5,2 2,6

300 200

Glaucoma scintillans N1

199

2. K

b,=4,625 100

individus en 0,5c.c.

7,8

individus

Volume

A Survival Kit in Phase Plane Analysis

8 6

K

P.aurelia N2

1.

b2=0,772

4 2

jours 3

4

5

6

7

Fig. 2 Data fitting the model for Glaucoma Scintillans and P. Aurelia

3 Two Species in Competition and the “Principle of Competitive Exclusion” The next step is the study of the differential system (

xP D "1 x  1 x2  ˛1 xy D x."1  1 x  ˛1 y/ yP D "2 y  2 y2  ˛2 xy D y."2  2 y  ˛2 x/

(3)

which describes the competition between two similar species for a limited resources as, for instance, food supply. We assume also that the two species share a limited territory. This environmental situation is called ecological niche. The inhibitory effect to the growth of a species due to the presence of the other is given by the coefficients ˛i . In order to understand what will happen we must study the phase portrait of this planar nonlinear dynamical system in the first quadrant. As usual we consider the critical points, the 0-isocline (that is the points in which yP D 0) and the 1-isocline (that is the points in which xP D 0), because they play a crucial role in the construction of the phase portrait. At first we consider a phase portrait with three singular points, namely the origin, the point A."1 = 1 ; 0/ which gives the saturation level of the first species in absence of the second one, and the point B.0; "2 = 2 / which gives the saturation level of the second species in absence of the first one. The 0-isocline is formed by the x-axis and the segment joining the points A."1 = 1 ; 0/ and C.0; "1 =˛1 /, while the 1-isocline is formed by the y-axis and the segment joining the points B.0; "2 = 2 / and D."2 =˛2 ; 0/ (Fig. 3). Under the condition that we have three singular points, these two segments do not intersect each other and divide the first quadrant in three regions. It easy to see, with standard sign computations, that the !-limit of any positive trajectory in the first quadrant is the singular point on the external segment. This means that only a species will survive tending to his saturation level, while the other must die out. This phenomenon is known as the “Principle of competitive exclusion”. It was first enunciated, in a slight different form, by Darwin in 1859 in his celebrated paper “The origin of species by natural selection” [7], and it is a kind of axiom in ecology. But if the above defined two segments intersect each other at a point P, it appears a fourth singular point P.

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B

B

C

C

O

A

O

D

A

D

Fig. 3 The principle of exclusion. Only a species will survive, while the other must die out

B

C P

O

D

A

Fig. 4 A case in which only one species will survive depending from the initial conditions. But ecologists do not like it

In order to understand what it is going on in this new situation it is necessary to investigate the stability of the point P. Using a standard linearization argument and after some straightforward calculations which we omit for sake of simplicity (a survival kit, by definition, cannot be “heavy”) it is possible to see that if 1 2 > a1 a2 the point P is asymptotically stable, and this leads to coexistence. On the other hand if 1 2 > a1 a2 the point P is a saddle, and the separatrices of the saddle divide the first quadrant in two regions. Coexistence it is not possible and in one region the first species will survive and the other must die out, while in the other region the second species will survive, and the first must die out (Fig. 4). However coexistence, as well as the existence of different regions of supremacy, is in contrast with the “Principle of competitive exclusion” and therefore it is not accepted by the ecologists. How can we fix this problem? We just observe that this is a problem, because the validation of a mathematical model depends on his applicability in the study of the real phenomenon. For this reason the first simple model was not acceptable because

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was giving an exponential growth of the population, while the logistic equation his supported by many experimental data. The answer to this question lies in the fact that we consider two species which are similar! “Mathematically” speaking this leads to the fact that the coefficients "i ; i and ai are similar and therefore the two segments joining respectively the points A."1 = 1 ; 0/ and C.0; "1 =˛1 / and the points B.0; "2 = 2 / and D."2 =˛2 ; 0/ tends to be parallel and hence do not intersect. In the model of the ecological niche the fourth singular point does not exist and the “Principle of competitive exclusion” has a rigorous mathematical proof, while if we consider the interaction between two species which are not similar the fourth singular point may arise leading the model to coexistence or different regions of supremacy.

4 The Predator-Prey Model This model plays a crucial role in the theory of dynamics of population and his surprising outcome was giving a decisive contribute in showing the importance of the application of phase portrait analysis in a mathematical model of a real phenomenon. We can say that after this intriguing result the whole theory of mathematical models was finally recognized in his importance and applicability, and for this reason we present this model in more details, giving also a brief story of the problem. Right after the World War I, the Italian biologist Umberto D’Ancona was studying the population variation of varies species of fishes that were interacting each other in the Mediterranean Sea. He observed the total catch of selachians (as for instance Sharks or similar fishes) versus the total catch of food fish, taking some data from the port of Fiume (a city at that time in Italy and now in Croatia with the name of Rijeka). There was a clear large increase in the percentage of selachians during the war, as well as a decrease in the percentage of the food fish. How this strange phenomenon could be explained? As selachians are predators and their prey was the food fish, we are in front of a struggle for life between competing species. It was reasonable to assume the fact that the war was reducing fishing and therefore the number of selachians was actually increasing. But why this is not happening for the prey, namely the food fish. And this is an important question because the food fish is actually also our food. No one, with very few exceptions, likes to eat sharks. The father in low of D’Ancona was the famous Italian mathematician Vito Volterra, and was natural to discuss this issue with him. And now it is time to present his model, which appeared in 1927 not in a mathematical journal but in a journal of marine biology [39]. We also note that in the introduction of this paper Volterra gives probably the best definition of a mathematical model and why this leads to differential equations.

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Consider the following system, in the HP-plane (

PP D bP C ˇHP P D aH  ˛HP H

(4)

where P denotes the predator and H the prey and the coefficients ˛; ˇ; ; ı are all positive. Let us briefly discuss this system. In absence of predator the prey grows without limit, because there is not a logistic term. This is not a problem, because the number of prey is reduced by the predators with the rate of predation given by the coefficient ˛. On the other hand, in absence of prey, the predator die out, but this cannot occur in virtue of the positive coefficient ˇ which measures the rate of growth due to the predation. We observe that actually ˇ D  ˛, were  is the so called “biological gain”. There is a clear difference if the struggle for life involves lions and zebras instead of whales and plankton. Again, we consider the critical points, the 0-isocline and the 1-isocline. There are two critical points: the origin and the point E.b=ˇ; a=˛/. The 0-isocline is formed by the H-axis and the line H D b=ˇ. The 1-isocline is formed by the y-axis and the line P D a=˛ . Clearly such lines intersect each other at the point E.b=ˇ; a=˛/ and trajectories are counterclockwise around it. Clearly the origin is a saddle points and, after a linearization, the point E.b=ˇ; a=˛/ is a center. This is a problem, because it is well known in the theory of linearization of almost linear systems that if the linearization gives a center any arbitrary small perturbation can change the stability. This means that the point can actually remain a center, but can became asymptotically stable, or asymptotically unstable as well. This fact will play a crucial role when we will prove that for a linear water wave no particle trajectory is closed unless the free surface is flat, which is the main motivation of these notes. Coming back to our system, this problem was actually solved by Volterra with a very elegant and smart proof which we omit, in order to keep the survival kit “light”. Hence we can say that all the trajectories in the first quadrant are cycles and therefore periodic. This result is interesting, but in principle it can be inferred without the help of phase portrait analysis, arguing as follows. Let us start at a generic point, which gives a certain amount of predators and prey (clearly such amount is to be considered in terms of “biomass”, otherwise one must explain what p  predators or 2 preys mean!). Predators are eating the preys and in this way reducing their number. But in this way at a certain moment there will be a lack of prey. Predators are starving and, with no more food, eventually will die out. This will reduce the number of predators and making the life of the food fish more “easy”. For this reason, assuming that the prey does not compete for the food supply since this is abundant, the prey will increase. Now the predators are in a smaller number and their preys are increasing. It is again easy for them to prey and in this way the number of predators increases again. One may argue that this will give a periodicity in the number of predators and preys. The mathematical analysis of their mutual

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interaction is just a confirmation. But for a more accurate analysis of the evolution of the model we must come back to the system. We consider any periodic solution, with a generic period T, and define the mean b namely value b P and H, Z

bD 1 H T

T 0

1 b PD T

.t/dt;

Z

T

P.t/dt:

(5)

0

From the equation PP D bP C ˇHP we can evaluate Z

T 0

Z

Z T P P.t/ dt D bT C ˇ H.t/dt: P.t/ 0

P P.t/ dt D ln P.T/ln P.0/ D 0, because we are on a T-periodic solution, 0 P.t/ Z T we get that bT D ˇ H.t/dt and therefore that T

Being

0

1 b D ˇ T

Z

T

b H.t/dt D H:

0

a With a same argument one can see that b PD . ˛ b and b Hence we proved that the mean values H P do not depend from the period and are actually the coordinates of the critical point E.b=ˇ; a=˛/. This last result should not surprise, because once that we proved that the mean values do not depend from the choice of the periodic solution we can read the singular point as a periodic solution and clearly one has bD 1 H T

Z

T 0

1 H.t/ dt D T

Z

T 0

b b dt D ˇ ˇ

and 1 b PD T

Z

T 0

P.t/ dt D b PD

1 T

Z

T 0

a a dt D : ˛ ˛

as well. We just observe that actually the biologist D’Ancona founded a kind of mean values in the port of Fiume. Now we are going to apply this important result. Assume that the condition of life becomes harder for both species, as for instance when fishing takes place, because in principle we cannot choose the kind of fish that

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we will get. the system becomes: (

PP D bP C ˇHP  BP P D aH  ˛HP  AH H

(6)

being A and B the positive coefficients which measures the effect of fishing. This gives (

PP D .b C B/P C ˇHP P D .a C A/H  ˛HP H

(7)

and we can study this new system as before, getting that the new coordinates of bCB aA  b increases, while b the new critical point E are ; . We get that H P ˇ ˛ decreases. Hence Volterra proved this crucial result, which is known as “Volterra’s effect”: conditions more difficult advantage the prey! And vice versa when, because of the war, a lot of people was obliged to stop fishing, this produced an increasing of the predators. The validation of the model was soon given by the fact that in the following years the percentage of selachians and food fish returned to the expected levels. As already mentioned, this spectacular result was giving a strong impulse to the whole theory of mathematical models (Figs. 5 and 6). Fig. 5 The classical predator-prey model

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Fig. 6 Hard condition of life: the Volterra’s effect

Finally we consider a situation in which the evolution equation of the prey has a logistic term. The system becomes (

PP D bP C ˇHP P D aH  ˛HP  H 2 H

(8)

and we get a new critical point Q.a= ; 0/ on the H-axis. We do not go in details, but we just note that it is not difficult to prove the following. If a= > b=ˇ the singular point R still is present but changes its stability becoming asymptotically stable. This means that in a long term we will have coexistence. Observe that if we consider fishes and ocean, it is natural to assume that in this situation a= must be very large, and therefore a= > b=ˇ. There are several predator-prey models in which the logistic term must be considered and actually we can have that a= < b=ˇ. Now the evolution of the system changes dramatically. The critical point R disappears and eventually the predator must die out, just because there is not enough food for survive. And the prey will tend to the saturation level. It is not necessary to stress the fact that this scenario is very important in biology and gives a clear support to the ecologists that remind us how dangerous is to change the environment.

5 A Mathematical Descriptions of Epidemics We are now presenting some mathematical models in epidemiology. Following a work of Iannelli [17], we can say that “A main point to stress when introducing mathematical models of epidemics (and of any other population phenomenon as

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well) is that we must consider oversimplified phenomenological situations, so that the basic mechanism that we want to investigate can be looked at freed from the encumbrance of a highly detailed description which – though justified by the need of realism – would actually lead to impossible mathematics and/or to the hiding of the essential features of the phenomenon” Nevertheless we observe that often simple models work as fundamental prototypes which show the basic aspect of the phenomena. This is actually the case of the two models, namely the SIS model and the SIR model, that we are going to discuss. At first we consider an isolated population and we assume that the size of such population is constant, say of N individuals. This is not restrictive as it can appear: indeed for epidemics usually one consider “the rapid spread of infectious disease to a large number of persons in a given population within a short period of time, usually two weeks or less”. And in a short period of time the population of Florence, as well as the population of London or New York can clearly considered as a constant. The spread of an epidemic is usually described by dividing the population into three main subclasses: – Susceptible: those individuals who are not sick and can be infected – Infective: those individuals who have the disease and can transmit it to others – Removed: those individuals who have been infective and now are immune, or dead (or isolated). If we call S.t/; I.t/; R.t/ respectively the number of individuals that, at the time t, belong to the above defined subclasses, we get the relation S.t/ C I.t/ C R.t/ D N

(9)

The SIS mode describes a disease that is not lethal and does not gives immunity, therefore a susceptible individual can be infected, but eventually recovers and becomes again susceptible, namely S.t/ ! I.t/ ! S.t/ and we do not have the class R.t/. In general the pathogen is a bacterium, as for instance in the case of gonorrhea, because a virus gives immunity. But we include in this model also common cold and flu, because the virus causing this kind of disease is unstable and roughly speaking changes is structure. Therefore the acquired immunity does not “work” for the new virus. The evolution system for the SIS model is the following (

SP D  IS C ıI IP D  IS  ıI

(10)

where  measures the rate at which susceptible catch the disease entering in contact with individuals of the infective class (this can occur just in a bus if the pathogen is resistant in the contact with air as the case of common cold, or in a sexual relationship if the pathogen is fleeting, as in the case of gonorrhea), while ı measures the rate at which infective recover and leave their class. Being R.t/ D 0, from (14)

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Fig. 7 A confirmation of the SIS model for the gonorrhea. One can see a relevant increase of infected in the women curve after year 1968. The reason is not what you might think, namely the sexual revolution, but it is due to the fact that in most cases gonorrhea is asymptomatic in women and after 1968 the medical checks for this kind of disease, as well as for most venereal diseases, drastically increased

we have that S.t/ D N  I.t/, and (16) becomes IP D C I.N  I/  ıI D . N  ı/I   I 2

(11)

If  N  ı  0 the disease cannot spread and epidemic goes extinct. If  N  ı > 0 Eq. (17) is just the logistic equation which has been previous presented. The saturation level, namely N  ı= is now called endemic level. It is crucial to take any possible action in order to reduce such level, or even to have N  ı= so that the disease cannot spread. Being N fixed we must work on the ratio ı= in order to have it as large as possible. This can be done increasing ı and reducing  . The meaning of this is very clear: more prevention and better treatments! It is amazing to see how this oversimplified model can give us what we expect (Fig. 7). On the other hand the SIR model describes a disease that is or lethal, as for instance HIV infection, or gives immunity, as for instance measles, mumps and other childhood diseases. A susceptible individual can be infected, but eventually or dies out or recovers with acquired immunity, namely S.t/ ! I.t/ ! R.t/. The evolution system for the SIR model is the following 8 ˆ P ˆ < S D  IS IP D  IS  ıI ˆ ˆ : RP D ıI

(12)

where  is defined as before, while ı is now called the removal rate. As far as we know, this model was presented by Kermack and McKendrick [18] when they were monitoring the pest which occurred in Bombay in 1926. We can study only the planar system given by the first two equations, because once that we

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solve the system, knowing S.t/ and I.t/, we can find R.t/ in virtue of (14). The main difference of this system compared with the previous ones lies in the fact that now we do not have isolated critical points, but the set of critical points is given by the whole positive x-axis. This is not a big issue because if an epidemics is occurring there must be infected individuals and we do not consider point of the positive x-axis as initial points. Is easy to see that S.t/ is always decreasing, and that, a part from the positive x-axis, the 0-isocline is given by the line x D ı= . This gives the fact that I.t/ increases for x > ı= and decreases for x < ı= . Therefore if we consider the positive semi-trajectory starting from the initial point P.x; y/ with x > ı= , we will intersect the 0-isocline at a level which is called “acme” of the epidemic, because it gives the maximum number of infected individuals. From this moment I.t/ decreases and eventually tends to 0: the epidemics goes extinct, and with some more not difficult calculation it is possible to see that, as expected, at the end of the epidemics the population is not depleted. On the other hand, if we consider the positive semi-trajectory starting from the initial point P.x; y/ with x < ı=; I.t/ decreases and eventually tends to 0. Arguing as before it is necessary to have N  ı= , and again this happens increasing ı and reducing  . But now there is a significative difference. We can always reduce  , that is to have more prevention, but we can increase ı only if we are in the case of a disease which gives immunity, because this means as before better treatments. In case of a lethal disease increasing ı, means to kill the infected individuals, and clearly this is not acceptable! But if the disease affects animals instead of humans, this is precisely what we are doing. In this light consider how was defeated the MCD (mad cow disease), or how is fighted the virus HPAI (high pathogenic avian influenza).

6 Phase Portrait of the Liénard Equation: Existence and Uniqueness of Limit Cycles We now discuss the problem of existence of periodic solutions for the Liénard equation xR C f .x/Px C g.x/ D 0

(13)

Such a problem has been widely investigated since the first results of Alfred-Marie Liénard, a French physicist and engineer, appeared in 1928 [21] and there is an enormous quantity of papers on this topic. It is well known that Liénard equation is equivalent to the system (

xP D y yP D f .x/y  g.x/

(14)

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in the phase plane, and to the system (

xP D y  F.x/ yP D g.x/

(15)

Rx in the Liénard plane, where F.x/ D 0 f .x/dx. For this reason, the problem of the existence of periodic solutions is bring back to a problem of existence of limit cycles for the previous systems. Among the existence results until 1960, the classical theorems of Filippov [11], Levinson-Smith [20] and Dragilev [8] may be considered as milestones, while in the last decades the number of results is dramatically increasing. All the results are based on the classical Poincaré-Bendixson Theorem, and in order to fulfill the assumptions of this theorem, once that one has proved that there is a unique singular point, which is unstable, it is necessary to produce a winding trajectory large enough. Let us discuss this problem in more details, even if we understand that in this way our “survival kit” will be no more “light”. The standard assumptions for this kind of problem are the following: – f is continuous, and g is locally Lipschitz continuous. This guarantees existence and uniqueness of solutions for system (15), and hence also for system (14). – xg.x/ > 0 for all x ¤ 0. Therefore the origin is the unique singular point and trajectories are clockwise. At first we consider the well known and paradigmatic example given by the Van Der Pol equation xR C .x2  1/Px C x D 0 with  > 0

(16)

which was introduced in 1926 by Dutch electrical engineer and physicist Balthasar Van der Pol while he was working at Philips [32]. Van der Pol found stable oscillations, which he called relaxation-oscillations, in electrical circuits employing vacuum tubes. This is actually the starting point of the whole theory of nonlinear oscillations! In the phase plane equation (16) is written as (

xP D y yP D .x2  1/y  x

(17)

We are going to prove the existence of at least a stable limit cycle, and later we will prove its uniqueness. Clearly the standard assumptions previously introduced are fulfilled, and once again we consider the critical point, the 0-isocline and the 1-isocline, because as already mentioned they play a crucial role in the construction of the phase portrait. The 1-isocline is given by the x-axis, while the 0-isocline is given by the function H.x/ D  .x2x1/ (Fig. 8).

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Fig. 8 The 0-isocline and the 1-isocline for the Van der Pol equation in the phase plane

In order to discuss the stability of the origin we consider the function D.x; y/ which gives the square distance of a generic point P.x; y/ from the origin, namely D.x; y/ D x2 C y2 . The rate of change of the distance is given by P y/ D 1 xPx C 1 yPy D  1 .x2  1/y2 D.x; 2 2 2

(18)

therefore in the strip given by jxj < 1 the distance from the origin is increasing, while outside the strip is decreasing, and thus we have that the origin is a source. In order to produce a winding trajectory the 0-isocline will play a crucial role. In this light let  be the graph of the function H.x/ D  .x2x1/ and take a point   P x;  .x2x1/ with x < 1. It is easy to see that the negative semi-trajectory intersecting the point P lies above  and therefore is bounded away from the x-axis. On the other hand the positive semi-trajectory intersects at first the line x D 1 in a point Q, and then again  in 0 < x < 1. This because the slope of such trajectory, namely y0 .x; y/ D .x2  1/ 

x y

(19)

is bounded in the strip given by jxj < 1 for y bounded away from 0, and therefore there are no vertical asymptotes. Now there are two possibilities: or the trajectory intersects the x-axis in x < 1 and then the y-axis in y < 0 or intersects the line x D 1, again because there are no vertical asymptotes. In the latter case the trajectory intersects the x-axis in x > 1 because, as already noted, in jxj > 1 the distance from the origin is decreasing, and moves in y < 0 without intersecting again  . This because, arguing as above one can see that the negative semi-trajectory intersecting  in x > lies below  and therefore are bounded away from the x-axis. Hence we proved that in both cases the positive semi-trajectory intersecting P eventually intersects the y-axis in y < 0. Now we

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y 3

y

2 6

1

5 –3

–2 –1

1

2

3

x 4

–1

3

–2

2 –3

1

m = 0.1 –2

y

–1

0

3

–1

2

–2

1

2

3

x

–3

1

–4 –3

–2 –1

0 –1 –2 –3

1

2

3 x

–5 –6 –7 m =1

m =1 Fig. 9 Limit cycles for the Van der Pol equation [1, p. 448]. As expected, for  small the phase portrait is similar to the one of the harmonic oscillator

have again two possibilities: or such trajectory intersects the x-axis in 1 < x < 0, and hence it is winding, or intersects the x D 1. Using the fact that the distance from the origin is decreasing we see that the trajectory intersects the x-axis in x < 1 and moves in y > 0. Being bounded away from  the trajectory intersects the line x D 1 in a point R below Q and hence it is winding as well. Being the origin a source we can apply the Poincaré-Bendixson Theorem and get the existence of at least a stable limit cycle (Fig. 9). We can obtain the same result with a different approach.

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Observe that Z

ˇ

y.ˇ/  y.˛/ D ˛

y0 .x/ dx D

Z

ˇ ˛

.x2  1/ 

x dx y.x/ Z

It is possible to choose ˛ < 1 < 1 < ˇ such that

ˇ ˛

(20)

.x2  1/ dx > 0

and if y.x/ is large enough we get that in the interval Œ˛; ˇ the quantity .x2  x 1/ C y.x/ is positive. Therefore, if we consider a point P.˛; y1 / with y1 large enough and define as  C .P/ the positive semi-trajectory starting from P, we can see in virtue of (20) that  C .P/ intersects the line x D ˇ in a point Q.ˇ; y2 / with 0 < y2 < y1 . This because if  C .P/ intersects the x-axis before x D ˇ, we can consider 0 < y2 < y1 and the negative semi-trajectory starting from Q.ˇ; y2 /, namely   .Q/; clearly   .Q/ intersects the line x D ˛ in a point P.˛; y1 / with y2 < y1 . Now we consider  C .Q/. We know that in jxj > 1 the distance from the origin is decreasing, hence  C .Q/ intersects the line x D ˇ in y < 0 at a point R.ˇ; y3 / with jy3 j < y2 . Now we consider  C .R/: arguing as above we can prove that  C .R/ intersects the line x D ˛ in a point S.˛; y4 / with y3 < y4 < 0. Finally we consider  C .S/. We are again in jxj > 1, and the distance from the origin is decreasing. Therefore  C .S/ intersects the line x D ˛ in y > 0 at a point T.˛; y5 / with 0 < y5 < jy4 j < jy3 j < y2 < y1 . This proves that  C .P/ is actually winding. As before, being the origin a source we can apply the Poincaré-Bendixson Theorem and get the existence of at least a stable limit cycle. We called the Van Der Pol equation paradigmatic just because the two methods presented in order to produce the existence of at least a stable limit cycle can applied also for the more general Liénard equation. More precisely, we have the following theorems, which generalize the above presented results for the Van Der Pol equation. Theorem 1 ([34]) Assume that (i) f is continuous, and g is locally Lipschitz continuous. xg.x/ > 0 for all x ¤ 0; (ii) f .0/ 0 for jxj > ı; g.x/ jg.x/j (iii) min lim sup ; lim sup < C1; x!C1 f .x/ x!1 f .x/ (iv) There exists h > ı and b > 0 such that f .x/ C jg.x/j > 0. Then there exists a periodic solution of Eq. (13). Proof Let us have a sketch of the proof. Consider the system (14). At first we note that the assumption f .0/ < 0 guarantees that the origin is a source, in virtue of a comparison with the Duffing equation. This is a standard argument and, in any case, the phase portrait of the Duffing equation will be discussed soon. At this point the proof follows the idea used for the Van Der Pol equation, namely to produce a negative semi-trajectory bounded away from the x-axis. The 0-isocline, namely the function H.x/ D  g.x/ , will play again a crucial role. Let  be the graph of f .x/

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Fig. 10 The phase portrait of Liénard equation under the assumption of Theorem 1

  the function H.x/ D  g.x/ and take a point P x;  g.x/ with x < ı, (or x > ı). f .x/ f .x/ In virtue of the sign condition on f .x/ and assumption (iii), it is easy to see that it is possible to choose a suitable point P in a way that   .P/ lies above  and therefore is bounded away from the x-axis. Now we follow  C .P/. Assumption (iv) avoids the possibility of having horizontal asymptotes, therefore  C .P/ intersects the x-axis twice, just as was happening for the Van Der Pol system (17). Hence we have a winding trajectory and we can apply the Poincaré-Bendixson Theorem to get the existence of at least a stable limit cycle for system (14), that is the existence of a stable periodic solution for Eq. (13) (Fig. 10). Theorem 2 ([35]) Assume that (i) f is continuous R x and g is locally Lipschitz continuous, xg.x/ > 0 for all x ¤ 0; limx!˙1 0 g.t/dt D C1; (ii) f .0/ < 0, and there exist ı 2 R such that f .x/ > 0 for jxj > ı;

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Zˇ (iii) there exist ˛  ı and ˇ  ı such that

f .x/dx > 0. ˛

Then there exists a periodic solution of Eq. (13). Proof Also in this case we give a sketch of the proof, which now is similar to the second approach presented above for the Van Der Pol equation. Consider System (14); again, the assumption f .0/ < 0 guarantees that the origin is a source. Assumption (iii) plays the same role of formula (20). On the other hand f is positive when jxj > ı, but we cannot say that the distance from the origin is decreasing as for the Van Der Pol equation. In order to get a similar result one must require the assumption (i) and this will be clear in the following discussion concerning the Duffing equation. In this situation we are able to produce a winding trajectory and apply the Poincaré-Bendixson Theorem to get the existence of at least a stable limit cycle for system (14), that is the existence of a stable periodic solution for Eq. (13). At this point we can observe that in general the methods used to attack this problem are basically two, and such methods were actually used for the Van Der Pol equation, as well as in the above presented Theorems 1 and 2. We can call the first one the “method of energy”, because one may consider the Liénard equation as perturbation of the Duffing equation xR C g.x/ D 0

(21)

which plays the role of the energy. Let us discuss in details this situation.

6.1 The Method of Energy The Duffing equation is equivalent in both planes to the system ( xP D y yP D g.x/

(22)

and it is well known that the level curves of the function H.x; y/ D Z

1 2 y C G.x/ 2

(23)

x

where G.x/ D

g.x/dx, are its solutions. 0

Therefore, by a straightforward calculation, the phase portrait of the Duffing equation is a global center if limx!˙1 G.x/ D C1, while it is a local center if G.x/ is bounded (We just note that for the Van Der Pol equation treated above H.x; y/ was the square distance from the origin D.x; y/). Here we follow the elegant and concise

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description of Lefschetz [19]. If we consider the level curve 1 2 y C G.x/ D K 2

(24)

in the dynamical interpretation as motion of a particle, the first term represents its kinetic energy and (17) expresses the law of conservation of energy as applied to the particle. For this reason, we may consider the level curves of the function H.x; y/ as energy levels. Coming back to the Liénard system (15), consider a generic point S D .xS ; yS /. Keeping the dynamical interpretation, we can say that this point lies on a certain level of energy. For sake of simplicity, we consider a generic point of the y-axis S D .0; yS /, which lies on the level energy 12 y2S D KS . Define  C .S/ as the positive semi-trajectory starting from S, and assume that C  .S/ moves around the origin and intersects again the y-axis in the same half-plane of S at a point R D .0; yR /. Clearly, such semi-trajectory is winding if jyR j < jyS j, unwinding if jyR j > jyS j and a cycle if jyR j D jyS j. In terms of energy, this means that in the first case we are losing energy, in the second one we are gaining energy and in the last one there is no increment. Therefore, one can investigate the variation of energy, even if in most papers this was not explicitly observed. In general, the situation is the following: we start at a point P D .xP ; yP / at the time t0 and follow  C .P/ for the time T until we reach the point Q D .xQ ; yQ / at the time t0 C T. The variation of energy is H.xQ ; yQ /  H.xP ; yP / D

1 2 1 y C G.xQ /  y2P  G.xP / D 2 Q 2

Z

t0 CT

P y/ dt H.x;

(25)

t0

We observe that, using the Liénard system (14), P y/ D yPy C g.x/Px D g.x/y C g.x/.y  F.x/ D g.x/F.x/ H.x; This well known result in the dynamical interpretation shows that when g.x/F.x/ > 0 we are losing energy, while when g.x/F.x/ < 0 we are gaining energy, and in order to have the existence of the limit cycle it is necessary that g.x/F.x/ change sign. More precisely, considering the positive semi-trajectory  C .P/ reaching the point Q, the variation of energy given by the integral in (25) may be split in four parts as follows Z

t0 CT t0

P y/dt D H.x;

Z

ˇ ˛

Z

g.x/F.x/ dx C y  F.x/ ˛

C ˇ

Z

g.x/F.x/ dx C y  F.x/

y2

F.x/ dyC y1

Z

yQ

F.x/ dy y3

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where ˛ < 0 and ˇ > 0 are necessary in order to make a correct change of variables. Such semi-trajectory will be winding if and only if the sum of the four integrals is negative. Z We observe that actually the integral F.x/dy plays a crucial role. This because Z ˇ g.x/F.x/ the integral dx may be considered arbitrarily small. y  F.x/ ˛ In principle there are two ways for which this can be achieved. The first one is letting the difference .ˇ  ˛/ be arbitrarily small. In virtue of the regularity of the g.x/F.x/ function , the integral will be arbitrarily small. On the other hand being y  F.x/ the 0-isocline the y-axis, the trajectory may be read as a function of y, and hence the Z integral F.x/dy is well defined outside the strip Œ˛; ˇ. This way cannot be further developed because in general we cannot evaluate the integral unless we have strong condition on F.x/ on the whole line. The second way is more efficient. The interval Œ˛; ˇ is now fixed and can be Z ˇ g.x/F.x/ large. However, the integral dx may be considered arbitrarily small y  F.x/ ˛ if we take Z a trajectory with jyj arbitrarily large. In this case we can evaluate the integral

F.x/dy, provided that outside the interval Œ˛; ˇ, there are conditions on

F.x/ which allow an estimation Z of the previous integral. For this reason, the role of

F.x/dy was already emphasized by Lefschetz [15,

p. 267], who called this integral taken along a path as the energy “dissipated” by the system. In order to work with energy, it is therefore necessary to consider trajectories arbitrarily large, and in this light we need sufficient conditions which guarantee that any positive semi-trajectory starting from a point P D .x; y/ with jyj arbitrarily large, intersects the vertical isocline y D F.x/, otherwise the method fails. Such conditions were at first introduced in [36], but we are not going to discuss them. We just present the classical Dragilev Theorem because it is based on this idea, even if not explicitly stated Theorem 3 (Dragilev [8]) Assume that (i) f is continuous, and g is locally Lipschitz continuous; xg.x/ > 0 for all x ¤ 0. limx!˙1 G.x/ D C1; (ii) xF.x/ < 0 for x ¤ 0 and jxj sufficiently small. (iii) There exist constants N; K1 ; K2 with K1 < K2 such that F.x/  K1 for x > N and F.x/ < K2 for x < N. Then system (15) admits at least one limit cycle.

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6.2 Intersection with the Vertical Isocline The second method to obtain a winding trajectory is strictly related with the intersection with the vertical isocline. More precisely, if there is P D .0; y/ with y ¤ 0 such that   .P/ does not intersect the curve y D F.x/ and  C .P/ is oscillatory, clearly such trajectory is winding. Such property was called “property K” in [38] where this problem has been investigated. We observe that, in order to get that  C .P/ is oscillatory, it is again necessary that  C .P/ intersects y D F.x/ for jyj large enough. This method seems more effective because no balance of energy is necessary, but actually, the requirement that   .P/ does not intersects the vertical isocline is a strong condition on the structure of the system, and in general this implies that F.x/ dominates G.x/ .This is precisely the case of Theorem 1. As it was already discussed, an example in which both methods may be used is 3 the Van Der Pol equation (16), and one can easily check that F.x/ is x3  x while 2 G.x/ is x2 and F.x/ actually dominates G.x/.

6.3 The Massera Theorem Finally we discuss the problem of uniqueness of limit cycles. Among the results we present only the classical Massera Theorem, because his proof is based on a clever and simple geometrical idea, which works in the case g.x/ D x. Other results involve more sophisticated mathematical background, and therefore cannot take place in this short note. Theorem 4 (Massera, [23]) The system (14) has at most one limit cycle which is stable, and hence Eq. (13) has at most one non trivial periodic solution which is stable, provided that f is continuous, g.x/ D x and f .x/ is monotone decreasing for x < 0, monotone increasing for x > 0. We just observe that Van Der Pol equation satisfies such assumptions, and hence has an unique stable non trivial periodic solution. The Theorem of Massera improved a previous result due to Sansone [29] in which there was the additional assumption jf .x/j < 2. This assumption comes from the fact that Sansone was using the polar coordinates. Such strong restriction on f is clearly not satisfied in the polynomial case and hence the Massera’s result is much more powerful. We must observe that Massera was proving the uniqueness of limit cycles regardless the existence because only the monotonicity properties and the continuity were required. It is easy to prove that, in order to fulfill the necessary conditions for the existence of limit cycles, the only cases to be considered are.

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– f .x/ has two zeros a and b; a < 0 < b. In this case the existence of limit cycles is granted, in virtue of Theorem 2 – f .x/ remains negative for x < 0, (or for x > 0), while intersects the x axis once in x > 0 ( or for x < 0). Now the existence of limit cycles is no more granted. Moreover this case does not cover the crucial polynomial case, which is still the most important. For related results, and a comparison of the theorems of Massera and Sansone, one can see [27, 28, 37]. Proof A crucial step in the proof is the fact that any limit cycle of system (14) must be star-like, which means that any halfray intersects the limit cycle only once. For some strange reason, this was not explicitly mentioned in the paper of Massera, but it is proved in the paper of Sansone, where for this result the additional assumption jf .x/j < 2 was not necessary. Roughly speaking, the key of the proof is the following: One performs a similarity transformation on the limit-cycle  and in this way obtain a family of simple closed curves k on the plane at both sides of  . This because  is star-like. Then one proves that the positive semi-trajectories of system (14) moving from the points of k stay in the interior of k for k > 1 and in the exterior of k for k < 1. Therefore we get that cannot exist any other limit-cycle other than  . Let us see what is going on in more details (Figs. 11 and 12). In the system (14) we consider a limit-cycle  and make the similarity transformation .x; y/ ! .kx; ky/. Therefore  is moved in a closed curve k which Fig. 11 The geometric idea of the Massera theorem

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Fig. 12 An example of a not star-like limit cycle. Guess why Massera proof it is no more working in this case

lies in the interior of  if k < 1 and in the exterior of  if k > 1. Take a point P.x; y/ on  and consider its tangent vector. This is given by the slope, which in this case is y0 .x; y/ D f .x/ 

x y

(26)

Due to the similarity transformation, the tangent vector at the point Pk .kx; ky/ on the closed curve k is parallel to the one at the point P. But the slope of the system in this point is y0 .kx; ky/ D f .kx/ 

kx x D f .kx/  ky y

(27)

The monotonicity assumption on f is now used to prove that the slope at P.x; y/, namely y0 .x; y/ D f .x/ 

x x  f .kx/  D y0 .kx; ky/ y y

(28)

that is the slope at Pk .kx; ky/, if k < 1. Clearly such inequality holds with  if k > 1. Therefore  C .Pk / enters k and lies in its interior if k > 1, while exits k and lies in its exterior if k < 1. Being Pk a generic point this shows that  is actually unique and stable. As already mentioned, this smart proof works only in the case g.x/ D x and up to now was not possible to have a similar geometrical proof in the general case. With this last result we can say that, finally, our survival kit is complete.

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7 Water Waves: Phase Plane Analysis Versus Classical Theory We are now ready to show how phase-plane analysis can be applied also in water waves. We discuss the motion of particle trajectories in linear water waves and follow the joint paper with Constantin [5] where it is proved that for a linear water wave no particle trajectory is closed, unless the free surface is flat. Each trajectory involves over a period a backward/forward movement of the particle, and the path is an elliptical arc (which degenerates on the flat bed) but with a forward drift. Notice that due to the mathematical intractability of the governing equations for water waves, the classical approach towards explaining this aspect of water waves consists in analyzing the particle motion after linearization of the governing equations. Within the linear water wave theory, the ordinary differential equations system describing the motion of the particles is nevertheless nonlinear and explicit solutions are not available. In the first approximation of this nonlinear system, all particle paths are closed. Support for this conclusion is given by the only known explicit solution with a non-flat free surface for the governing equations in water of infinite depth [13], solution for which all particle paths are circles of diameters decreasing with the distance from the free surface. But as already noted in the discussion of the predator-prey model, once that after a linearization we find closed trajectory this property cannot be inferred to the nonlinear systems. This was actually the starting point of the paper, where instead of first approximation phase plane analysis was called to “help”. However we observe that the presented result can actually be viewed as a more detailed version of the classical theory for the trajectories of particles below a water wave, as it was summarized by LonguetHiggin’s [22]: In progressive gravity waves of very small amplitude it is well known that the orbits of the particles are either elliptical or circular. In steep waves, however, the orbits become quite distorted, as shown by the existence of a mean horizontal drift or mass-transport in irrotational waves.

More precisely, we show that within linear water wave theory, the particle paths are almost closed and the more we approach the free surface, the more pronounced the deviation from a closed orbit becomes. This is in agreement with Stokes’ observation [30]: “It appears that the forward motion of the particles is not altogether compensated by their backward motion; so that, in addition to their motion of oscillation, the particles have a progressive motion in the direction of the propagation of the waves : : :” We follow the above mentioned paper [5] and begin our analysis from the governing equations 8 dx ˆ < D M cosh.ky/ cos.x  ct/ dt (29) dy ˆ : D M sinh.ky/ sin.x  ct/ dt

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because this is the standard approach for people working in this field area. Notice that from the previous part of the paper one can see that MD

!h0 sinh.kh0 /

(30)

Without actually solving the system we are interested in the principal features of its solutions. At first we consider the classical first approximation in terms of the small parameter M. We restrict our attention to a the fixed time interval Œ0; T, where T > 0 is the wave period. Since y belongs to a set bounded a priori, from (29) we readily obtain that x.t/  x0 D O.M/;

y.t/  y0 D O.M/;

t 2 Œ0; T

(31)

where O.M/ denotes an expression of order M. Using the mean-value theorem, we may write (29) on Œ0; T as 8 dx ˆ < dt dy ˆ : dt

D M cosh.ky0 / cos.kx0  !t/ C O.M 2 / D M sinh.ky0 / sin.kx0  !t/ C O.M 2 /

(32)

(we just observe that actually ! D kc). Neglecting terms of second order in M, we find that 8 dx ˆ <  M cosh.ky0 / cos.kx0  !t/ dt (33) dy ˆ :  M sinh.ky0 / sin.kx0  !t/ dt so that by integration we obtain 8 dx ˆ < dt dy ˆ : dt

 x0 

M !

cosh.ky0 / sin.kx0  !t/

 x0 C

M !

sinh.ky0 / cos.kx0  !t/

(34)

Thus Œy.t/  y0 2 M2 Œx.t/  x0 2 C  2 2 2 ! cosh .ky0 / sinh .ky0 /

(35)

which is the equation of an ellipse: to a first-order approximation the water particles move in closed elliptic orbits, the centre of the ellipse being .x0 ; y0 /. We some calculations, which we omit for sake of simplicity, one can see that the sense of the particle motion round its orbit is clockwise as viewed with respect to the positive x-direction (the propagation direction). By (34), to a first approximation

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a surface water particle traces an elliptical orbit with the vertical axis of length M ! sinh.kh0 / D h0 , equal to the height of the wave, while its horizontal axis is slightly larger. The same type of motion occurs for the water particles below the surface but both axes of the elliptical orbits become smaller and smaller with depth. Since cosh2 .r/  sinh2 .r/ D 1, all the ellipses have the same distance 2M ! between their foci, but the lengths of their axes decrease as we go deep into the water: the ellipses are confocal if they are coaxially superimposed upon each other. Observe that the ratio of the height to its length for such an ellipse is tanh.ky0 / < 1 so that the flattening of the orbits becomes more pronounced as the depth increases, until at the bottom they are completely flattened into horizontal lines: at the bottom y D 0 the vertical axis is zero but the major axis equals M ! so that the ellipse degenerates into a straight line and the water particles simply oscillate backwards and forwards on the flat bed. So far with the classical first approximation approach. But once again we stress the fact that after a linearization we find closed trajectory this property cannot be inferred to the nonlinear systems. And in this problem we observe that actually we made two linearizations, the first one in order to be able to write the governing equations and the second in order to recognize the principal features of its solutions. Hence the obtained result, even if in accordance with the picture of the motion below a deep water wave provided by Gerstner’s explicit solution in water of infinite depth [13], where all particles move on circles, cannot be accepted. But in any case Fig. 13 was present in several text-books and also (but not any longer: : :) in Wikipedia. It is time to study the exact solution to (29), and see how phase plane analysis can make the job. To study the exact solution to (29) it is convenient to re-write it in a moving frame with scaled independent variables: the transformation X D k.x  ct/; Fig. 13 The first approximation of a particle paths in a linear wave

Y D kY

(36)

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maps system (29) in 8 dX ˆ < D kM cosh.Y/ cos.X/  kc dt dY ˆ : D kM sinh.Y/ sin.X/ dt

(37)

Notice that in view of (30), kM D k

"!h0 kh0 D "! < kc D ! sinh.kh0 / sinh.kh0 /

(38)

since s > sinh.s/ for s > 0 while " < 1 within the framework of linear theory. We just observe that the autonomous system (37) meets the standard regularity assumptions for the uniqueness of the Cauchy problem so that its trajectories do not intersect. Denote A.X; Y/ D kM cosh.Y/ cos.X/  kc; B.X; Y/ D kM sinh.Y/ sin.X/

(39)

Since both functions A and B are periodic of period 2 in the X-variable, it suffices to restrict our investigation of (37) to the strip  X 

1Y Y ? we have  1 so that ˛ is well defined. The M cosh.Y/ smooth function ˛ is even, it assumes its infimum Y ? at X D 0, and satisfies lim ˛.X/ D C1

(42)

X!˙1

Once that we know the 0-isocline and the 1-isocline, we can say that in the halfstrip considered there is an unique singular point Q.0; Y ? /. Moreover we are able to determine the sign of the two components of the vector field given by system (37). For X 2 . 2 ; / the function A.X; Y/ is negative while B.X; Y) is positive. For X 2 .0; 2 / the function A.X; Y/ is negative below the graph of X ! ˛.X/ and positive above it, while B.X; Y/ is positive here. Using the symmetry with respect to the Yaxis, we obtain the corresponding signs for X 2 .; 0/. At this point, with some more work and using some standard phase-plane argument, it is possible to see that in the phase portrait of system (37) if we take a point P.; Y/ and consider  C .P/ there are two possibilities, depending from the value of Y.  C .P/ goes to +1 for Y > ˇ, while  C .P/ reaches the Y-axis for Y < ˇ. Clearly it is not possible to determine such ˇ, but this is not important in this framework, while we can say that if we consider Pˇ .; ˇ/, actually  C .Pˇ / tends to the singular point Q and therefore it is a separatrix, that is, a phase curve that marks the boundary between phase curves with different properties. Now for 0 < X <  we consider the points R.X; Y/ above such separatrix. With a similar argument we can see that there are points R such that   .R/ intersects the Y-axis and points R such that   .R/ intersects the line X D . Therefore we get the existence of a second separatrix which in this case starts from the singular point Q. Using the mirror symmetry we have other two separatrices and we can argue that the singular point is actually a saddle point. A different approach, in order to study the property of Q, is presented in the paper, considering the Hamiltonian structure of (37) and using more sophisticated tools as Morse theory. But the qualitative approach was the first one used to attack the problem, and without any doubt deserves a place in this note. The phase portrait of (37) is thus complete and appears in Fig. 14. Knowing the phase curves .X.t/; Y.t// of (37), the particle trajectories in the linear wave are simply given by x.t/ D

X.t/ C ct; k

y.t/ D

Y.t/ k

(43)

From the above phase plane analysis of (37) we deduce that in order to have a physically realistic solution it is necessary that kh0 .1 C /  Y ? cosh1 .

sinh.kh0 / / "kh0

(44)

Indeed, (44) is necessary to prevent the existence of particle paths that grow indefinitely with time. Relation (37) can be interpreted as follows: given the average

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Fig. 14 Phase portrait in the moving frame

water depth h0 > 0, in order to represent a realistic model of water waves of small amplitude " > 0 with wavenumber k > 0; it is sufficient that "

tanh.kh0 / kh0

(45)

since the function s ! tanh.s/ is decreasing for s > 0. Such inequality gives a s quantitative meaning to the statement “" < 1” small. From now on we will assume his validity. But how the phase-plane analysis plays a crucial role for the solution of this problem? The answer to this question is given by the following Lemma, which his proved in the paper. Lemma 1 Given Y0 2 Œ0; ˇ/, let  D .Y0 / > 0 be the time needed for the solution.X.t/; Y.t// of (37) with initial data .; Y0 / to intersect the line X D . Then the phase curve .X.t/; Y.t// corresponds via (43) to a periodic solution of (29) if and only if  D 2 . ck In view of the Lemma, the statement of non existence of periodic solutions of (29), and therefore the non existence of closed orbits for the water particles follows once we show that .Y0 / >

2 ck

Y0 2 Œ0; ˇ/

(46)

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Fig. 15 Particle paths in the moving frame

This inequality can be checked by direct computation for Y0 D 0, because in this case the trajectory lies on the X-axis and it is determined by the solution of the differential equation dX D kM cos.X/  kc dt

(47)

with initial data X.0/ D . We omit such direct computation which in any case is in the paper. Consider now the case when Y0 2 .0; ˇ/ since for Y0 D ˇ;  C .Pˇ / is actually the separatrix and therefore never reaches X D  in finite time. If we follow  C .P/ we easily see that such positive semi-trajectory intersects the line X D 2 at some point . 2 ; Y1 / with Y1 2 .Y0 ; Y ? / and then reaches the Y-axis (Fig. 15). Notice that  C .P/ lies below the line Y D Y1 for X 2 . 2 ;  whereas for X 2 Œ0; 2 / it lies above it. We work only for X > 0 just because using the mirror symmetry we have a similar result in Œ; 0. This is the key result because now in virtue of the second equation of system one can argue that .Y0 / is more than the time necessary if we move on the line Y D Y1 from X D  to X D . But arguing as before this time can be determined by direct computation on the line Y D Y1 because in this case it is determined by the solution of the differential equation dX D kM cosh.Y1 / cos.X/  kc dt

(48)

with initial data X.0/ D . And also in this case, with the direct computation what we omit for sake of simplicity, one can check that this time is actually more than 2 ck . With the help of phase plane analysis we proved that in time  the particle traces a loop that fails to close-up: there is a small forward drift, and in the paper it is also shown, in virtue of the above mentioned direct computations, that this forward drift is minimal on the flat bed (Fig. 16). We finish this note just recalling that after this paper other results in water waves theory were proved using phase plane analysis, see for instance Constantin et al. [4, 6], Ehrnström and Villari [9], Ehrnström et al. [10], Henry [15, 16] and Matioc [24]. These results agree with experimental results

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Fig. 16 Particle trajectory above the flat bed

in Umeyama [31], with numerical results in Nachbin and Ribeiro-Junior [25] and with results from nonlinear equations without approximation in Constantin [2, 3], Henry [14] and in Okamoto and Shåji [26]. And, as far as we know, more people is now attacking this kind of problems using this classical technique. Acknowledgements I thank Dr. Francesco Mugelli for his friendly help in the writing of this note.

References 1. Boyce, W.E., DiPrima, R.C.: Elementary Differential Equations and Boundary Value Problems, 3rd edn. Wiley, New York (1977) 2. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006) 3. Constantin, A.: The flow beneath a periodic travelling surface water wave. J. Phys. A: Math. Theor. 48 (2015). Art. No. 143001 4. Constantin, A., Strauss, W.: Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 63, 533–557 (2010) 5. Constantin, A., Villari, G.: Particle trajectories in linear water waves. J. Math. Fluid Mech. 10, 1–18 (2008) 6. Constantin, A., Ehrnström, M., Villari, G.: Particle trajectories in linear deep-water waves. Nonlinear Anal. Real World Appl. 9, 1336–1344 (2008) 7. Darwin, C.: On the Origin of Species by Means of Natural Selection. John Murray, London (1859) 8. Dragilev, A.V.: Periodic solution of a differential equation of nonlinear oscillation. Prikladnaya Mat. I. Mek. 16, 85–88 (1952; Russian) 9. Ehrnström, M., Villari, G.: Linear water waves with vorticity: rotational features and particle paths. J. Differ. Equ. 244, 1888–1909 (2008) 10. Ehrnström, M., Escher, J., Villari, G.: Steady water waves with multiple critical layers: interior dynamics. J. Math. Fluid Mech. 14, 407–419 (2012) 11. Filippov, A.F.: A sufficient condition for the existence of a stable limit cycle for a second order equation (Russian). Mat. Sb. 30, 171–180 (1952) 12. Gause, G.F.: Verifications sperimentales de la theorie mathematique de la lutte pour la vie. Actualites scientifiques et industrielles, vol. 277. Hermann et C. editeurs, Paris (1935) 13. Gerstner, F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809) 14. Henry, D.: The trajectories of particles in deep-water Stokes waves. Int. Math. Res. Not., 13 pp. (2006). Art. ID 23405 15. Henry, D.: Particle trajectories in linear periodic capillary and capillary-gravity water waves. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365, 2241–2251 (2007)

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16. Henry, D.: Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves. J. Nonlinear Math. Phys. 14, 1–7 (2007) 17. Iannelli, M.: The mathematical description of epidemics: some basic models and problems. In: Da Prato, G. (ed.) Mathematical Aspects of Human Diseases, pp. 15–25. Giardini, Pisa (1992) 18. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. A 115, 700–721 (1927) 19. Lefschetz, S.: Differential Equations: Geometric Theory. Interscience, New York (1957) 20. Levinson, N., Smith, O.K.: A general equation for relaxation oscillations. Duke Math. J. 9, 382–403 (1942) 21. Liénard, A.: Étude des oscillations entretenues. Revue génér. de l’électr. 23, 901–902, 906–954 (1928) 22. Longuet-Higgins, M.S.: The trajectories of particles in steep, symmetric gravity waves. J. Fluid Mech. 94, 497–517 (1979) 23. Massera, J.L.: Sur un Théoreme de G. Sansone sur l’equation de Liénard. Boll. Unione Mat. Ital. 9(3), 367–369 (1954) 24. Matioc, A.-V.: On particle trajectories in linear deep-water waves. Commun. Pure Appl. Anal. 11, 1537–1547 (2012) 25. Nachbin, A., Ribeiro-Junior, R.: A boundary integral formulation for particle trajectories in Stokes waves. Discret. Cont. Dyn. Syst. A 34, 3135–3153 (2014) 26. Okamoto, H., Shoji, M.: Trajectories of fluid particles in a periodic water wave. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 370, 1661–1676 (2012) 27. Rosati, L., Villari, G.: On Massera’s Theorem concerning the uniqueness of a periodic solution for the Liénard equation. When does such a periodic solution actually exists? Bound. Value Probl. 2013, 144 (2013) 28. Sabatini, M., Villari, G.: On the uniqueness of limit cycles for Liénard equation: the legacy of G. Sansone. Le Matematiche LXV, 201–214 (2010) 29. Sansone, G.: Soluzioni periodiche dell’equazione di Liénard. Calcolo del periodo. Rend. Sem. Mat. Univ. e politecnico Torino 10, 155–171 (1951) 30. Stokes, G.G.: On the theory of oscillatory waves. Trans. Camb. Philos. Soc. 8, 441–455 (1847) 31. Umeyama, M.: Eulerian Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 370, 1687–1702 (2012) 32. Van Der Pol, B.: On “relaxation oscillations” I. Lond. Edinb. Dublin Philos. Mag. (7) 2, 978– 992 (1926) 33. Verhulst, P.F.: Notice sur la loi que la population poursuitvdans son accroissement. Correspondance Mathématique et Physiquev 10, 113–121 (1838) 34. Villari, G.: Periodic solutions of Liénard’s equation. J. Math. Anal. Appl. 86, 379–386 (1982) 35. Villari, G.: On the existence of periodic solutions for Liénard’s equation. Nonlinear Anal. TMA 7, 71–78 (1983) 36. Villari, G.: On the qualitative behaviour of solutions of Liénard equation. J. Differ. Equ. 67, 269–277 (1987) 37. Villari, G.: An improvement of Massera’s theorem for the existence and uniqueness of a periodic solution for the Liénard equation. Rend. Istit. Mat. Univ. Trieste 44, 187–195 (2012) 38. Villari, G., Zanolin, F.: On a dynamical system in the Liénard Plane. Necessary and sufficient conditions for the intersection with the vertical isocline and applications. Funkcial. Ekvac. 33, 19–38 (1990) 39. Volterra, V.: Variazioni e fluttuazioni del numero di individui in specie animali conviventi. Memorie del Regio Comitato Talassografico Italiano, Mem. CXXXI, 1–142 (1927)

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Editors in Chief: J.-M. Morel, B. Teissier; Editorial Policy 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications – quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Besides monographs, multi-author manuscripts resulting from SUMMER SCHOOLS or similar INTENSIVE COURSES are welcome, provided their objective was held to present an active mathematical topic to an audience at the beginning or intermediate graduate level (a list of participants should be provided). The resulting manuscript should not be just a collection of course notes, but should require advance planning and coordination among the main lecturers. The subject matter should dictate the structure of the book. This structure should be motivated and explained in a scientific introduction, and the notation, references, index and formulation of results should be, if possible, unified by the editors. Each contribution should have an abstract and an introduction referring to the other contributions. In other words, more preparatory work must go into a multi-authored volume than simply assembling a disparate collection of papers, communicated at the event. 3. Manuscripts should be submitted either online at www.editorialmanager.com/lnm to Springer’s mathematics editorial in Heidelberg, or electronically to one of the series editors. Authors should be aware that incomplete or insufficiently close-to-final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Parallel submission of a manuscript to another publisher while under consideration for LNM is not acceptable and can lead to rejection. 4. In general, monographs will be sent out to at least 2 external referees for evaluation. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. Volume Editors of multi-author works are expected to arrange for the refereeing, to the usual scientific standards, of the individual contributions. If the resulting reports can be

forwarded to the LNM Editorial Board, this is very helpful. If no reports are forwarded or if other questions remain unclear in respect of homogeneity etc, the series editors may wish to consult external referees for an overall evaluation of the volume. 5. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. – For evaluation purposes, manuscripts should be submitted as pdf files. 6. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files (see LaTeX templates online: https://www.springer.com/gb/authors-editors/book-authorseditors/manuscriptpreparation/5636) plus the corresponding pdf- or zipped ps-file. The LaTeX source files are essential for producing the full-text online version of the book, see http://link.springer.com/bookseries/304 for the existing online volumes of LNM). The technical production of a Lecture Notes volume takes approximately 12 weeks. Additional instructions, if necessary, are available on request from [email protected]. 7. Authors receive a total of 30 free copies of their volume and free access to their book on SpringerLink, but no royalties. They are entitled to a discount of 33.3 % on the price of Springer books purchased for their personal use, if ordering directly from Springer. 8. Commitment to publish is made by a Publishing Agreement; contributing authors of multiauthor books are requested to sign a Consent to Publish form. Springer-Verlag registers the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor Jean-Michel Morel, CMLA, École Normale Supérieure de Cachan, France E-mail: [email protected] Professor Bernard Teissier, Equipe Géométrie et Dynamique, Institut de Mathématiques de Jussieu – Paris Rive Gauche, Paris, France E-mail: [email protected] Springer: Ute McCrory, Mathematics, Heidelberg, Germany, E-mail: [email protected]