Integrable Systems [1 ed.] 2022932445, 9781786308276

Table of contents :
Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Preface
Chapter 1. Symplectic Manifolds
1.1. Introduction
1.2. Symplectic vector spaces
1.3. Symplectic manifolds
1.4. Vectors fields and flows
1.5. The Darboux theorem
1.6. Poisson brackets and Hamiltonian systems
1.7. Examples
1.8. Coadjoint orbits and their symplectic structures
1.9. Application to the group SO(n)
1.9.1. Application to the group SO(3)
1.9.2. Application to the group SO(4)
1.10. Exercises
Chapter 2. Hamilton–Jacobi Theory
2.1. Euler–Lagrange equation
2.2. Legendre transformation
2.3. Hamilton’s canonical equations
2.4. Canonical transformations
2.5. Hamilton–Jacobi equation
2.6. Applications
2.6.1. Harmonic oscillator
2.6.2. The Kepler problem
2.6.3. Simple pendulum
2.7. Exercises
Chapter 3. Integrable Systems
3.1. Hamiltonian systems and Arnold–Liouville theorem
3.2. Rotation of a rigid body about a fixed point
3.2.1. The Euler problem of a rigid body
3.2.2. The Lagrange top
3.2.3. The Kowalewski spinning top
3.2.4. Special cases
3.3. Motion of a solid through ideal fluid
3.3.1. Clebsch’s case
3.3.2. Lyapunov–Steklov’s case
3.4. Yang–Mills field with gauge group SU(2)
3.5. Appendix (geodesic flow and Euler–Arnold equations)
3.6. Exercises
Chapter 4. Spectral Methods for Solving Integrable Systems
4.1. Lax equations and spectral curves
4.2. Integrable systems and Kac–Moody Lie algebras
4.3. Geodesic flow on SO(n)
4.4. The Euler problem of a rigid body
4.5. The Manakov geodesic flow on the group SO(4)
4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem
4.7. The Lagrange top
4.8. Quartic potential, Garnier system
4.9. The coupled nonlinear Schrödinger equations
4.10. The Yang–Mills equations
4.11. The Kowalewski top
4.12. The Goryachev–Chaplygin top
4.13. Periodic infinite band matrix
4.14. Exercises
Chapter 5. The Spectrum of Jacobi Matrices and Algebraic Curves
5.1. Jacobi matrices and algebraic curves
5.2. Difference operators
5.3. Continued fraction, orthogonal polynomials and Abelian integrals
5.4. Exercises
Chapter 6. Griffiths Linearization Flows on Jacobians
6.1. Spectral curves
6.2. Cohomological deformation theory
6.3. Mittag–Leffler problem
6.4. Linearizing flows
6.5. The Toda lattice
6.6. The Lagrange top
6.7. Nahm’s equations
6.8. The n-dimensional rigid body
6.9. Exercises
Chapter 7. Algebraically Integrable Systems
7.1. Meromorphic solutions
7.2. Algebraic complete integrability
7.3. The Liouville–Arnold–Adler–van Moerbeke theorem
7.4. The Euler problem of a rigid body
7.5. The Kowalewski top
7.6. The Hénon–Heiles system
7.7. The Manakov geodesic flow on the group SO(4)
7.8. Geodesic flow on SO(4) with a quartic invariant
7.9. The geodesic flow on SO(n) for a left invariant metric
7.10. The periodic five-particle Kac–van Moerbeke lattice
7.11. Generalized periodic Toda systems
7.12. The Gross–Neveu system
7.13. The Kolossof potential
7.14. Exercises
Chapter 8. Generalized Algebraic Completely Integrable Systems
8.1. Generalities
8.2. The RDG potential and a five-dimensional system
8.3. The Hénon–Heiles problem and a five-dimensional system
8.4. The Goryachev–Chaplygin top and a seven-dimensional system
8.5. The Lagrange top
8.6. Exercises
Chapter 9. The Korteweg–de Vries Equation
9.1. Historical aspects and introduction
9.2. Stationary Schrödinger and integral Gelfand–Levitan equations
9.3. The inverse scattering method
9.4. Exercises
Chapter 10. KP–KdV Hierarchy and Pseudo-differential Operators
10.1. Pseudo-differential operators and symplectic structures
10.2. KdV equation, Heisenberg and Virasoro algebras
10.3. KP hierarchy and vertex operators
10.4. Exercises
References
Index
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Integrable Systems

To my Professor Pierre van Moerbeke

Integrable Systems

Ahmed Lesfari

First published 2022 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2022 The rights of Ahmed Lesfari to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group. Library of Congress Control Number: 2022932445 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-827-6

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1. Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . .

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1.1. Introduction . . . . . . . . . . . . . . . . . . . . 1.2. Symplectic vector spaces . . . . . . . . . . . . . 1.3. Symplectic manifolds . . . . . . . . . . . . . . . 1.4. Vectors fields and flows . . . . . . . . . . . . . . 1.5. The Darboux theorem . . . . . . . . . . . . . . . 1.6. Poisson brackets and Hamiltonian systems . . . 1.7. Examples . . . . . . . . . . . . . . . . . . . . . . 1.8. Coadjoint orbits and their symplectic structures 1.9. Application to the group SO(n) . . . . . . . . . 1.9.1. Application to the group SO(3) . . . . . . . 1.9.2. Application to the group SO(4) . . . . . . . 1.10. Exercises . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Hamilton–Jacobi Theory . . . . . . . . . . . . . . . . . . . . .

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2.1. Euler–Lagrange equation . . . . 2.2. Legendre transformation . . . . 2.3. Hamilton’s canonical equations 2.4. Canonical transformations . . . 2.5. Hamilton–Jacobi equation . . . 2.6. Applications . . . . . . . . . . . 2.6.1. Harmonic oscillator . . . . . 2.6.2. The Kepler problem . . . . . 2.6.3. Simple pendulum . . . . . . 2.7. Exercises . . . . . . . . . . . . .

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Chapter 3. Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Hamiltonian systems and Arnold–Liouville theorem . 3.2. Rotation of a rigid body about a fixed point . . . . . . . 3.2.1. The Euler problem of a rigid body . . . . . . . . . . 3.2.2. The Lagrange top . . . . . . . . . . . . . . . . . . . 3.2.3. The Kowalewski spinning top . . . . . . . . . . . . 3.2.4. Special cases . . . . . . . . . . . . . . . . . . . . . . 3.3. Motion of a solid through ideal fluid . . . . . . . . . . . 3.3.1. Clebsch’s case . . . . . . . . . . . . . . . . . . . . . 3.3.2. Lyapunov–Steklov’s case . . . . . . . . . . . . . . . 3.4. Yang–Mills field with gauge group SU (2) . . . . . . . 3.5. Appendix (geodesic flow and Euler–Arnold equations) 3.6. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Spectral Methods for Solving Integrable Systems . . . . 103 4.1. Lax equations and spectral curves . . . . . . . . . . . . . . . 4.2. Integrable systems and Kac–Moody Lie algebras . . . . . . 4.3. Geodesic flow on SO(n) . . . . . . . . . . . . . . . . . . . . 4.4. The Euler problem of a rigid body . . . . . . . . . . . . . . . 4.5. The Manakov geodesic flow on the group SO(4) . . . . . . 4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem 4.7. The Lagrange top . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Quartic potential, Garnier system . . . . . . . . . . . . . . . 4.9. The coupled nonlinear Schrödinger equations . . . . . . . . 4.10. The Yang–Mills equations . . . . . . . . . . . . . . . . . . . 4.11. The Kowalewski top . . . . . . . . . . . . . . . . . . . . . . 4.12. The Goryachev–Chaplygin top . . . . . . . . . . . . . . . . 4.13. Periodic infinite band matrix . . . . . . . . . . . . . . . . . 4.14. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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103 104 107 108 109 114 115 115 118 119 119 121 122 122

Chapter 5. The Spectrum of Jacobi Matrices and Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.1. Jacobi matrices and algebraic curves . . . . . . . . . . . . . . . . . 5.2. Difference operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Continued fraction, orthogonal polynomials and Abelian integrals . 5.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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129 135 137 140

Chapter 6. Griffiths Linearization Flows on Jacobians . . . . . . . . . 143 6.1. Spectral curves . . . . . . . . . . . . 6.2. Cohomological deformation theory 6.3. Mittag–Leffler problem . . . . . . . 6.4. Linearizing flows . . . . . . . . . .

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Contents

6.5. The Toda lattice . . . . . . . . 6.6. The Lagrange top . . . . . . . 6.7. Nahm’s equations . . . . . . . 6.8. The n-dimensional rigid body 6.9. Exercises . . . . . . . . . . . .

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Chapter 7. Algebraically Integrable Systems . . . . . . . . . . . . . . . 159 7.1. Meromorphic solutions . . . . . . . . . . . . . . . . . . 7.2. Algebraic complete integrability . . . . . . . . . . . . . 7.3. The Liouville–Arnold–Adler–van Moerbeke theorem . 7.4. The Euler problem of a rigid body . . . . . . . . . . . . 7.5. The Kowalewski top . . . . . . . . . . . . . . . . . . . . 7.6. The Hénon–Heiles system . . . . . . . . . . . . . . . . 7.7. The Manakov geodesic flow on the group SO(4) . . . 7.8. Geodesic flow on SO(4) with a quartic invariant . . . . 7.9. The geodesic flow on SO(n) for a left invariant metric 7.10. The periodic five-particle Kac–van Moerbeke lattice . 7.11. Generalized periodic Toda systems . . . . . . . . . . . 7.12. The Gross–Neveu system . . . . . . . . . . . . . . . . 7.13. The Kolossof potential . . . . . . . . . . . . . . . . . . 7.14. Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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159 164 171 173 175 191 200 206 210 212 213 214 215 215

Chapter 8. Generalized Algebraic Completely Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. The RDG potential and a five-dimensional system . . . . . . . . 8.3. The Hénon–Heiles problem and a five-dimensional system . . . 8.4. The Goryachev–Chaplygin top and a seven-dimensional system 8.5. The Lagrange top . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 9. The Korteweg–de Vries Equation . . . . . . . . . . . . . . . 241 9.1. Historical aspects and introduction . . . . . . . . . . . . . . . . 9.2. Stationary Schrödinger and integral Gelfand–Levitan equations 9.3. The inverse scattering method . . . . . . . . . . . . . . . . . . . 9.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 10. KP–KdV Hierarchy and Pseudo-differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.1. Pseudo-differential operators and symplectic structures . . . . . . . . 275 10.2. KdV equation, Heisenberg and Virasoro algebras . . . . . . . . . . . . 279

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10.3. KP hierarchy and vertex operators . . . . . . . . . . . . . . . . . . . . 281 10.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 References Index

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Preface

This book is intended for a wide readership of mathematicians and physicists: students pursuing graduate, masters and higher degrees in mathematics and mathematical physics. It is devoted to some geometric and topological aspects of the theory of integrable systems and the presentation is clear and well-organized, with many examples and problems provided throughout the text. Integrable Hamiltonian systems are nonlinear ordinary differential equations that are described by a Hamiltonian function and possess sufficiently many independent constants of motion in involution. The problem of finding and integrating Hamiltonian systems has attracted a considerable amount of attention in recent decades. Besides the fact that many integrable systems have been the subject of powerful and beautiful theories of mathematics, another motivation for their study is the concepts of integrability that are applied to an increasing number of physical systems, biological phenomena, population dynamics and chemical rate equations, to mention but a few applications. However, it still seems hopeless to describe, or even to recognize with any facility, the Hamiltonian systems which are integrable, even though they are exceptional. Chapter 1 is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some interesting properties of one-parameter groups of diffeomorphisms or of flow, Lie derivative, interior product or Cartan’s formula, as well as the study of a central theorem of symplectic geometry, namely, Darboux’s theorem. We also show how to determine explicitly symplectic structures on adjoint and coadjoint orbits of a Lie group, with particular attention given to the group SO(n). Chapter 2 deals with the study of some notions concerning the Hamilton–Jacobi theory in the calculus of variations. We will establish the Euler–Lagrange differential equations, Hamilton’s canonical equations and the Hamilton–Jacobi partial differential equation and explain how it is widely used in practice to solve some

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problems. As an application, we will study the geodesics, the harmonic oscillator, the Kepler problem and the simple pendulum. In Chapter 3, we study the Arnold–Liouville theorem: the regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following differential geometric fact; a compact and connected n-dimensional manifold on which there exist n vector fields that commute and are independent at every point is diffeomorphic to an n-dimensional real torus, and there is a transformation to so-called action-angle variables, mapping the flow into a straight line motion on that torus. We give a proof as direct as possible of the Arnold–Liouville theorem and we make a careful study of its connection with the concept of completely integrable systems. Many problems are studied in detail: the rotation of a rigid body about a fixed point, the motion of a solid in an ideal fluid and the Yang–Mills field with gauge group SU (2). In Chapter 4, we give a detailed study of the integrable systems that can be written as Lax equations with a spectral parameter. Such equations have no a priori Hamiltonian content. However, through the Adler–Kostant–Symes (AKS) construction, we can produce Hamiltonian systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form. We outline an algebraic-geometric interpretation of the flows of these systems, which are shown to describe linear motion on a complex torus. The relationship between spectral theory and completely integrable systems is a fundamental aspect of the modern theory of integrable systems. This chapter surveys a number of classical and recent results and our purpose here is to sketch a motivated overview of this interesting subject. We present a Lie algebra theoretical schema leading to integrable systems based on the Kostant–Kirillov coadjoint action. Many problems on Kostant–Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac–Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the AKS theorem, and the van Moerbeke–Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it) of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. This chapter describes a version of the general scheme, and shows in detail how several important classes of examples fit into the general framework. Several examples of integrable systems of relevance in mathematical physics are carefully discussed: geodesic flow on SO(n), the Euler problem of a rigid body, Manakov geodesic flow on the group SO(4), Jacobi geodesic flow on an ellipsoid, the Neumann problem, the Lagrange top, a quartic potential or Garnier system, coupled nonlinear Schrödinger equations, Yang–Mills equations, the Kowalewski spinning top, the Goryachev–Chaplygin top and the periodic infinite band matrix.

Preface

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The aim of Chapter 5 is to describe some connections between spectral theory in infinite dimensional Lie algebras, deformation theory and algebraic curves. We study infinite continued fractions, isospectral deformation of periodic Jacobi matrices, general difference operators, Cauchy–Stieltjes transforms and Abelian integrals from an algebraic geometrical point of view. These results can be used to obtain insight into integrable systems. In Chapter 6, we present in detail the Griffiths’ approach and his cohomological interpretation of the linearization test for solving integrable systems without reference to Kac–Moody algebras. His method is based on the observation that the tangent space to any deformation lies in a suitable cohomology group and on algebraic curves, higher cohomology can always be eliminated using duality theory. We explain how results from deformation theory and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. These conditions are cohomological and the Lax equations turn out to have a natural cohomological interpretation. Several nonlinear problems in mathematical physics illustrate these results: the Toda lattice, Nahm’s equations and the n-dimensional rigid body. In Chapter 7, the notion of algebraically completely integrable Hamiltonian systems in the Adler–van Moerbeke sense is explained, and techniques to find and solve such systems are presented. These are integrable systems whose trajectories are straight line motions on Abelian varieties (complex algebraic tori). We make, via the Kowalewski–Painlevé analysis, a study of the level manifolds of the systems, which are described explicitly as being affine part of Abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of Abelian integrals. We describe an explicit embedding of these Abelian varieties that complete the generic invariant surfaces into projective spaces. Many problems are studied in detail: the Euler problem of a rigid body, the Kowalewski top, the Hénon–Heiles system, Manakov geodesic flow on the group SO(4), geodesic flow on SO(4) with a quartic invariant, geodesic flow on SO(n) for a left invariant metric, the periodic five-particle Kac–van Moerbeke lattice, generalized periodic Toda systems, the Gross–Neveu system and the Kolossof potential. In Chapter 8, we discuss the study of generalized algebraic completely integrable systems. There are many examples of differential equations that have the weak Painlevé property that all movable singularities of the general solution have only a finite number of branches, and some interesting integrable systems appear as coverings of algebraic completely integrable systems. The invariant varieties are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. These systems are Liouville integrable and by the Arnold–Liouville theorem, the compact connected manifolds invariant by the real flows are tori, the real parts of complex affine coverings of Abelian varieties. Most of these systems of differential equations possess solutions that are Laurent series of t1/n (t being complex time) and whose coefficients depend rationally on certain

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algebraic parameters. We discuss some interesting examples: Ramani–Dorizzi– Grammaticos (RDG) potential, the Hénon–Heiles system, the Goryachev–Chaplygin top, a seven-dimensional system and the Lagrange top. Chapter 9 covers the stationary Schrödinger equation, the integral Gelfand–Levitan equation and the inverse scattering method used to solve exactly the Korteweg–de Vries (KdV) equation. The latter is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena. In Chapter 10, we study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro, Heisenberg and nonlinear evolution equations such as the KdV, Boussinesq and Kadomtsev–Petviashvili (KP) equations play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato–Date–Jimbo–Miwa–Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations and the Gelfand–Dickey flows, the Heisenberg and Virasoro algebras are given. The study of the KP and KdV hierarchies, the use of tau functions related to infinite dimensional Grassmannians, Fay identities, vertex operators and the Hirota’s bilinear formalism led to obtaining remarkable properties concerning these algebras such as, for example, the existence of an infinite family of first integrals functionally independent and in involution. It is well known that when studying integrable systems, elliptic functions and integrals, compact Riemann surfaces or algebraic curves, Abelian surfaces (as well as the basic techniques to study two-dimensional algebraic completely integrable systems) play a crucial role. These facts, which may be well known to the algebraic reader, can be found, for example, in Adler and van Moerbeke (2004); Fay (1973); Griffiths and Harris (1978); Lesfari (2015b) and Vanhaecke (2001). I would like to thank and am grateful to P. van Moerbeke and L. Haine, from whom I learned much of this subject through conversations and remarks. I would also like to thank the editors for their interest, seriousness and professionalism. Finally my thanks go to my wife and our children for much encouragement and undeniable support, who helped bring this book into being. Ahmed L ESFARI September 2021

1 Symplectic Manifolds

1.1. Introduction This chapter is devoted to the study of symplectic manifolds and their connection with Hamiltonian systems. It is well known that symplectic manifolds play a crucial role in classical mechanics, geometrical optics and thermodynamics, and currently have conquered a rich territory, asserting themselves as a central branch of differential geometry and topology. In addition to their activity as an independent subject, symplectic manifolds are strongly stimulated by important interactions with many mathematical and physical specialties, among others. The aim of this chapter is to study some properties of symplectic manifolds and Hamiltonian dynamical systems, and to review some operations on these manifolds. This chapter is organized as follows. In the second section, we begin by briefly recalling some notions about symplectic vector spaces. The third section defines and develops explicit calculation of symplectic structures on a differentiable manifold and studies some important properties. The forth section is devoted to the study of some properties of one-parameter groups of diffeomorphisms or flow, Lie derivative, interior product and Cartan’s formula. We review some interesting properties and operations on differential forms. The fifth section deals with the study of a central theorem of symplectic geometry, namely Darboux’s theorem: the symplectic manifolds (M, ω) of dimension 2m are locally isomorphic to (R2m , ω). The sixth section contains some technical statements concerning Hamiltonian vector fields. The latter form a Lie subalgebra of the space vector field and we show that the matrix associated with a Hamiltonian system forms a symplectic structure. Several properties concerning Hamiltonian vector fields, their connection with symplectic manifolds, Poisson manifolds or Hamiltonian manifolds as well as some interesting examples are studied in the seventh section. We will see in the eight section how to determine a symplectic structure on the orbit of the coadjoint representation of a Lie group. Section nine is dedicated to the explicit determination of symplectic structures Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

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on adjoint and coadjoint orbits of a Lie group SO(n). Some exercises are proposed in the last section. 1.2. Symplectic vector spaces D EFINITION 1.1.– A symplectic space (E, ω) is a finite dimensional real vector space E with a bilinear form ω : E × E −→ R, which is alternating (or antisymmetric), that is, ω(x, y) = −ω(y, x), ∀x, y ∈ E, and non-degenerate, that is, ω(x, y) = 0, ∀y ∈ E =⇒ x = 0. The form ω is referred to as symplectic form (or symplectic structure). The dimension of a symplectic vector space is always even. We show (using a reasoning similar to the Gram–Schmidt orthogonalization process) that any symplectic vector space (E, ω) has a base (e1 , ..., e2m ) called symplectic basis (or canonical basis), satisfying the following relations: ω(em+i , ej ) = δij and ω(ei , ej ) = ω(em+i , em+j ) = 0. Note that each em+i is orthogonal to all base vectors except ei . In terms of symplectic basic vectors (e1 , ..., e2m ), the matrix (ωij ) where ωij ≡ ω(ei , ej ) has the form ⎞ ω11 ... ω12m  0 −Im ⎟ ⎜ .. . . .. , = ⎠ ⎝ . . . Im 0 ω2m1 ... ω2m2m ⎛

where Im denotes the m × m unit matrix.

m E XAMPLE 1.1.– R2m with the form ω(x, y) = k=1 (xm+k yk −xk ym+k ), x ∈ R2m , y ∈ R2m , is a symplectic vector space. Let (e1 , ..., em ) be an orthonormal basis of Rm . Then, ((e1 , 0), ..., (em , 0), (0, e1 ), ..., (0, em )) is a symplectic basis of R2m . Let (E, ω) be a symplectic vector space and F a vector subspace of E. Let F ⊥ = {x ∈ E : ∀y ∈ F, ω(x, y) = 0} be the orthogonal (symplectic) of F . D EFINITION 1.2.– The subspace F is isotropic if F ⊂ F ⊥ , coisotropic if F ⊥ ⊂ F , Lagrangian if F = F ⊥ and symplectic if F ∩ F ⊥ = {0}. If F , F1 and F2 are subspaces of a symplectic space (E, ω), then dim F + dim F ⊥ = dim E, F1 ⊂ F2 =⇒ F2⊥ ⊂ F1⊥ ,

(F ⊥ )⊥ = F,

(F1 ∩ F2 )⊥ = F1⊥ + F2⊥ ,

F1⊥ ∩ F2⊥ = (F1 + F2 )⊥ ,

F is coisotropic if and only if F ⊥ is isotropic and F is Lagrangian if and only if F is isotropic and coisotropic.

Symplectic Manifolds

3

1.3. Symplectic manifolds D EFINITION 1.3.– Let M be an even-dimensional differentiable manifold. A symplectic structure (or symplectic form) on M is a closed non-degenerate differential 2-form ω on M . The non-degeneracy condition means that: ∀x ∈ M , ∀ξ = 0, ∃η: ω (ξ, η) = 0, (ξ, η ∈ Tx M ). The pair (M, ω) (or simply M ) is called a symplectic manifold. At a point p ∈ M , we have a non-degenerate antisymmetric bilinear form on the tangent space Tp M , which explains why the dimension of the manifold M is even.

m E XAMPLE 1.2.– R2m with  the 2-form dxk ∧ dyk is a symplectic  ω =  k=1 ∂ ∂ ∂ manifold. The vectors ∂x1 , ..., ∂xm , ∂y1 , ..., ∂y∂m , p ∈ R2m , p

p

p

p

2m 2m m constitute a symplectic

m basis of the tangent space T R = R . Similarly, C with i the form ω = 2 k=1 dzk ∧ dz k is a symplectic manifold. This form coincides with the previous form by means of the identification Cm R2m , zk = xk + iyk . Riemann surfaces, Kählerian manifolds and complex projective manifolds are symplectic manifolds. Another class of symplectic manifolds consists of the coadjoint orbits (see section 1.8).

We will see that the cotangent bundle T ∗ M (i.e. the union of all cotangent spaces of M ) admits a natural symplectic structure. The phase spaces of the Hamiltonian systems studied below are symplectic manifolds and often they are cotangent bundles equipped with the canonical structure. T HEOREM 1.1.– Let M be a differentiable manifold of dimension m and T ∗ M its cotangent bundle. Then T ∗ M possesses a symplectic structure

mand in a local coordinate (x1 , . . . , xm , y1 , . . . , ym ), the form ω is given by ω = k=1 dxk ∧ dyk . P ROOF.– Let (U, ϕ) be a local chart in the neighborhood of p ∈ M , and consider the

m application ϕ : U ⊂ M −→ Rm , p −→ ϕ(p) = k=1 xk ek , where ek are the vectors basis of Rm . Consider the canonical projections T M −→ M , and T (T ∗ M ) −→ T ∗ M , of tangent bundles, respectively, to M and T ∗ M on their bases. We note π ∗ : T ∗ M −→ M , the canonical projection and dπ ∗ : T (T ∗ M ) −→ T , its linear

M m tangent application. We have ϕ∗ : T ∗ M −→ R2m , α −→ ϕ∗ (α) = k=1 (xk ek + yk εk ), where εk are the basic forms of T ∗ Rm and α denotes αp ∈ T ∗ M . So, if α is a ∗ ∗ ∗ 2m 1-form on M and ξα is a vector

m tangent to T M , then dϕ : T (T M ) −→ T R = 2m ∗ R , ξα −→ dϕ (ξα ) = k=1 (βk ek + γk εk ), where βk , γk are the components of ξα in the local chart of R2m . Consider λα (ξα ) = α(dπ ∗ ξα ) = α(ξ), where ξ is

4

Integrable Systems

a tangent vector to M . Let (x1 , ..., xm , y1 , ..., ym ) be a system of local coordinates compatible with a local trivialization of the tangent bundle T ∗ M . Let us show that:

λα (ξα ) = α

m 

 βk e k

k=1

=

m 

⎛ (xk ek + yk εk ) ⎝

m 

⎞ βj e j ⎠ =

j=1

k=1

m 

βk yk .

k=1

Indeed, let (x1 , ...,

xmm ) be a system of local coordinates around p ∈ M . Since ∀α ∈ T ∗ M , α = coordinates y1 , ..., ym by k=1 αk dxk , then by defining local

m yk (α) = yk , k = 1, ..., m, the 1-form λ is written as λ = k=1 yk dxk . The form λ ∗ on the cotangent bundle T

M (doing correspondence λα to α) is called Liouville m m form. We have λ(α) = k=1 yk (α)dxk (α), λ(α)(ξα ) = k=1 yk (α)dxk (α) 





m m m j=1 βj ej + γj εj = k=1 yk βk = λα (ξα ), λ = k=1 yk dxk . The symplectic ∗ structure of T M is given by the exterior derivative of λ, that is, the 2-form ω = −dλ. The forms λ and ω are called canonical forms on T ∗ M . We can visualize all this with the help of the following diagram: T ∗ (T ∗ M ) ↑λ λα (ξ)

R ←− T (T ∗ M ) −→ ↓dπ ∗ α(ξ)

R ←−

TM

−→

ϕ∗

T ∗M ↓π ∗

−→ R2m

M

−→ Rm

ϕ

The form ω is closed: dω = 0 since d ◦ d = 0 and it is non-degenerate. To show this last property, just note that the form is well defined independently of the chosen coordinates but we can also show it using a direct calculation. Indeed, let ξ = (ξ1 , ..., ξ2m ) ∈ Tp M and η = (η1 , ..., η2m ) ∈ Tp M . We have ω(ξ, η) =

m  k=1

dxk ∧ dyk (ξ, η) =

m 

(dxk (ξ)dyk (η) − dxk (η)dyk (ξ)) .

k=1

Since dxk (ξ) = ξm+k is the (m + k)th component of ξ and dyk (ξ) = ξk is the kth component of ξ, then

ω(ξ, η) =

m  k=1

 (ξm+k ηk − ηm+k ξk ) = (ξ1 ...ξ2m )

O −I I O



⎞ η1 ⎜ .. ⎟ ⎝ . ⎠, η2m ⎛

with O the null matrix and I the unit matrix of order m. Then, for all x ∈ M and for , ..., ξ2m ) = 0, it exists η = (ξm+1 , ..., ξ2m , −ξ1 , ..., −ξm ) such that: all ξ = (ξ1

m  2 ω(ξ, η) = k=1 ξm+k − ξk2 = 0, because ξk = 0, ∀k = 1, ..., 2m. In the local

Symplectic Manifolds

5

coordinate

n system (x1 , ..., xm , y1 , ..., ym ), this symplectic form is written as ω = k=1 dxk ∧ dyk , which completes the proof.  A manifold M is said to be orientable if there exists on M an atlas such that the Jacobian of any change of chart is strictly positive or if M has a volume form (i.e. a differential form that does not vanish anywhere). For example, Rn is oriented by the volume form dx1 ∧ ... ∧ dxn . The circle S 1 is oriented by dθ. The torus T 2 = S 1 × S 1 is oriented by the volume form dθ ∧ dϕ. All holomorphic manifolds are orientable. T HEOREM 1.2.– (a) A closed differential 2-form ω on a differentiable manifold M of dimension 2m is symplectic, if and only if, ω m is a volume form. (b) Any symplectic manifold is orientable. (c) Any orientable manifold of dimension two is symplectic. However, in even dimensions larger than 2, this is no longer true. P ROOF.– (a) This is due to the fact that the non-degeneracy of ω is equivalent to the fact that ω m is never zero. (b) We have ω = dx1 ∧ dxm+1 + · · · + dxm ∧ dx2m in a system of symplectic charts (x1 , ..., x2m ). Therefore, ω m = dx1 ∧dxm+1 ∧...∧dxm ∧ m(m−1) dx2m = (−1) 2 dx1 ∧ dx2 ∧ ... ∧ dx2m , which means that the 2m-form ω m is a volume form on the manifold M and therefore this one is orientable. The orientation associated with the differential form ω is the canonical orientation of R2m . (c) This results from the fact that any differential 2-form on a 2-manifold is always closed.  T HEOREM 1.3.– Let α be a differential 1-form on the manifold M and α∗ λ the reciprocal image of the Liouville form λ on the cotangent bundle T ∗ M . Then, α∗ λ = α. P ROOF.– Since α : M −→ T ∗ M , we can consider the reciprocal image that we note α∗ : T ∗ T ∗ M −→ T ∗ M , of λ : T ∗ M −→ T ∗ T ∗ M (Liouville form), such that, for any vector ξ tangent to M , we have the relation α∗ λ(ξ) = λ(α)(dαξ). Since dα is an application T M −→ T T ∗ M , then α∗ λ(ξ) = λ(α)(dαξ) = λα (dαξ) = αdπ ∗ dα(ξ) = αd(π ∗ α)(ξ) = α(ξ), because π ∗ α(p) = p where p ∈ M and the result follows.  A submanifold N of a symplectic manifold M is called Lagrangian if for all p ∈ N , the tangent space Tp N coincides with the following configuration space {η ∈

Tp M : ωp (ξ, η) = 0, ∀ξ ∈ Tp N }. On this space, the 2-form dxk ∧ dyk that defines the symplectic structure is identically zero. Lagrangian submanifolds are considered among the most important submanifolds of symplectic manifolds. Note that dim N = 1 2 dim M and that for all vector fields X, Y on N , we have ω(X, Y ) = 0. E XAMPLE 1.3.– If (x1 , ..., xm , y1 , ..., ym ) is a local coordinate system on an open U ⊂ M , then the subset of U defined by y1 = · · · = ym = 0 is a Lagrangian submanifold of M . The submanifold α(M ) is Lagrangian in T ∗ M if and only if the form α is closed because 0 = α∗ ω = α∗ (−dλ) = −d(α∗ λ) = −dα.

6

Integrable Systems

Let M be a differentiable manifold, T ∗ M its cotangent bundle with the symplectic form ω, sα : U −→ T ∗ M , p −→ α(p), a section on an open U ⊂ M . From the local expression of ω (theorem 1.6), we deduce that the null section of the bundle T ∗ M is a Lagrangian submanifold of T ∗ M . If sα (U ) is a Lagrangian submanifold of T ∗ M , then sα is called the Lagrangian section. We have (theorem 1.8), s∗α λ = α, and according to the previous example, sα (U ) is a Lagrangian submanifold of T ∗ M if and only if the form α is closed. Let (M, ω), (N, η) be two symplectic manifolds of the same dimension and f : M −→ N , a differentiable application. We say that f is a symplectic morphism if it preserves the symplectic forms, that is, if f satisfies f ∗ η = ω. When f is a diffeomorphism, we say that f is a symplectic diffeomorphism or f is a symplectomorphism. T HEOREM 1.4.– (a) A symplectic morphism is a local diffeomorphism. (b) A symplectomorphism preserves the orientation. P ROOF.– (a) Indeed, since the 2-form ω is non-degenerate, then the differential df (p) : Tp M −→ Tp N , p ∈ M , is a linear isomorphism and according to the local inversion theorem, f is a local diffeomorphism. Another proof is to note that f ∗ η m = (f ∗ η)m = ω m . The map f has constant rank 2m because ω m and η m are volume forms on M and N , respectively. And the result follows. (b) It is deduced from (a) that the symplectic diffeomorphisms or symplectomorphisms preserve the volume form and therefore the orientation. The Jacobian determinant of the transformation is +1.  R EMARK 1.1.– Note that the inverse f −1 : N −→ M of a symplectomorphism f : M −→ N is also a symplectomorphism. Let (M, ω), (N, η) be two symplectic manifolds, pr1 : M × N −→ M , pr2 : M × N −→ N , the projections of M × N on its two factors. The forms pr1∗ ω + pr2∗ η and pr1∗ ω − pr2∗ η on the product M × N are symplectic forms. Take the case where dim M = dim N = 2m and consider a differentiable map f : M −→ N , as well as its graph defined by the set A = {(x, y) ∈ M × N : y = f (x)}. The application g defined by g : M −→ A, x −→ (x, f (x)) is a diffeomorphism. The set A is a 2m-dimensional Lagrangian submanifold of (M × N, pr1∗ ω − pr2∗ η) if and only if the reciprocal image of pr1∗ ω − pr2∗ η by application g is the identically zero form on M . For the differentiable map f to be a symplectic morphism, it is necessary and sufficient that the graph of f is a Lagrangian submanifold of the product manifold (M × N, pr1∗ ω − pr2∗ η). T HEOREM 1.5.– (a) Let f : M −→ M be a diffeomorphism. Then, the application f ∗ : T ∗ M −→ T ∗ M is a symplectomorphism. (b) Let g : T ∗ M −→ T ∗ M be a diffeomorphism such that: g ∗ λ = λ. Then, there is a diffeomorphism f : M −→ M such that: g = f ∗ .

Symplectic Manifolds

7

P ROOF.– (a) Let us show that f ∗∗ ω = ω. We have f ∗∗ λ(α)(ξα ) = λ(f ∗ (α))(df ∗ ξα ) = f ∗ (α)dπ ∗ df ∗ (ξα ) = α(df dπ ∗ df ∗ (ξα )), and therefore, f ∗∗ λ(α)(ξα ) = α(d(f ◦ π ∗ ◦ f ∗ )(ξα )). Since f ∗ α = αf −1 (p) and π ∗ f ∗ α = f −1 (p), then f ◦ π ∗ ◦ f ∗ (α) = p = π ∗ α, that is, f ◦ π∗ ◦ f ∗ = π∗

[1.1]

and f ∗∗ λ(α)(ξα ) = α(dπ ∗ (ξα )) = λα (ξα ) = λ(α)(ξα ). Consequently, f ∗∗ λ = λ and f ∗∗ ω = ω. (b) Since g ∗ λ = λ, then g ∗ λ(η) = λ(dgη) = ω(ξ, dgη) = λ(η) = ω(ξ, η). Moreover, we have g ∗ ω = ω, hence ω(dgξ, dgη) = ω(ξ, η) = ω(ξ, dgη) and ω(dgξ −ξ, dgη) = 0, ∀η. Since the form ω is non-degenerate, we deduce that dgξ = ξ and that g preserves the integral curves of ξ. On the null section of the tangent bundle (i.e. on the manifold), we have ξ = 0 and then g|M is an application f : M −→ M . Let us show that: f ◦ π ∗ ◦ g = π ∗ = f ◦ π ∗ ◦ f ∗ . Indeed, taking the differential, we get df ◦dπ ∗ ◦dg(ξ) = df ◦ dπ ∗ (ξ) = df (ξp ), because dg(ξ) = ξ and ξp ≡ dπ ∗ (ξ)), hence df ◦ dπ ∗ ◦ dg(ξ) = ξp = dπ ∗ (ξ). Therefore, df ◦ dπ ∗ ◦ dg = dπ ∗ , f ◦ π ∗ ◦ g = π ∗ . Since f ◦ π ∗ ◦ f ∗ = π ∗ (according to [1.1]), so g = f ∗ .  T HEOREM 1.6.– Let I : Tx∗ M −→ Tx M , ωξ1 −→ ξ, where ωξ1 (η) = ω (η, ξ), ∀η ∈ Tx M . Then I is an isomorphism generated by the symplectic form ω. P ROOF.– Denote by I −1 the map I −1 : Tx M −→ Tx∗ M , ξ −→ I −1 (ξ) ≡ ωξ1 , with I −1 (ξ)(η) = ωξ1 (η) = ω(η, ξ), ∀η ∈ Tx M . The form ω being bilinear, then we have I −1 (ξ1 + ξ2 )(η) = I −1 (ξ1 )(η) + I −1 (ξ2 )(η), ∀η ∈ Tx M . To show that I −1 is bijective, it suffices to show that it is injective (because dim Tx M = dim Tx∗ M ). The form ω is non-degenerate, and it follows that KerI −1 = {0}. Hence, I −1 is an isomorphism and consequently I is also an isomorphism.  1.4. Vectors fields and flows  Let M be a differentiable manifold of dimension m. Let T M = x∈M Tx M , be the bundle tangent to M (union of spaces tangent to M at all its points x). This bundle has a structure of a differentiable manifold of dimension 2m and allows us to immediately transfer to manifolds the theory of ordinary differential equations. D EFINITION 1.4.– A vector field (also called a tangent bundle section) on M is an application, noted by X, which at any point x ∈ M associates a tangent vector Xx ∈ Tx M . In other words, it is an application: X : M −→ T M , such that if π : T M −→ M , is the natural projection, then we have π ◦ X = idM .

8

Integrable Systems

Figure 1.1. Vector field

Figure 1.2. Tangent space

Note that the diagram M

X

−→ T M  idM ↓π M

is commutative. In a local coordinate system (x1

, ..., xm ) in a neighborhood U ⊂ M , m the vector field X is written in the form X = k=1 fk (x) ∂x∂ k , x ∈ U , where the functions f1 , . . . , fm : U −→ R, are the components of X with respect to (x1 , ..., xm ). A vector field X is differentiable if its components fk (x) are differentiable functions. This definition of differentiability does not obviously depend on the choice of the local coordinate system.

Indeed, if (y1 , ..., ym ) is another local m ∂ coordinate system in U , then X = ∈ U , where k=1 hk (x) ∂yk , x h1 , . . . , hm : U −→ R, are the components of X in relation to (y , ..., ym ) and the 1

m k result follows from the fact that hk (x) = l=1 ∂x∂y ), x ∈ U . To the vector field l fl (x X corresponds to a system of differential equations dx1 dxm = f1 (x1 , ..., xm ) , ..., = fm (x1 , ..., xm ) . dt dt

[1.2]

D EFINITION 1.5.– A differentiable vector field X over M is called a dynamical system. A vector field is written locally in the form [1.2].

Symplectic Manifolds

9

D EFINITION 1.6.– An integral curve (or trajectory) of a vector field X is a differentiable curve γ : I −→ M , t −→ γ(t), such that ∀t ∈ I, dγ(t) dt = X (γ(t)), where I is an interval of R.

m ∂ If k=1 fk (x) ∂xk is the local expression of X, then the integral curves (or trajectories) of X are the solutions γ(t) = {xk (t)} of [1.2]. We assume in the following that the vector field X is differentiable (of class C ∞ ) and with compact support (i.e. X is zero outside of a compact of M ). This will especially be the case if the manifold M is compact. Given a point x ∈ M , we denote by gtX (x) (or quite simply gt (x)) the position of x after a displacement of a duration t ∈ R.

Figure 1.3. Flow

We therefore have an application gtX : M −→ M , t ∈ R, which is a diffeomorphism (a one-to-one differentiable mapping with a differentiable inverse), by virtue of the theory of differential equations. More precisely, to the vector field X we associate a one-parameter group of diffeomorphisms gtX on M , that is, a differentiable application (of class C ∞ ): M × R −→ M , verifying a group law: (i) ∀t ∈ R, gtX : M −→ M is a diffeomorphism of M on M . (ii) ∀t, s ∈ R, X gt+s = gtX ◦ gsX . Condition (ii) means that the correspondence t −→ gtX is a homomorphism of the additive group R in the group of diffeomorphisms from M to  −1 X M . It implies that g−t = gtX , because g0X = idM is the identical transformation that leaves each point invariant. D EFINITION 1.7.– The one-parameter group of diffeomorphism gtX on M is called d X flow. It admits the vector field X for velocity field dt gt (x) = X gtX (x) , with the X initial condition: g0 (x) = x.  d X Obviously, dt gt (x)t=0 = X(x). So through these formulas gtX (x) is the curve on the manifold,  which passes through x such that the tangent at each point is the vector X gtX (x) . We will now see how to construct the flow gtX over the whole variety M . T HEOREM 1.7.– The vector field X generates a unique one-parameter group of diffeomorphism of M . X P ROOF.– a) Construction equation  X  of gt for small t. For x fixed, the differential d X X g (x) = X g (x) , function of t with the initial condition: g (x) = x, admits a t 0 dt t

10

Integrable Systems

unique solution gtX defined in the neighborhood of the point x0 and depending on the initial condition C ∞ . So gtX is locally a diffeomorphism. Therefore, for each point x0 ∈ M, we can find a neighborhood U (x0 ) ⊂ M , a positive real number ε ≡ ε (x0 ) such that for all t ∈ ]−ε, ε[, the differential equation in question with its initial condition admits a unique solution differentiable gtX (x) defined in U (x0 ) and X verifying the group relation gt+s (x) = gtX ◦ gsX (x), with t, s, t + s ∈ ]−ε, ε[. Indeed, X let us pose x1 = gt (x), t fixed and consider the solution of the differential equation X satisfying in the neighborhood of the point x0 to the initial condition gs=0 = x1 . This solution satisfies the same differential equation and coincides at a point gtX (x) = x1 , X with the function gt+s . Therefore, by uniqueness of the solution of the differential equation, the two functions are locally equal. Therefore, the application gtX is locally a diffeomorphism. We recall that the vector field X is supposed to be differentiable (of class C ∞ ) and with compact support K. From the open cover of K formed by U (x), we can extract a finite subcover (Ui ), since K is compact. Let us denote εi by the numbers ε corresponding to Ui and put ε0 = inf (εi ), gtX (x) = x, x ∈ / K. Therefore, the differential equation in question admits a unique solution gtX on X M × ]−ε0 , ε0 [ verifying the group relation: gt+s = gtX ◦ gsX , the inverse of gtX being X X g−t and so gt is a diffeomorphism for t small enough. b) Construction of gtX for all t ∈ R. According to (a), it suffices to construct gtX for t ∈ ]−∞, −ε0 [∪]ε0 , ∞[. We will see that the applications gtX are defined according to the multiplication law in the form t = k ε20 + r,  ofε0the  group. Note that t can be written ∗ with k ∈ Z and r ∈ 0, 2 . Let us consider, for t ∈ R+ and for t ∈ R∗− , X X ε0 ◦ · · · ◦ g ε0 ◦ g gtX = g X r , 2  2  

X X X ε0 ◦ · · · ◦ g ε0 ◦ g gtX = g− r , − 2  2  

k−times

k−times

X X respectively. The diffeomorphisms g± ε0 and gr were defined in (a), and we deduce 2

that for any real t, gtX is a diffeomorphism defined globally on the manifold M . 

C OROLLARY 1.1.– Every solution of the differential equation dx(t) dt = X(x(t)), x ∈ M , with the initial condition x (for t = 0), can be extended indefinitely. The value of the solution gtX (x) at the instant t is differentiable with respect to t and x. With a slight abuse of notation, we can write the preceding differential equation in the form of the system of differential equations [1.2] with the initial conditions x1 , ..., xm for t = 0. With the vector field X, we associate the first-order differential of the operator LX . We refer here to the differentiation of functions in the direction   d field X. We have LX : C ∞ (M ) −→ C ∞ (M ), F −→ LX F (x) = dt F gtX (x) t=0 , x ∈ M . Here, C ∞ (M ) designates the set of functions F : M −→ R of class C ∞ . The operator LX is linear: LX (α1 F1 + α2 F2 ) = α1 LX F1 + α2 LX F2 , (α1 , α2 ∈ R), and satisfies Leibniz’s formula: LX (F1 F2 ) = F1 LX F2 + F2 LX F1 . Since LX F (x) only depends on the values of F in the neighborhood of x, we can therefore apply the

Symplectic Manifolds

11

operator LX without the need to extend them to the whole manifold M . Let (x1 , ..., xm ) be local coordinates on M . In this coordinate system, the vector X is given by its components f1 , . . . , fm and the flow gtX is given by the system of differential equations [1.2]. So the derivative of the function F = F (x1 , ..., xm ) in ∂F ∂F the direction X is LX F = f1 ∂x + · · · + fm ∂x . In other words, in the coordinates 1 m ∂ + · · · + fm ∂x∂m . (x1 , ..., xm ) the operator LX has the form LX = f1 ∂x 1 D EFINITION 1.8.– We say that two vector fields X1 and X2 on a manifold M commute (or are commutative) if and only if the corresponding flows commute, gtX1 1 ◦ gtX2 2 (x) = gtX2 2 ◦ gtX1 1 (x),

∀x ∈ M.

Figure 1.4. Commutative flows

T HEOREM 1.8.– Two vector fields X1 and X2 on a manifold M commute if and only if, [LX1 , LX2 ] ≡ LX1 LX2 − LX2 LX1 = 0. P ROOF.– a) Let us first show that the condition is necessary. Note that,   ∂ 2   X2 F gt2 ◦ gtX1 1 (x) − F gtX1 1 ◦ gtX2 2 (x)  ∂t1 ∂t2 t2 =t1 =0 = (LX1 LX2 − LX2 LX1 ) F (x),

∀F ∈ C ∞ (M ),

Indeed, according to the definition of LX2 , we find,    ∂ X2 X1 = LX2 F gtX1 1 (x) . F gt2 ◦ gt1 (x)  ∂t2 t2 =0

∀x ∈ M.

12

Integrable Systems

Thus,     ∂2 ∂ X2 X1 X1  F gt2 ◦ gt1 (x)  = LX2 F gt1 (x)  , ∂t1 ∂t2 ∂t1 t2 =t1 =0 t1 =0   ∂ X1 = G gt1 (x)  where G ≡ LX2 F, ∂t1 t1 =0 = LX1 G(x) by definition of LX1 , = LX1 LX2 F (x) . Likewise, we have    ∂2 F gtX1 1 ◦ gtX2 2 (x)  = LX1 F gtX2 2 (x) , ∂t2 ∂t1 t1 =0 and   ∂2 F gtX1 1 ◦ gtX2 2 (x)  = LX2 LX1 F (x) . ∂t2 ∂t1 t2 =t1 =0 Therefore,     ∂2 ∂2 X1 X2 X1 X2  F gt1 ◦ gt2 (x)  − F gt1 ◦ gt2 (x)  ∂t2 ∂t1 ∂t2 ∂t1 t1 =0 t2 =t1 =0     2  ∂ X2 X1 X1 X2 F gt2 ◦ gt1 (x) − F gt1 ◦ gt2 (x)  = , ∂t1 ∂t2 t2 =t1 =0 = LX1 LX2 F (x) − LX2 LX1 F (x) . t1 So if X1 and X2 commute on M , that is, gX ◦ gtX2 2 (x) = gtX2 2 ◦ gtX1 1 (x), ∀x ∈ 1 M , then according to the above formula, (LX1 LX2 − LX2 LX1 ) F (x) = 0, ∀F ∈ C ∞ (M ), ∀x ∈ M . Consequently, LX1 LX2 = LX2 LX1 .

b) Let us show that gtX1 1 ◦ gtX2 2 (x) = gtX2 2 ◦ gtX1 1 (x), ∀x ∈ M, that is, that the X1 X2 condition is sufficient, or that: F gt1 ◦ gt2 (x) = F gtX2 2 ◦ gtX1 1 (x) , ∀F ∈ C ∞ (M ), ∀x ∈ M . Let us pose ξ = gtX1 1 ◦ gtX2 2 (x), ζ = gtX2 2 ◦ gtX1 1 (x), and

Symplectic Manifolds

13

develop in Taylor series the function F (ξ) − F (ζ) at the neighborhood of t1 = t2 = 0. We have, 

  ∂ (F (ξ) − F (ζ))  ∂t1 t1 =t2 =0  2    2   ∂ t1 ∂  +t2 (F (ξ) − F (ζ))  + (F (ξ) − F (ζ))  2 ∂t2 2 ∂t1 t1 =t2 =0 t1 =t2 =0     2 2 2   ∂ t2 ∂  + (F (ξ) − F (ζ))  + t1 t2 (F (ξ) − F (ζ))  2 2 ∂t2 ∂t1 ∂t2 t1 =t2 =0 t1 =t2 =0 3 3 2  2 +o t1 , t2 , t1 t2 , t1 t2 . F (ξ) − F (ζ) = F (x) − F (x) + t1

Let us calculate the different terms. We have     ∂ ∂ X1 X2  F (ξ) = F gt1 ◦ gt2 (x)  ∂t1 ∂t1 t1 =t2 =0 t1 t2 =0    = = Lx1 F gtX2 2 (x)  = Lx1 F (x), t2 =0

and     ∂ ∂ F (ζ) = F gtX2 2 ◦ gtX1 1 (x)  , ∂t1 ∂t1 t1 =t2 =0 t1 =t2 =0    ∂  X1 = G gt1 (x)  where G = F gtX2 2  , ∂t1 t2 =0 t1 =0    = Lx1 G(x) = Lx1 F gtX2 2  = Lx1 F (x). t2 =0

  = 0. By symmetry, we also have Therefore, we have ∂t∂1 (F (ξ) − F (ζ)) t1 =t2 =0   ∂ = 0. Likewise, we have ∂t2 (F (ξ) − F (ζ)) t1 =t2 =0

  ∂ 2   X1 ∂2 F gt1 ◦ gtX2 2 (x) (F (ξ) − F (ζ)) = 2 2 ∂t1 ∂t1 t1 =t2 =0    −F gtX2 2 ◦ gtX1 1 (x) 

t1 =t2 =0

.

14

Integrable Systems

Now

∂ ∂t1 F



gtX1 1 ◦ gtX2 2 (x) =

∂ ∂t1 F



 gtX1 1 (y) = LX1 F gtX1 1 (y) , where y =

gtX2 2 (x), so   ∂ ∂ 2  X1 F gt1 ◦ gtX2 2 (x) = LX1 F gtX1 1 (y) = LX1 LX1 F gtX1 1 (y) , 2 ∂t1 ∂t1  = LX1 LX1 F gtX1 1 ◦ gtX2 2 (x) −→ LX1 LX1 F (x). t1 =t2 =0

 Likewise, we have ∂t∂1 F gtX2 2 ◦ gtX1 1 (x)  LX1 G gtX1 1 (x) , where G = F gtX2 2 , hence

=

∂ ∂t1 G



gtX1 1 (x)

=

  ∂ ∂ 2  X2 X1 X1 X1 (x) = (x) , F g ◦ g L G g (x) = L L G g X X X t t t t 1 1 1 1 1 2 1 ∂t21 ∂t1  = LX1 LX1 F gtX2 2 ◦ gtX1 1 (x) −→ LX1 LX1 F (x). t1 =t2 =0

  (F (ξ) − F (ζ)) = 0. It follows, by symmetry, that t1 =t2 =0   (F (ξ) − F (ζ)) = 0. Moreover, we deduce from the necessary condition

Thus, ∂2 ∂t22

∂2 ∂t21

t1 =t2 =0

and from the fact that the vector fields X1 and X2 commute the following relation: ∂2 (F (ξ) − F (ζ))|t1 =t2 =0 ∂t1 ∂t2   ∂ 2   X1 X2 X2 X1 F gt1 ◦ gt2 (x) − F gt2 ◦ gt1 (x)  = , ∂t1 ∂t2 t1 =t2 =0 =

∂2 (LX2 LX1 − LX1 LX2 ) F (x) = 0. ∂t1 ∂t2

    Therefore, F gtX1 1 ◦ gtX2 2 (x) − F gtX2 2 ◦ gtX1 1 (x) = o t31 , t32 , t21 t2 , t1 t22 . Consider the times t1 and t2 of the order ε. We find a difference between the two new points of the manifold, depending on whether we apply the field  X1 before the 3 field X2 or the inverse, of the order of ε . F gtX1 1 ◦ gtX2 2 (x) −    F gtX2 2 ◦ gtX1 1 (x) = o ε3 . Now, if t1 and t2 are arbitrary fixed times, let us square the space between the two paths with squares of sides ε. Each square represents the small space traveled during a small time ε, either according to the field X1 or according to the field X2 . We have found that when the space between two paths

Symplectic Manifolds

15

differs from that of a square, we get a difference ε3 . By modifying the path traveled by a square in successive stages, we obtain   t t   1 2 F gtX1 1 ◦ gtX2 2 (x) − F gtX2 2 ◦ gtX1 1 (x) ≤ 2 o ε3 , ε by the fact that we have tε1 × tε2 steps intermediaries. This is valid for all ε; just take   ε small enough, tending to zero, so that t1ε2t2 o ε3 = t1 t2 o (ε) −→ 0.  ε→0

Figure 1.5. Proof of the commutativity of flows

Since every first-order linear differential operator is given by a vector field, LX2 LX1 − LX1 LX2 being a first-order linear differential operator, the latter also corresponds to some vector field that we denote by X3 . D EFINITION 1.9.– The Poisson bracket or commutator of two vector fields X1 and X2 on the manifold M , denoted by X3 = {X1 , X2 } or X3 = [X1 , X2 ], is the vector field X3 for which LX3 = LX2 LX1 − LX1 LX2 . E XAMPLE 1.4.– The Poisson bracket transforms the vector space of vector fields over a manifold into Lie algebra. Let X be a vector field on a differentiable manifold M . We have shown (theorem 1.7) that X generates a unique one-parameter group of diffeomorphism gtX (which we d X also denote by gt ) on M , solution to the differential equation: dt gt (p) = X(gtX (p)), X p ∈ M , with the initial condition g0 (p) = p. Let ω be a k-form differential on M . respect to X is the k-form differential D EFINITION 1.10.– The Lie  derivative of ωg∗with (ω(gt (p)))−ω(p) d ∗  gt ω t=0 = limt→0 t . defined by LX ω = dt t In general, for t = 0, we have     d ∗ d ∗ ∗ d ∗   gt ω = gt+s ω  gs ω  = gt = gt∗ (LX ω). dt ds ds s=0 s=0

[1.3]

For all t ∈ R, the application gt : R −→ R being a diffeomorphism then dgt and dg−t are the applications, dgt : Tp M −→ Tgt (p) M , dg−t : Tgt (p) M −→ Tp M .

16

Integrable Systems

D EFINITION 1.11.– The Lie derivative of a vector field Y in the direction X is defined (p) d g−t Y t=0 = limt→0 g−t (Y (gt (p)))−Y . by LX Y = dt t In general, for t = 0, we have     d d d  = g−t = g−t (LY ). g−t Y = g−t−s Y  g−s Y  dt ds ds s=0 s=0 D EFINITION 1.12.– The interior product of a k-form differential ω by a vector field X on the differentiable manifold M is a (k − 1)-form differential, iX ω, defined by (iX ω)(X1 , ..., Xk−1 ) = ω(X, X1 , ..., Xk−1 ), where X1 , ..., Xk−1 are vector fields. It is easy to show that if ω is a k-differential form, λ a differential form of any degree, X and Y two vector fields, f a linear map and a a constant, then iX+Y ω = iX ω + iY ω, iX iX ω = 0,

iaX ω = aiX ω,

iX iY ω = −iY iX ω,

iX (f ω) = f (iX ω),

iX (ω ∧ λ) = (iX ω) ∧ λ + (−1)k ω ∧ (iX λ),

iX f ∗ ω = f ∗ (if X ω),

where f ∗ ω denote the pull-back by f . E XAMPLE 1.5.– Let us calculate the expression of the interior product in local m ∂ coordinates. If X = j=1 Xj (x) ∂x is the local expression of the vector field on the j

manifold M of dimension m and ω = i1 0 are constants. Moreover, if we consider the continued fraction, ψ(z) =

γ0 λ1 z − μ 1 −

γ1 γ λ2 z−μ2 − λ z−μ2 − 3

3

..

.

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Integrable Systems

and realize an equivalent transformation γ0

ψ(z) = z−

μ1 λ1

−

γ1 λ1 λ2 μ

z− λ2 − 2

z−

γ2 λ2 λ3 μ3 − λ3

..

. γ

μ

j we reconstruct the Γ-fraction corresponding to dσ(x) and put λj λj+1 = a2j , λjj = bj . It follows that there is a one-to-one correspondence between the set of Jacobi matrices and all the orthogonal polynomial systems on R. If we choose the 0 orthogonal polynomials Pn = γn−1 Bn−1 (x), as the basis of the vector space j=1

consisting of all polynomials, then the Jacobi matrix represents the multiplication by x. With the Jacobi matrix, we associate an operator T on a separable Hilbert space E as follows, T e0 = b0 e0 + a0 e1 , T ej = aj−1 ej−1 + bj ej + aj ej+1 , j = 1, 2, ... where (e1 , ...) is an orthonormal basis in E. The operator T is symmetric. Indeed, for any two finite vectors u and v, we have T u, v = u, T v, according to the symmetry

∞ 1 of the Jacobi matrix. Moreover, if the (Carleman’s) condition: j=0 aj = +∞, is satisfied, then the operator T is self-adjoint and its spectrum is simple with e0 a generating element. In this case, the information about the spectrum of T is contained in function, + ∞   dσ(x) ϕ(z) = (T − zI)−1 e0 , e0 = , [5.10] −∞ z − x defined at z ∈ / σ(T ) where σ(x) = Ix e0 , e0  and Ix is the resolution of the identity of the operator T . Recall that the infinite continued fraction converges if the limit [5.9] exists. If the operator T is self-adjoint, then the continued fraction ϕ(z) converges uniformly in any closed bounded domain of z without common points with the real axis to the analytic function defined by [5.10]. If the support of dσ(x) is bounded,

Ak (z) then the sequence B converges uniformly to a holomorphic function near z = k (z) ∞. Moreover, if a Jacobi matrix is bounded, that is, if there exists ρ > 0 such that ∀j, |aj | ≤ ρ3 , |bj | ≤ ρ3 , then the associated Γ-fraction converges uniformly on the following domain {z : |z| ≥ ρ} and the support of dσ(x) is included in [−ρ, ρ]. In the case of a periodic Jacobi matrix, this one is obviously bounded and therefore the associated Γ-fraction converges near z = ∞. In addition, the function ϕ(z) is written in the form [5.10] (Cauchy–Stieltjes transform of dσ(x)), which shows that ϕ(z) has zero of first order at z = ∞ and, for any point z belonging to the upper-half plane, the imaginary part of ϕ(z) is non-positive.

We will now extend the Jacobi matrix Γ to the infinite symmetric, tridiagonal and N -periodic Jacobi matrix A [5.2] and use the results obtained previously. We consider ϕ(z) [5.8] as being the associated N -periodic Γ-fraction. The latter converges near the infinite point z = ∞. After an analytic prolongation, the function ϕ(z) coincides

The Spectrum of Jacobi Matrices and Algebraic Curves

139

with a0 f1 where f1 is a meromorphic function on the genus N − 1 hyperelliptic curve C [5.4]. The latter is branched at the 2N real zeros ξ1 , ξ2 ,...,ξ2N of the polynomial P 2 (z) − 4α2 . The interval [ξ2j−1 , ξ2j ], 1 ≤ j ≤ N , is called the stable band and the interval [ξ2j , ξ2j+1 ], 1 ≤ j ≤ N − 1, is called the unstable band. T HEOREM 5.2.– Each zero σ1 < σ2 < · · · < σN −1 of Δk,l [5.5] belongs to the jth finite unstable band [λ2j , λ2j+1 ], 1 ≤ j ≤ N − 1. The function ϕ(z) can be expressed (see below) by means of Abelian integrals on the hyperelliptic curve C [5.4]. For N = 1, Bk (x) is the kth Tschebyscheff polynomial of the second type. For N > 1, a new phenomenon related to discrete measures was ΔN,1 discovered by authors including Kato. We have seen that ϕ(z) = a0 f1 = a0 ΔN,N h, belonging to L(D + P − Q). We then have the following theorem: T HEOREM 5.3.– The function ϕ(z) can be explicitly written by means of Abelian integrals on the hyperelliptic curve C [5.4] as follows:

ϕ(z) =

N −1  j=1

N Res(ϕ(z), σj− )  (−1)N +1 + z − σj 2πi j=1

+

ξ2l

ξ2l−1

" P 2 (x) − 4α2 dx, (z − x)ΔN,N (x) [5.11]

αh(σj− )+(−1)N a20 .Λ  , l=j (σj −σl )

where Res(ϕ(z), σj− ) = ⎛ ⎜ ⎜ ⎜ ⎜ Λ ≡ det ⎜ ⎜ ⎜ ⎝

b2 − σj a2 0 .. . 0

√

The

differentials

a2

···

0

b3 − σj a3 .. . a 3

··· obtained

..

.

..

..

in

the

and 0 .. . 0

. . aN −2 0 aN −2 bN −1 − σj previous

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

section,

a√

ΔN,N (x) P 2 (x)−4α2

P 2 (x)−4α2

dx,

b ΔN,N (x) dx, (a and b are constants), are positive measures on each stable band [ξ2j−1 , ξ2j ]. Therefore, expression [5.11] means that ϕ(z) can be obtained by the Cauchy–Stieltjes transform of dσ =

N −1  j=1

Res(ϕ(z), σj− ).δ(x

(−1)N +1 − σj )dx + . 2πi

= discrete measure + continuous measure,

" P 2 (x) − 4α2 dx, ΔN,N (x)

140

Integrable Systems

 ∞ dσ as follows, ϕ(z) = −∞ z−x . The function ϕ(z) belongs to L(D + P − Q) where + + D = σ1 + · · · + σN −1 is contained in C+ = {p ∈ C : |h| > 1} (theorem 5.1). From expression [5.11], we have D = σj−1 + · · · + σj−l + σj+l+1 + · · · + σj+N −1 , where j1 < j2 < ... < jl denote the numbers for which Res(ϕ(z), σj− ) > 0 and jl+1 < jl+2 < ... < jN −1 the numbers for which Res(ϕ(z), σj− ) = 0. Hence, P 2 (σj− ) − 4α2 − / Res(ϕ(z), σj ) = 0 or − . l=j (σj − σl ) 5.4. Exercises E XERCISE 5.1.– Prove the statement made in section 5.2 that the symplectic structure is given by formula [5.7]. E XERCISE 5.2.– Prove theorem 5.2. E XERCISE 5.3.– Let Ik (z) be the infinite part of continued fraction [5.8] starting with nth elements of sequences an and bn . In the notation of section 5.3, express Ik (z) in terms of polynomials Ak and Bk . E XERCISE 5.4.– We have shown (section 5.3) how, with the Jacobi matrix, we can associate an operator T on a separable Hilbert space. Assume that limk→∞ |bk | = ∞, a2

1 k−1 lim sup |bk−1 bk | < 4 . Show that the operator T has a discrete spectrum.

E XERCISE 5.5.– Let λ1 , ..., λn , n > 3, be n real numbers. Show that there exists a periodic Jacobi matrix: ⎛ ⎜ ⎜ ⎜ ⎜ Γ=⎜ ⎜ ⎜ ⎝

b1 a1

0

···

a1 b2 a2 . .. . 0 a2 . . .. .. .. . . . aN · · · · · · aN −1

⎞ aN .. ⎟ . ⎟ ⎟ ⎟ , 0 ⎟ ⎟ ⎟ aN −1 ⎠ bN

ai = 0,

with the λi as periodic spectrum if and only if they can be ordered so that either (i) λ1 > λ2 ≥ λ3 > λ4 ≥ λ5 > · · · , and in this case the matrix Γ can be taken to have all ai positive or (ii) λ1 = λ2 > λ3 ≥ λ4 > λ5 ≥ · · · , and in this case the matrix Γ will have some ai negative. An alternative statement in this case is that there exists a periodic Jacobi matrix Γ with all ai positive and the λi as antiperiodic spectrum (hint: see (Ferguson 1980)).

The Spectrum of Jacobi Matrices and Algebraic Curves

141

E XERCISE 5.6.– Prove the assertions made in section 5.3 that if the operator T is self-adjoint, then the continued fraction ϕ(z) [5.8] converges uniformly in any closed bounded domain of z without common points with the real axis, to the analytic function defined by [5.10].

6 Griffiths Linearization Flows on Jacobians

The aim of this chapter is to present the Griffiths linearization method of studying integrable systems, summarizing the situations discussed in Chapter 4. Griffiths has found necessary and sufficient conditions on the matrix B, without reference to the Kac–Moody algebras, so that the flow of the Lax form [4.1] can be linearized on the Jacobian variety Jac(C) for the spectral curve C defined by [4.2]. These conditions are cohomological and we will see that the Lax equations turn out to have a very natural cohomological interpretation. These results are exemplified by the Toda lattice, the Lagrange top, Nahm’s equations and the n-dimensional rigid body. 6.1. Spectral curves Suppose that for every p(h, z) belonging to the curve of affine equation [4.1], that is, C = {(h, z) : det(A − zI) = 0},

[6.1]

with dim ker(A − zI) = 1 (i.e. the corresponding eigenspace of A is one-dimensional) and generated by a vector v(t, p) ∈ V where V Cn is an n-dimensional vector space. There is then a family of holomorphic mappings that send (h, z) ∈ C to ker(A − zI): ft : C −→ PV,

p −→ Cv(t, p).

[6.2]

(We call this the eigenvector map associated with the Lax equation). We set: Lt = ft∗ (OPV (1)) ∈ Picd (C) ∼ = Jac(C),

L = L0

[6.3]

where d = deg ft (C); OPV (1) is the hyperplane line bundle on PV and Picd (C) the Picard variety of C. Let us recall that it is the set of straight bundles of degree d on C. Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

144

Integrable Systems

By continuity, the degree ofLt does not  vary with time t. Let H be the hyperplane class of PV . We have deg Lt = C ft∗ H = ft (C) H. This expression is the Poincaré dual of the class [C] of C and coincides with the degree of C. Hence, deg Lt = deg(C). While t varies, Lt moves in Picd (C). Therefore, if we fix a line bundle L0 ∈ Picd (C), the line 1 1 bundle L−1 0 ⊗ Lt moves in the Jacobian variety Jac(C) = H (C, OC )/H (C, Z) −1 0 ∗ H (C, ΩC ) /H1 (C, Z), that is, the mapping L −→ L0 ⊗ L induces a morphism Picd (C) Jac(C). The motion of the line bundle L−1 0 ⊗ Lt depends on the choice of the matrix B. A question arises: determine necessary and sufficient conditions on the matrix B so that the flow t −→ Lt ∈ Jac(C),

[6.4]

can be linearized on the Jacobian variety Jac(C). 6.2. Cohomological deformation theory As we have pointed out, Griffiths has found necessary and sufficient conditions of a cohomological nature on B that the flow t −→ Lt ∈ Jac(C) be linear. His method is based on the observation that the tangent space to any deformation lies in a suitable cohomology group and on algebraic curves, and higher cohomology can always be eliminated using duality theory. In fact, by applying more or less standard cohomological techniques from deformation theory (Arbarello et al. 1985), we may give necessary and sufficient conditions that the map t −→ Lt be linear. Let f : C −→ X,

[6.5]

be a non-constant holomorphic map where C is a given smooth algebraic curve and X is a complex manifold. We define the normal sheaf of C in X by the exact sequence f∗

0 −→ ΘC −→ f ∗ ΘX −→ Nf −→ 0

[6.6]

where ΘC , ΘX are the respective tangent sheaves and f∗ is the differential of f . Then the Kodaira–Spencer tangent space (Arbarello et al. 1985) to the moduli space of the map (6.5) is given by H 0 (C, Nf ). If ft : C −→ X, f0 = f , is a deformation of [6.5], then f˙ ∈ H 0 (C, Nf ) the corresponding infinitesimal deformation at t = 0, that is, in local product coordinates (z, t) on ∪t Ct and w = (w1 , w2 , ..., wn ) of X, ft is given by (t, ξ) −→ w(t, ξ), then the section f˙ ∈ H 0 (C, Nf ) is locally given by ∂w(t,ξ)  modulo ∂w(0,ξ) . The corresponding cohomological sequence of [6.6] is ∂t  ∂z t=0

∂

H 0 (ΘC ) −→ H 0 (f ∗ ΘX ) −→ H 0 (Nf ) −→ H 1 (ΘC ). Here, H 1 (ΘC ) is the tangent space to the moduli space of C as an abstract curve and ∂(f˙) ≡ C˙ ∈ H 1 (ΘC ) is the tangent to the family of curves {Ct }. Thus, the tangent space to deformations of [6.5] where the curve C remains fixed is given by H 0 (f ∗ ΘX )/H 0 (ΘC ) ⊂ H 0 (Nf ). Since

Griffiths Linearization Flows on Jacobians

145

the isospectral curve C is independent of t, this is the situation that we are interested in. Here, we take again the vector space V of dimension n and assume that X = PV (projective space) and consider the Euler sequence p

i

0 −→ OPV −→ V ⊗ OPV (1) −→ OPV −→ 0

[6.7]

This is an exact sequence of vector bundles, and it remains exact after pulling back to C via f ∗ (combining this with [6.6]). We have a diagram of exact sequences (L = f ∗ OPV (1)): 0 ↓ OC ↓v V ⊗L ↓ f∗

0 −→ ΘC −→ f ∗ ΘPV −→ Nf −→ 0 ↓ 0 The associated cohomology diagram contains the following piece: H 0 (C, V ⊗ L) ↓τ j

δ

H 0 (C, ΘC ) −→ H 0 (C, f ∗ ΘPV ) −→ H 0 (C, Nf ) −→ H 1 (C, ΘC ) ↓δ H 1 (C, OC ) Consider the family of holomorphic maps ft : C −→ PV . Locally, choose a coordinate ξ on C and also a position vector mapping (t, ξ) −→ v(t, ξ) ∈ V \{0}, that is, a local lift vt of ft to V \{0}, such that ft (ξ) = C.v(t, ξ) ⊂ V . Note that vt is a time-dependent map C −→ V \{0}. This lift is not canonical and exists only locally, but we are going to use it to define an object denoted by v, ˙ which will be independent of the lift and therefore will be globally well defined. Since OPV is the tautological bundle of PV , the fiber of f ∗ OPV (−1) at a point p ∈ C may be identified with the space Cvt (p), which defines the maps f ∗ OPV (−1)V ⊗ OC and vt : OC −→ V ⊗ Lt , φ −→ φvt , where v0 coincides with the application v mentioned in the previous diagram. If v2 is another lift given by v2(t, ξ)  = κ(t, ξ)v(t, ξ), κ = 0, then we have ∂v(t,ξ)  ˙v2 = κv˙ + κv. modulo v(t, ξ). The latter quantity is ˙ Let us set, v(ξ) ˙ =  ∂t

t=0

well defined of the representative position mapping of v, that is, since the inclusion v OC → V ⊗ L, L = f ∗ OPV (1), is locally given by OC  φ −→ φ.v, it follows

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Integrable Systems

that v˙ ∈ H 0 (C, V ⊗ L/OC ) = H 0 (C, f ∗ ΘPV ) is well defined and independent of the choice of the lift. Then we have j(v) ˙ =f

[6.8]

We are interested in the tangent vector L˙ ≡



dLt  dt t=0

∈ H 1 (C, OC ).

T HEOREM 6.1.– We have L˙ = δ(v), ˙ where v˙ is the infinitesimal variation of ft : C −→ PV and in particular, L˙ = 0, if and only if v˙ = τ (w) for some w ∈ H 0 (V ⊗ L) where τ is the map in the above diagram. P ROOF.– Let (w0 , ..., wn ) be homogeneous coordinates in PV associated with a fixed basis of V and let Ui be the corresponding open cover of PV .

We shall write wk ov = n ∂ k v . The quantity v˙ as a vector field on PV may be written as i=0 v˙ i ∂w i . In order to calculate the action of the connecting morphism δ on v, ˙ we must take a counterimage of v˙ under the morphism p of [6.7]; this is the n-ple (v˙ 0 , ..., v˙ n ). Now we must apply ˘ the Cech differential associated with the cover f −1 (U i) of C and invert the map i. Since the latter is f −→ (f w0 , ..., f wn ), we get the cocycle vv˙ kk − vv˙ ii that describes a class in H 1 (C, OC ). On the other hand, the transition functions of the line bundle vk k ˙ Lt may be written as gik (t) = w wi oft = vi , and the cocycle corresponding to L is −1 v˙ k v˙ i therefore εik = g˙ ik gik = vk − vi . The second assertion results from the exactness of the column in the diagram above. 

N

N N −k k k We write B(t, h) = = h1 , where we have k=0 Bk (t)h k=0 Bk (t)h0 1 regarded h as an affine coordinate in the P , which is the base of the covering π : C −→ P1 , while h0 , h1 are homogeneous coordinates. Recall that B(t, h) ∈ H 0 (C, Hom(V, V (N ))), where V is the sheaf of sections of the trivial bundle C × V , V (D) = V ⊗ OC (D). Here, B(t, h) is a holomorphic section of the bundle Hom(V, V ) ⊗ OC (N ), OC (N ) = π ∗ OP1 (N ), that is, we are viewing h = [h0 , h1 ] as a homogeneous coordinate on P1 pulled up to C. Let D = (hN 0 ) be 0 the divisor N.π −1 (∞) on the curve C. Then B/hN ∈ H (C, Hom(V, V (D))) and 0 v ∈ H 0 (V ⊗ L), where V (D) ∼ = V (N ) are the sections of V ⊗ OC (D) (here B/hN 0 is a matrix in Hom(V, V ) with meromorphic functions in H 0 (C, OC (D)) as entries, O that  is, we are viewing h1 /h0 as a function in H (C, OC (D))). Hence, B .v ∈ H 0 (C, V ⊗ L(D)) and the cohomological interpretation of the Lax hN 0 equation is given by the following theorem: T HEOREM 6.2.– We have  B .v . v˙ = τ hN 0

[6.9]

In addition, L˙ = 0 if and only if there is a meromorphic function ϕ ∈ H 0 (C, OC (D)) such that hBN .v + ϕv ∈ H 0 (C, V ⊗ L(D)) is holomorphic. 0

Griffiths Linearization Flows on Jacobians

147

P ROOF.– We consider a diagram of exact sequence of sheaves on the curve C, 0 0 0 ↓ ↓ ↓ 0 −→ OC −→ OC (D) −→ OD (D) −→ 0 ↓v ↓v ↓ 0 −→ V ⊗ L −→ V ⊗ L(D) −→ V ⊗ L ⊗ OD (D) −→ 0 ↓τ ↓τ ↓ 0 −→ f ∗ ΘPV −→ f ∗ ΘPV (D) −→ f ∗ ΘPV ⊗ OD (D) −→ 0 ↓ ↓ ↓ 0 0 0 whence we get the cohomology diagram, 0 ↓ H 0 (C, OC (D)) ↓v

0 ↓ −→

i

j

i

j

δ

1 H 0 (C, OD (D)) −→ ↓σ

0 −→ H 0 (C, V ⊗ L) −→ H 0 (C, V ⊗ L(D)) −→ H 0 (C, V ⊗ L ⊗ OD (D)) ↓τ ↓τ ↓τ 0 −→ H 0 (C, f ∗ ΘPV ) −→ H 0 (C, f ∗ ΘPV (D)) −→ H 0 (C, f ∗ ΘPV ⊗ OD (D)) ↓δ ↓δ δ

1 −→ H 1 (C, OC )

−→

H 1 (C, OC (D))

Then the meaning of formula [6.9] is

B hN 0

.v ∈ H 0 (V ⊗ L(D)), τ



B hN 0

˙ v). .v = i( ˙

Working in C2 with coordinates (h, z), A(t, h) and B(t, h) are polynomials in h ∈ C whose coefficients are holomorphic functions of t. We write B(t, h) = hBN where 0

B ∈ H 0 (V, V (N )) is considered a homogeneous polynomial in h0 , h1 . Since the tangent space to any algebro-geometric moduli space is computed cohomologically, the answer to the question above is expressed in terms of an H 1 . The cohomology H 1 can always be reduced to the cohomology H 0 using the duality. Near the point p = (h, z) ∈ C, differentiating with regard to t the eigenvalue problem Av(t, p) = ˙ + Av˙ = z v. zv(t, p) leads to Av ˙ Using the Lax equation: A˙ = [B, A], we obtain A(v˙ − Bv) = z(v˙ − Bv). Since generically the eigenvalues have multiplicity 1, we have Bv = v˙ + λv,

[6.10]

for a some λ. Hence τ (Bv) = v˙ ∈ V ⊗ L/C.v. The existence of ϕ is equivalent to the existence of b ∈ H 0 (C, OC (D)) with i(b) = hBN .v + ϕv ∈ H 0 (C, V ⊗ L(D)), and 0 then by the above commutative diagram, we get L˙ = δ v˙ = δτ (b) = 0. 

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Since Bv = v˙ + λv, that is, Bv = v˙ + λj v, where λj is the principal part of the Laurent series expansion of λ at p, then given the curve C defined by [4.1] and p ∈ C, Griffiths defines [Laurent tail(B)]p ≡ {principal part of the Laurent series expansion of λ at p}, and shows that the Lax flow can be linearized on the Jacobian variety Jac(C) if and only if for every p ∈ (h)∞ (divisor of the poles of h), we have d [Laurent tail(B)]p ∈ linear combination {[Laurent tail(B)]p ; Laurent dt tail at p of any meromorphic function f on C such that : (f ) ≥ n(h)∞ }. Equation [4.1] is invariant under the substitution B −→ B + P (h, A), P (h, g) ∈ C[h, g], which shows that B is not unique its natural place is somewhere in

n and that k a cohomology group. Let B(t, h) = B h be a polynomial of degree n. Let k=0 k

D = h−1 (∞) = j nj pj , nj ≥ 0 (where h is seen as a meromorphic function) be a positive divisor on C and let zj be a local coordinate around pj . B must be interpreted as an element of H 0 (C, Hom(V, V (D)) where V is the sheaf of sections of the

trivial bundle C × V and V (D) = V ⊗ OC (D). A section of OD (D) is written as ϕ = ϕj ,

−1 k ϕj = k=−nj ak zj ; it is a principal part (Laurent tail) centered on pj . 6.3. Mittag–Leffler problem The Mittag–Leffler problem can be formulated as follows: given a principal part ϕj , find conditions for a function ϕ ∈ H 0 (C, OC (D)) such that ϕ − ϕj is holomorphic around pj . The answer is provided by the following theorem:

T HEOREM 6.3.– Let D = j aj pj . Given the Laurent tail {ϕj }, there exists ϕ ∈ H 0 (C, OC (D)) such that ϕ − ϕj is holomorphic near pj if and only if  Respj (ϕj .ω) = 0, [6.11] j

for every holomorphic differential ω on C. P ROOF.– The exact sheaf sequence of Mittag–Leffler attached to the divisor D is 0 −→ OC −→ OC (D) −→ OD (D) −→ 0, and the last term OD (D) is a skyscraper sheaf that may be identified with the collections {ϕj } of the Laurent tail. We deduce the cohomology sequence of this exact sequence as well as its dual res

δ

1 H 0 (C, OC (D)) −→ H 0 (C, OD (D)) −→ H 1 (C, OC ) −→ H 1 (C, OC (D)) −→ 0,

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149

δ∗

1 H 1 (C, OC (−D)) ←− H 0 (C, OD (D))∗ ←− H 0 (C, ΩC ) ←− H 0 (C, OC (−D)),

where H 0 (C, ΩC ) is the space of holomorphic 1-forms on C. The problem is therefore equivalent to the resolution of the equation δϕ = 0 and because of the duality this amounts to solving the system of linear equations: δ1 ϕ, ω = 0, ∀ω ∈ H 0 (C, ΩC ). Consider an open cover of C by small discs Uj centered in pj . We suppose that on every Uj there exists a meromorphic function fj such that: Respj (fj ) = ϕj , if Uj contains pj (otherwise, we will choose the zero function). Then the cocycle δ1 (ϕ)ik = {fi − fk } represents an element of H 1 (C, OC ). The corresponding (0, 1)-form under

the Dolbeault isomorphism H 1 (C, OC ) H∂0,1 (C) is h = j ∂(zj fj ), where {zj } is

a partition of the unity (with supp zj ⊂ Uj such that j zj = 1 in a neighborhood  of pj ) and we define h(pj ) = 0. Recall that, by definition, δ1 ϕ, ω = C φ, where φ is a Dolbeault representative of the cup-product  δϕ.ω.  introduced

With the notations above, we therefore have δ1 ϕ, ω = C h∧ω = C j ∂(zj fj ω) = C j d(zj fj ω). Let Vj () be a disc of radius  and center pj . We have + δ1 ϕ, ω = lim

→0

 C\(∪j Vj ())

j

+



d(zj fj ω) = − lim

→0

∂Vj ()

d(zj fj ω),

j



so δ1 ϕ, ω = −2πi j Respj (fj ω) = −2πi j Respj (ϕj ω), and ∀ω ∈ H 0 (C, ΩC ), δ1 ϕ, ω = 0 is equivalent to [6.11]. The number of independent equations of this system of g linear equations with d unknowns is equal to g − dim I(−D). Now, two meromorphic functions having the same principal part differ only by a constant, so dim H 0 (C, OC (D)) = deg D − g + 1 + dim I(−D), which is the Riemann–Roch theorem. The theorem results from [6.11] and the fact that the above sequences are exact.  6.4. Linearizing flows The residue of B, denoted by ρ(B) ∈ H 0 (C, OD (D), is the collection of Laurent tails {λj } given above (recall that λj is the principal part of the Laurent series expansion of λ at p). We shall say that the flow Lt is linear if there exists a complex 2 t number a such that ddtL2t = a dL dt . The Griffiths theorem is as follows:  dLt  T HEOREM 6.4.– (a) We have L˙ = δ1 (ρ(B)). (b) Let = dt t=0 0 Im res ⊂ H (C, OD (D) be the Laurent tails of meromorphic functions in H 0 (C, OD (D). Then the flow Lt [6.4] in Picd (C) is linear if and only if ˙ = 0 mod.(ρ(B), Im res). ρ(B)

[6.12]

P ROOF.– (a) By the commutative diagram above, we let E ∈ H 0 (C, V ⊗ L(D)) satisfy τ (E) = i(w) for some w ∈ H 0 (f ∗ OPV ). In particular, take E = hBN .v and 0

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Integrable Systems

w = v˙ as in theorem 6.2. By commutativity τ j(E) = jτ (E) = ji(v) ˙ = 0 and so there exists λ ∈ H 0 (C, OD (D) such that σ(λ) = j(E). Since (from the diagram above) H 0 (C, OD (D) occurs in the top right corner as well as the bottom left corner, 2 t we get L = δ(v) ˙ = δ1 (λ) = δ(ρ(B)). (b) Note that the equality ddtL2t = a dL dt holds 1 ˙ in the fixed vector space H (C, OC ) and ρ(B) = 0 in the fixed vector space H 0 (C, OD (D)) ∼ = Ck , where k = degD. Let φ ∈ H 0 (C, OC (D)). We deduce from ˙ ˙ ¨ = aL. ˙ ρ(B) = aρ(B) + Resφ, that δ1 (ρ(B)) = aδ1 (ρ(B)), and from (a), L ¨ ˙ ˙ Conversely, if L = aL, then δ1 (ρ(B)) − aδ1 (ρ(B)) = 0, and therefore there exists ˙ − aρ(B) = Resφ, which completes the proof.  φ ∈ H 0 (C, OC (D)) such that ρ(B)

= The condition [6.12] is equivalent to j Respj (ρ˙ j (B))ω)

t j Respj (ρj (B))ω), ω ∈ H 0 (C, ΩC ). If this is satisfied, then the linear flow on Jac(C) is given by the bilinear map   (t, ω) −→ t Respj (ρj (B))ω) = t Respj (λj ω). [6.13] j

j

6.5. The Toda lattice The Toda lattice (Toda 1967) is a system of n particles connected by nonlinear springs with a restoring force depending exponentially on displacement. Such a system has been known for some time as a discrete version of the Korteweg–de Vries equation1 and is by the following Hamiltonian

N governed N H = 12 j=1 yj2 + j=1 exj −xj+1 . The Hamiltonian equations are x˙ j = yj , y˙ j = −exj −xj+1 + exj−1 −xj . Flaschka variables (Flaschka 1974): aj = 12 exj −xj+1 , bj = − 12 yj can be used to express the symplectic structure ω [5.7] in terms of xj and

N da yj as follows: ajj = dxj − dxj+1 , 2dbj = −dyj , then ω = − 12 j=2

N

N dyj i=j (dxi − dxi+1 ) = 12 j=2 dx∗j ∧ dyj∗ . We will study the integrability of this problem with the Griffiths approach. There are two cases: (i) The non-periodic case, that is, x0 = −∞, xN +1 = +∞, where the masses are arranged on a line.

Figure 6.1. Toda lattice (non-periodic case)

In terms of the Flaschka variables above, Toda’s equations take the following form: a˙ j = aj (bj+1 − bj ), b˙ j = 2(a2j − a2j+1 ), with aN +1 = a1 and bN +1 = b1 . To 3

u 1 In short, the KdV equation: ∂u − 6u ∂u + ∂x∂3 =0 . This is an infinite-dimensional completely ∂t ∂x integrable system. For more information on this equation, see Chapter 9.

Griffiths Linearization Flows on Jacobians

151

show that this system is completely integrable, one should find N first integrals independent involution each other. From the second equation, we have 

. and in

N ˙ N = j=1 bj j=1 bj = 0, and we normalize the bj ’s by requiring that

N j=1 bj = 0 (applying this fact to [5.7] leads to the original symplectic form

N ω = 12 j=2 dxj ∧ dyj ). This is a first integral for the system and to show that it is completely integrable, we must find N − 1 other integrals that are functionally independent and in involution. We further define N × N matrices A and B with ⎛ ⎞ 0 b1 a1 0 · · · aN ⎜ ⎜ .. .. ⎟ ⎜ −a1 ⎜ a1 b2 . . ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎜ ⎟ . . . , B = A = ⎜ 0 .. .. .. ⎜ . ⎟ 0 ⎟ ⎜ ⎜ ⎜ . ⎜ . ⎟ .. ⎝ .. ⎝ .. . bN −1 aN −1 ⎠ aN aN · · · 0 aN −1 bN ⎛

a1 · · · . 0 ..

···

..

..

.

..

.

.

..

.. . . · · · · · · −aN −1

−aN .. . .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ aN −1 ⎠ 0

The proposed system is equivalent to the Lax equation A˙ = [B, A]. From theorem 4.1, the quantities Ik = k1 trAk , 1 ≤ k ≤ N , are first integrals of motion. To be ˙ k−1 ) = tr([B, A].Ak−1 ) = tr(BAk − ABAk−1 ) = 0. more precise, I˙k = tr(A.A Note that I1 is the first already known integral. Since these N first integrals are shown to be independent and in involution with each other, the system in question is thus completely integrable. (ii) The periodic case, that is, yj+N = yj , xj+N = xj , the connected masses will be arranged on a circle.

Figure 6.2. Toda lattice (periodic case)

We show that in this case, the spectrum of the periodic Jacobi matrix ⎛

b1

a1 0 . b ..

···

⎜ ⎜ a1 2 ⎜ ⎜ . .. ... ... A=⎜ 0 ⎜ ⎜ . .. ⎝ .. . bN −1 aN h · · · 0 aN −1

⎞ aN h−1 ⎟ .. ⎟ . ⎟ ⎟ , 0 ⎟ ⎟ ⎟ aN −1 ⎠ bN

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Integrable Systems

remains invariant in time. The matrix B depending on the spectral parameter h is ⎛

0

a1 · · · . 0 ..

···

⎜ ⎜ −a1 ⎜ ⎜ .. B = ⎜ ... . . . . . . . ⎜ ⎜ . . . .. .. ⎝ .. aN h · · · · · · −aN −1

⎞ −aN h−1 ⎟ .. ⎟ . ⎟ ⎟ .. ⎟, . ⎟ ⎟ aN −1 ⎠ 0

and the rest follows from the general theory. Note that if aj (0) = 0, then aj (t) = 0 for all t. Since A (h) = A(h−1 ), then P (h, z) = det(A(h) − zI) = P (h−1 , z). Therefore, the application σ : C −→ C,

(h, z) −→ (h−1 , z),

[6.14]

is an involution on the spectral curve C. We choose ⎛

⎞ b1 a1 ⎜ a1 b2 ⎟ 0 ... aN ⎜ ⎟ ⎟ ⎜ .. . . .. ⎟ −1 ⎜ . .. A(h) = ⎝ . . . ⎠ h + ⎜ ⎟ ⎜ ⎟ ⎝ bN −1 aN −1 ⎠ 0 ... 0 bN aN ⎛ ⎞ 0 ... 0 ⎜ ⎟ + ⎝ ... . . . ... ⎠ h. ⎛

⎞

aN ... 0 The matrix A is meromorphic (previously we considered it to be a polynomial in h) but we will see that we can also the theory explained in this chapter to this /Nadopt −1 situation. We have P (h, z) = − j=1 aj .(h + h−1 ) + z N + c1 z N −1 + · · · + cN . Let /N −1 us assume that j=1 aj = 0 and let P (h, z) z N + c1 z N −1 + · · · + cN Q(h, z) ≡ /N −1 = h + h−1 + , /N −1 j=1 aj j=1 aj = h + h−1 + d0 z N + d1 z N −1 + · · · + dN . In P2 (C), the affine algebraic curve of equation Q(h, z) = 0 is singular at infinity for n ≥ 4. We will compute the genus of the normalization C of this curve. Note that C is a double covering of P1 (C) branched into 2N points coinciding with the fixed points of involution σ [6.14], that is, points where h = ±1. According to the Riemann–Hurwitz formula (see Appendix 2), the genus g of the curve C is

Griffiths Linearization Flows on Jacobians

153

  1 g = 2 g(P1 (C)) − 1 + 1 + 2N 2 = N − 1. Consider the covering C −→ P (C) 1 below and set z (∞) = P + Q, where P and Q are located on two separate sheets. From the equation Q(h, z) = 0, the divisor of h is (h) = N P − N Q. The divisor D is written D = N P + N Q, hence B ∈ H 0 (C, Hom(V, V (D)).

Figure 6.3. Divisor D

The residue ρ(B) ∈ H 0 (C, OD (D) satisfies the conditions of theorem 6.4 and consequently the linear flow is given by the application [6.13]. To compute the residue ρ(B) of B, we will determine a set of holomorphic eigenvectors, using the van Moerbeke–Mumford method described above. Let

us calculate the residue in Q g and the result will be similarly deduced in P. Let E = j=1 rj be a general divisor of degree g such that: ∀k,dim L(E + (k − 1)P − kQ) = 0. According to the Riemann–Roch theorem, dim L(E + kP − kQ) ≥ 1, hence dim L(E + kP − kQ) = 1, for all k. Let (fk ) ∈ L(E + kP − kQ) = H 0 (C, OC (E + kP − kQ)), 1 ≤ k ≤ N , be a base with fN = h. We can choose a vector v of the following form v = (f1 , ..., fN ) , such that v is an eigenvector of A, that is, Av = zv, (h, z) ∈ C. Hence, V = h−1 v is a holomorphic eigenvector. Without restricting generality, we take N = 3. The system Av = zv is explicitly written b1 f1 + a2 f2 + a3 = zf1 , a1 f1 + b2 f2 + a2 h = zf2 , a3 hf1 + a2 f2 + b3 h = zh. By multiplying each equation of this system by h−1 , everything becomes holomorphic except the last equation, that is, a3 f1 = z + Taylor. Recall that the residue ρ(B) of B is the section of OD (D) induced by λ in the equation Bv = v˙ + λv. In other words, Bv = (B)v + Taylor, and therefore ⎛

a1 f2 a3 h − h ⎝ − a1 f1 + a2 h a3 f1 − a2hf2

⎞

⎛ ⎞ 0 ⎠ = ⎝ 0 ⎠ + Taylor. z

˙ = 0. The same conclusion holds for the residue Hence, ρ(B) = h−1 z and ρ(B) in P. Then, the flow in question linearizes on the Jacobian variety of C. 6.6. The Lagrange top We consider the Lagrange top discussed in sections 3.2.2 and 4.7. It is proved in Ratiu and van Moerbeke (1982) that equations [3.3] or [3.4] (corresponding to the

154

Integrable Systems

   ˙ Lagrange case) may be written in the particular form (Γ + M h + ch2 ) = [Γ + M h + ch2 , Ω + Lh] of a Lax equation with a parameter if, and only if, λ1 = λ2 ≡ α, l1 = l2 = 0. We set A(h) = Γ + M h + ch2 ∈ so(3).

[6.15]

We have P (h, z) = det(zI − A) = z(z + |A|2 ), where |A|2 is the sum of the squares of the entries of A. By [6.15], we have |A|2 = γ0 + γ1 h + γ2 h2 + γ3 h3 + γ4 h4 ,

γ0 ≡ |Γ|2 , γ4 ≡ |c|2 .

[6.16]

The spectral curve is reducible with one component (z = 0) corresponding to the zero eigenvalue (z = 0) of any matrix in so(3). The other component z 2 +|A(h)|2 = 0 is by [6.16] an elliptic curve, which is smooth and can be realized by (h, z) −→ h, as two-sheeted branched covering of P1 with sheet interchange given by (h, z) −→ (h, −z). The spectral curve is a two-sheeted covering ϕ : C −→ P = P1 (h), branched over four points hj with all hj = ∞. Since B(h) = Ω + Lh = ΛM + Lh, we have α+β D = ϕ−1 (∞) = P + Q, where z = α+β h + · · · near P, z = − h + · · · near Q, 0 (α ≡ λ1 + λ2 , β = λ3 ). Then the residue ρ(B) ∈ H (OD (D)) is given by ρ(B) = 1 1 ˙ h + · · · near P, ρ(B) = − h + · · · near Q. We deduce that ρ(B) = 0 and [6.13] is satisfied. Therefore, the flow is linearized on the Jacobian variety of C. 6.7. Nahm’s equations Nahm’s equations (Nahm 1981) involve three functions (Tj (t))j=1,2,3 with values in the algebra u(n) formed by the complex antihermitian matrices of order n. These equations, which arise in the study of  monopoles, are T˙1 = [T2 , T3 ], T˙2 = [T3 , T1 ], T˙3 = [T1 , T2 ]. We have T˙j = 12 k,l jkl [Tk , Tl ], where jkl is the Levi–Civita symbol,

jkl

⎧ ⎨ +1 if (j, k, l) = (1, 2, 3), (3, 1, 2) where (2, 3, 1) = −1 if (j, k, l) = (3, 2, 1), (1, 3, 2) where (2, 1, 3) ⎩ 0 if j = k, k = l where l = j

Nahm’s equations are equivalent (Hitchin 1983) to the Lax equation A˙ = [B, A], where A = (T1 + iT2 ) − 2iT3 h + (T1 − iT2 )h2 , B =−

1 dA = iT3 − (T1 − iT2 )h. 2 dh

[6.17]

Griffiths Linearization Flows on Jacobians

155

Let C be the spectral curve (in the twistor space T P1 , the tangent bundle to P1 and geometrically may be thought of as the space of oriented straight lines in R3 ) associated with these equations. It is a smooth curve given by the equation P (h, z) = det((T1 + iT2 ) − 2iT3 h + (T1 − iT2 )h2 − zI) = 0, and its genus is (n − 1)2 . The coefficients of the polynomial P (h, z) are independent n of t and are invariants of Nahm’s equations. We have D = h−1 (∞) = j=1 Pj . Let zj =

λj ξj

+

1 ξj

+ Taylor, where ξj =

1 h

is a local coordinate around pj . According to

2) 2) [6.17], we have around pj , A = (T1 +iT2 )−2i Tξj3 + (T1 −iT , B = iT3 − (T1 −iT . The ξj ξ2 j

eigenvectors vj satisfy Avj = zj vj , whence (T1 + iT2 − 2iT3 h + (T1 − iT2 )h2 )vj = (λj h2 + h + Taylor)vj , and Bvj = iT3 − (T1 − iT2 )hvj . Hence, (T1 − iT2 )h2 vj = zj vj + o(h) and Bvj = −(T1 − iT2 )hvj + o(1). We thus have (T1 −iT 2 )vj (pj ) =

λj ˙ λj vj (pj ), while the residue is given by ρ(B) = j zj . Therefore, ρ B = 0, and [6.13] linearizes the flow in question on the Jacobian variety of C. 6.8. The n-dimensional rigid body Let J = diag(λ1 , ..., λn ), λj > 0, be the matrix representing the tensor of inertia of a rigid body in a principal axis system, and let Ω(t) ∈ so(n) be the skew-symmetric matrix associated with the angular velocity vector of the rigid body in the usual way. Define M = ΩJ + JΩ ∈ so(n), the equations of motion of the rigid body can be written as M˙ = [M, Ω]. These equations are Hamiltonian on each adjoint orbit of so(n) defined by initial conditions with Hamiltonian H(M ) = 21 (M, Ω) = − 14 T r(M Ω) (the cases n = 3 and n = 4 have and will be studied in several places). With Manakov’s trick (Manakov 1976), these equations   ˙  are equivalent to a Lax equation with parameter ( M + J 2 h) = [M + J 2 h, Ω + Jh].

−1 Hence, D = h (∞) = i pi is the divisor with pi being the n distinct points lying over h = ∞. If zi is a local coordinate on C near pi (say zi

= h−1 ), then from λi equation [6.10] and taking B = Ω + Jh, we obtain ρ(B) = i zi . Since λi are ˙ = 0 so the flow is linear on Jac(C). We note that constant, one clearly has ρ(B) 2 since A = M + J h where M + M = 0, J 2 − J 2 = 0, we have n P (h, z) = (−1) P (−h, −z). Thus, there is an involution of the spectral curve σ : C −→ C, (h, z) −→ (−h, −z). We note that Ω moves on an adjoint orbit Oν ⊂ so(n) and to linearize the flow in question we need 12 dim Oν integrals of motion that are in involution where for general ν, dim Oν =

n(n − 1)  n  − , 2 2

[6.18]

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(see Mironov (2010) for more information). Let g(C) be the genus of the spectral curve C and g(C0 ) the genus of the quotient C0 = C/σ of C by the involution σ. Since g(C) = (n−1)(n−2) , then by the Riemann–Hurwitz formula, 2  = (n−2)2 1 n(n − 1)  n  n ≡ 0 mod.2 4 g(C0 ) = g(C)− [6.19] − = (n−1)(n−3) 2 2 2 n ≡ 1 mod.2 4 Associated with the double covering C −→ C0 is the Prym variety P rym(C/C0 ) and since σ(ρ(B)) = −ρ(B), the flow in question actually occurs on this complex torus. From [6.19], it follows that 1 dim P rym(C/C0 ) = 2



n(n − 1)  n  − = 2 2

=

n(n−2) 4 (n−1)2 4

n ≡ 0 mod.2 n ≡ 1 mod.2

In comparison with [6.18], we obtain dim P rym(C/C0 ) = 12 dim Oν , and the motion linearizes on a torus P rym(C/C0 ) of exactly the right dimension. 6.9. Exercises E XERCISE 6.1.– Use the Griffiths linearization method to study the Jacobi geodesic flow on an ellipsoid and Neumann problem (section 4.6). .

.

.

.

E XERCISE 6.2.– Consider the system x1 = y1 , y 1 = 2x1 x2 , x2 = y2 , y 2 = x21 + 6x22 , corresponding to a particular case of Hénon–Heiles Hamiltonian: H = 12 (y12 + y22 ) − x21 x2 − 2x32 , where x1 , x2 , y1 , y2 are canonical coordinates and momenta, respectively (for detailed information on the Hénon–Heiles system in general, see section 7.6. For ease of calculation, we chose here a particular case of this system with a slight change of sign). a) Show that this system admits a Lax pair of the form A˙ = [A, B], where

A(h) =

y12 −



x21 2

y1 x1 − y2 h

+ 2x22 h + x22 h2 −

h3 2

−x21 + 2hx2 + h2 −y1 x1 + y2 h

 ,

h ∈ C∗ ,

0 1 , h ∈ C∗ . Determine a second first integral F of this 2x2 − h2 0 system, such that H and F are functionally independent and in involution. 

and B(h) =

b) Using the Griffiths linearization method, show that the Hamiltonian flows corresponding to H and F are linear.

Griffiths Linearization Flows on Jacobians

157

E XERCISE 6.3.– Let gl(n, C) be a Lie algebra of all n × n matrices and G the graded Lie algebra of all Laurent polynomials in complex variable h with coefficients in gl(n, C) and under the bracket operation: [Az m , Bz n ] = [A, B]z m+n . Let us denote A(h, t) ≡ Ah and assume that Ah is a generic solution of Lax equation [4.1] (i.e. Ah has simple eigenvalues for all but a finite number of values h). Let X be a compact Riemann surface corresponding to the spectral curve C of Ah . a) Prove that there exists a holomorphic line bundle E(t) (i.e. an eigenbundle) over X with the property that if (h, z) ∈ C and z is a simple eigenvalue for Ah , then the eigenspace E(h, z, t) = {x ∈ Cn : Ah x = zx} of Ah equals the fiber E(t)|(h,z) of E(t) over (h, z). b) Let Q(h, z) be a polynomial and R(h, z) the polynomial part of the Laurent polynomial Q(h, Ah ). Show that the Jacobian flow of the Lax equation A˙ h = [Ah , R(h, Ah )] is linear on the Jacobian torus. E XERCISE 6.4.– Using the method developed in this chapter, study again the Kowalewski spinning top and its Lax representation. E XERCISE 6.5.– We use the notation from section 6.4, with D = np1 + np2 . Show that there exist local coordinates z1 and z2 near p1 and p2 such that the residue of B at D is given by ρ(B) = z12 − z12 . 1

2

E XERCISE 6.6.– Consider the Hamiltonian system: x˙ j = xj (xj+1 − xj−1 ), xn+j = xj ,

1 ≤ j ≤ n. This is a special case of the Lotka–Volterra system: x˙ j = αj xj + n 1 k=1 ajk xj xk , 1 ≤ j ≤ n. The system that interests us here corresponds to the βj case αj = βj = 0 and (aij ) skew-symmetric. The Hamiltonian function is H = x1 + x2 + · · · + xn , the Poisson bracket is defined by {xj , xj+1 } = xj xj+1 with all other brackets equal to zero and the system is written in the Hamiltonian form: x˙ j = {xj , H}, 1 ≤ j ≤ n. a) Show that F = (x1 − log x1 ) + (x2 − log x2 ) + · · · + (xn − log xn ) is a first integral of this differential system and the latter admits the Lax pair A˙ = [A, B], where ⎛

⎞ √ √ 0 x1 · · · xn h−1 ⎜ √ ⎟ .. √ ⎜ x1 0 ⎟ x2 . ⎜ ⎟ ⎜ .. ⎟ . . √ . . A(h) = ⎜ . ⎟, . x2 . ⎜ ⎟ ⎜ . ⎟ .. .. √ ⎝ .. . . xn−1 ⎠ √ √ xn h · · · xn−1 0

h ∈ C∗ ,

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⎛

0 0 0 0 ⎜ ⎜ ⎜ √ 0 ⎜ − x1 x 2 B(h) = ⎜ .. .. ⎜ . . ⎜ ⎜ .. ⎝√ . xn−1 xn h √ x 1 xn h 0

√

x1 x2 0 0 .. .

0 ···

··· ..

.

..

.

0

√ − xn−2 xn−1

√ − xn−1 xn h−1

0

⎞

0 √ − x1 xn h−1 ⎟ ⎟ .. ⎟ . ⎟ ⎟. √ ⎟ xn−2 xn−1 ⎟ 0 0

⎟ ⎠

b) Using the Griffiths linearization method with D = np1 + np2 , show that the system above linearizes on the Jacobi variety Jac(C) where C is the spectral curve associated with the system. Show that the system in question  4 linearizes also on a Prym variety associated with spectral curve C of dimension n2 .

7 Algebraically Integrable Systems

This chapter presents an excellent introduction to the problems, techniques and results of algebraic complete integrability. We will mainly focus on algebraic integrability in the sense of Adler–van Moerbeke, where the fibers of the momentum map are affine parts of complex algebraic tori (Abelian varieties). It is well known that most of the problems of classical mechanics are of this form. Many important problems will be studied: Euler and Kowalewski tops, the Hénon–Heiles system, geodesic flow on SO(n), the Kac–van Moerbeke lattice, generalized periodic Toda systems, the Gross–Neveu system, the Kolossof potential, as well as other systems. 7.1. Meromorphic solutions Consider the system of nonlinear differential equations dz1 dzn = f1 (t, z1 , ..., zn ) , ..., = fn (t, z1 , ..., zn ) , dt dt

[7.1]

where f1 , ..., fn are functions of n + 1 complex variables t, z1 , ..., zn and which apply a domain of Cn+1 into C. The Cauchy problem is the search for a solution (z1 (t),0 ..., zn0(t))  in a neighborhood of a point t0 , passing through the given point t0 , z1 , ..., zn , that is, satisfying the initial conditions z1 (t0 ) = z10 , ..., zn (t0 ) = zn0 . The system [7.1] can be written in a vector form in Cn , dz dt = f (t, z(t)), by putting z = (z1 , ..., zn ) and f = (f1 , ..., fn ). In this case, the Cauchy problem will be to that when determine the solution z(t), such that z(t0 ) = z0 = (z10 , ..., zn0 ). We know   the functions f1 , ..., fn are holomorphic in the neighborhood of the t0 , z10 , ..., zn0 , then the Cauchy problem admits a unique holomorphic solution. A question arises: can the Cauchy problem   admit some non-holomorphic solution in the neighborhood of point t0 , z10 , ..., zn0 ? When the functions f1 , ..., fn are holomorphic, the answer is negative. Other circumstances may arise for the Cauchy problem concerning the system of differential equations [7.1], when the holomorphic hypothesis relative to Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

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the functions f1 , ..., fn is no longer satisfied in the neighborhood of a point. In such a case, it can be seen that the behavior of the solutions can take on the most diverse aspects. In general, the singularities of the solutions are of two types: mobile or fixed, depending on whether or not they depend on the initial conditions. Important results have been obtained by Painlevé (1975). Suppose, for example, that the system [7.1] is written in the form dz1 P1 (t, z1 , ..., zn ) dzn Pn (t, z1 , ..., zn ) = , ..., = , dt Q1 (t, z1 , ..., zn ) dt Qn (t, z1 , ..., zn ) where Pk (t, z1 , ..., zn ) =

 0≤i1 ,...,in ≤p

Qk (t, z1 , ..., zn ) =

(k)

Ai1 ,...,in (t)z1i1 ...znin , 1 ≤ k ≤ n,

 0≤j1 ,...,jn ≤q

(k)

Bj1 ,...,jn (t)z1j1 ...znjn , 1 ≤ k ≤ n,

are polynomials with several indeterminate z1 , ..., zn and algebraic coefficients in t. We know (i) that the fixed singularities are constituted by four sets of points. The first (k) (k) is the set of singular points of the coefficients Ai1 ,...,in (t), Bj1 ,...,jn (t) intervening in the polynomials Pk (t, z1 , ..., zn ) and Qk (t, z1 , ..., zn ). In general, this set contains t = ∞. The second set consists of the points αj such that Qk (t, z1 , ..., zn ) = 0, (k) which occurs if all of the coefficients Bj1 ,...,jn (t) vanish for t = αj . The third is the set of points βl such that for some values (z1 , ..., zn ) of (z1 , ..., zn ), we have Pk (βl , z1 , ..., zn ) = Qk (βl , z1 , ..., zn ) = 0. Then the second members of the above system are presented in the indeterminate form 00 at the points (βl , z1 , ..., zn ). Finally, there is the set of points γn such that there exist u1 , ..., un , for which Rk (γn , u1 , ..., un ) = Sk (γn , u1 , ..., un ) = 0, where Rk and Sk are polynomials in u1 , ..., un obtained from Pk and Qk by setting z1 = u11 , . . . , zn = u1n . Each of these sets contains only a finite number of elements. The system in question has a finite number of fixed singularities. (ii) The mobile singularities of solutions of this system are algebraic mobile singularities: poles and (or) algebraic critical points. There are no essential singular points for the solution (z1 , ..., zn ). Considering the system of differential equations [7.1], can we find sufficient conditions for the existence and uniqueness of meromorphic solutions? We will establish a theorem of existence and uniqueness for the solution of the Cauchy problem concerning the system of differential equations [7.1] using the method of indeterminate coefficients. The solution will be explained in the form of a Laurent series. The problem of convergence will therefore arise. This will be solved by the method of major functions. There are many theoretical and practical problems, or differential equations, of which the second member is not holomorphic. In the

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161

following, we will consider the Cauchy problem concerning the normal system [7.1], where f1 , ..., fn do not depend explicitly on t, that is, dz1 dzn = f1 (z1 , ..., zn ) , ..., = fn (z1 , ..., zn ) . dt dt

[7.2]

We suppose that f1 , ..., fn are rational functions in z1 , ..., zn and that the system [7.2] is weight-homogeneous. That is, there exist positive integers s1 , ..., sn such that fi (αs1 z1 , ..., αsn zn ) = αsi +1 fi (z1 , ..., zn ), 1 ≤ i ≤ n, for each non-zero constant α. In other words, the system [7.2] is invariant under the transformation t → α−1 t, z1 → αs1 z1 , ..., zn → αsn zn . Note that if the determinant  ∂fi − δij fi , [7.3] Δ ≡ det zj ∂zj 1≤i,j≤n is not identically zero, then the choice of the numbers s1 , ..., sn is unique. In what follows, we will assume that t0 = z0 = 0, which does not affect the generality of the results. T HEOREM 7.1.– Suppose that zi =

∞ 1  (k) k zi t , t si

1 ≤ i ≤ n,

z (0) = 0

[7.4]

k=0

(si ∈ Z, some si > 0) is the formal solution (Laurent series), obtained by the method of undetermined coefficients of the weight-homogeneous system [7.2]. Then the (0) coefficients zi satisfy the nonlinear equation (0)

si zi

(0)

+ fi (z1 , ..., zn(0) ) = 0, (1)

[7.5]

(2)

where 1 ≤ i ≤ n, while zi , zi , ... each satisfy a system of linear equations of the form (L − kI)z (k) = some polynomial in the z (j) , 0 ≤ j ≤ k,  (k) (k) ∂fi (0) (z ) + δ s where z (k) = (z1 , ..., zn ) and L ≡ ∂z ij i j

1≤i,j≤n

[7.6] is the Jacobian

matrix (Kowalewski matrix) of [7.5]. Moreover, the formal series [7.4] are convergent. P ROOF.– By substituting [7.4] into [7.2], taking into account the weight-homogeneity of the system, we obtain ∞ 

(k −

k=0

(k) si )zi tk−si −1

= fi

∞  k=0

(k) z1 tk−s1 , ...,

∞  k=0

 zn(k) tk−sn

,

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Integrable Systems

= fi

t

−s1

(0) (z1

+

=t

fi

(k) z1 tk ), ..., t−sn (zn(0)

+

k=1

−si −1

∞ 

(0) z1

+

∞ 

 zn(k) tk )

k=1

∞ 

(k) z1 tk , ..., zn(0)

+

k=1

∞ 

zn(k) tk

,

 .

k=1

Then, the second member is developed as follows: ∞ 

(k)

(0)

(k − si )zi tk = fi (z1 , ..., zn(0) ) +

k=0

n ∞   ∂fi (0) (k) (z1 , ..., zn(0) ) zj tk ∂z j j=1 k=1

+

∞ 



tk

k=2

(α,τ )∈Dk

n 9 1 ∂ α fi (0) (τ ) (0) (z , ..., z ) (zj j )αj , n α! ∂z α 1 j=1

n /n where α = (α1 , ..., αn ), τ = (τ1 , ..., τn ), |α| = j=1 αj , α! = j=1 αj ! and

n Dk = {(α, τ ) : τj > 0, ∀, |α| > 2, j=1 αj τj = k}. By identifying the terms that have the same power at the first and the second member, we obtain, successively for k = 0, the expression [7.5], for k = 1, (L − I) c(1) = 0, and for k ≥ 2, 

(L − kI)z (k)

i

=−

 (α,τ )∈Dk

n 9 1 ∂ α fi (0) (τ ) (0) (z , ..., z ) (zj j )αj , n 1 α! ∂z α j=1

[7.7]

n where τj > 0, j=1 αj τj = k, which leads to [7.6]. The solution obtained by the method of indeterminate coefficients is formal because we obtain it by performing on various series, which we assume a priori convergent, various operations whose validity remains to be justified. The theorem will therefore be established as soon as we have verified that these series are convergent. This will be done using the majorant method (Adler and van Moerbeke 1989). Note that free parameters either appear in the system [7.5] of n equations with n unknown, when the latter admits a continuous set of solutions, or because of the fact that λi ≡ k ∈ N∗ , 1 ≤ i ≤ n, is an eigenvalue of the matrix L. The coefficients can be seen as rational functions on 0an affine  variety V , 1 fibered over the indicial locus: 3n (0) (0) (0) s z = 0 . Let n z + f , ..., z ∈ V and fix a compact subset K n i i i 0 1 i=1 of V , containing an open neighborhood of n0 . Note that K can be equipped with the 0 topology  of  the plan. Let   complex 1  (τn )     (τ2 )  (τ1 ) A = 1 + max z1 (n0 ) , z2 (n0 ) , ..., zn  (n0 ) , where 1 ≤ τi ≤ λn , 1 ≤ i ≤ n and λn denotes the largest eigenvalue of the matrix L.  αLet B and  C be two  ∂ fi  constants with C > A, such that in the compact K we have  ∂z α (n0 ) ≤ α!B |α| ,    −1  (L(n0 ) − kIn )  ≤ C, k ≥ λn + 1. From [7.7], we deduce that  

/n  (τ ) αj   (k) zi (n0 ) ≤ C (α,τ )∈D B |α| j=1 zj j  , k ≥ λn + 1. Consider the series

Algebraically Integrable Systems

163

∞ Φ(t) = At + k=2 βk tk , where /n βk αare real numbers inductively defined by β1 ≡ A and βk ≡ C (α,τ )∈D B |α| j=1 βτjj , k ≥ 2. Note that the series Φ(t) is an upper  

∞ (k)  (1)  bound for k=1 zi tk , 1 ≤ i ≤ n. Indeed, we have zi  ≤ A. Suppose that    (j)  zi  ≤ βj , j < k, ∀i. Then, for k ≥ λn + 1,     (k) zi (n0 ) ≤ C



B |α|

(α,τ )∈D

n   9  (τj ) αj zj  ≤ C j=1

 (α,τ )∈D

B |α|

n   9  αj   β τ j  = βk . j=1 2

(nΦ(t)) . It results from the definition of the numbers βk that Φ (t) = At + CB 2 1−BnΦ(t)  (0) By writing the formal series [7.4] in the form, zi (t) = t1si zi + gi (t) , we see that the root " 1 + nABtz − (1 − 2nAB(1 + 2nBC)t + n2 A2 B 2 t2 ) Φ(t) = , 2nB(1 + nBC)

provides the required majorant for the functions gi (t), which therefore converge for sufficiently small t.  R EMARK 7.1.– The series [7.4] is the only meromorphic solution in the sense that this (k) solution results from the fact that the coefficients zi are unequivocally determined using the adopted method of calculation. R EMARK 7.2.– The result of the previous theorem applies to the following quasi-homogeneous differential equation of order n: dn z =f dtn



dz dn−1 z z, , ..., n−1 dt dt

,

[7.8] n−1

d z 0 dz 0 with f being a rational function in z, dz dt , ..., dtn−1 and z(t0 ) = z1 , dt (t0 ) = z2 ,..., n−1 d z 0 dtn−1 (t0 ) = zn . Indeed, equation [7.8] reduces to a system of n first-order differential dn−1 z equations by setting z(t) = z1 (t), dz dt (t) = z2 (t), ..., dtn−1 (t) = zn (t). We thus obtain

dz1 = z2 , dt

dz2 dzn−1 = z3 , ..., = zn , dt dt

dzn = f (z1 , z2 , ..., zn ) . dt

Such a system constitutes a particular case of the normal system [7.2].

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7.2. Algebraic complete integrability We will work with complexes instead of real ones. Concepts such as Liouville integrability, involution, commutativity of vector fields and so on can be defined as in the real case. On the other hand, difficulties arise: we know that there are no compact holomorphic submanifolds in the complex space Cm (maximum principle), therefore the complex tori that we can get in Arnold–Liouville’s theorem are not compact. So the problem of compactification of invariant varieties arises. In addition, the solutions of the system in question are not uniform (single-valued). First, we will recall some results, then we will define and explain the concept of algebraic complete integrability of Hamiltonian systems in more detail. The definition of the algebraic complete integrability of a Hamiltonian system varies according to the literature and is usually found (with some minor variants) in any modern text on integrable systems. For integrable systems treated in several chapters (which are important and are often encountered elsewhere), we will have to consider the affine space Cm . The integrable systems that we will deal with here are complex integrable systems where M is an affine space Cm , the algebra that we consider is just that of the polynomial functions and we focus on algebraic complete integrability in the sense of Adler–van Moerbeke. Consider a Hamiltonian completely integrable system XH : z˙ = J

∂H ≡ f (z), z ∈ Rm , ∂z

m = 2n + k,

[7.9]

(J(z) polynomial in z), with n + k functionally independent invariants H1 , ..., Hn+k ∂Hn+j of which k invariants (Casimir functions) lead to zero vector fields J ∂z (z) = 0, 1 ≤ j ≤ k, the n = (m − k)/2 remaining ones are in involution (i.e. {Hi , Hj } = 0), which give rise to n commuting vector fields. According to the Arnold–Liouville n+k 3 theorem 3.1, if the invariant manifolds {z ∈ Rm : Hi (z) = ci }, are compact, i=1

then for most values of ci ∈ R, their connected components are diffeomorphic to real tori Rn /Lattice, and the flows gtX1 (x),...,gtXn (x) defined by the vector fields XH1 ,...,XHn are straight-line motions on these tori. Now consider z ∈ Cm and t ∈ C. Let Δ ⊂ Cm be a non-empty Zariski open set. By the functional independence of the first integrals, the map (momentum mapping) ϕ ≡ (H1 , ..., Hn+k ) : Cm −→ Cn+k , is a generic submersion (i.e. dH1 (z), are linearly independent) on Δ. Let Ω = ϕ (Cm \Δ), # ..., dHn+k (z) $ that is, n+k Ω = c = (ci ) ∈ C : ∃z ∈ ϕ−1 (c) with dH1 (z) ∧ ... ∧ dHn+k (z) = 0 , be the set of critical values of the map ϕ and denote by Ω the Zariski closure of Ω in Cn+k . # $ P ROPOSITION 7.1.– The set defined by Γ = z ∈ Cm : ϕ(z) ∈ Cn+k \Ω is everywhere dense in Cm for the usual topology.   P ROOF.– Indeed, it suffices to show that the set Γ = ϕ−1 Cn+k \Ω , is a non-empty Zariski open set in Cm . Since a polynomial mapping between affine algebraic sets is continuous for the Zariski topology, then the above set is indeed a Zariski open set

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165

in Cm and it is non-empty. Suppose this one is empty, that is ϕ(Cm ) ⊂ Ω. Since the map ϕ is submersive on a non-empty open set of Zariski Δ ⊂ Cm , then ϕ(Δ) is open in Cn+k . According to Sard’s theorem for varieties, Cn+k \Ω is a non-empty Zariski open set, and therefore everywhere dense for the usual topology in Cn+k . So ϕ(Δ) ∩ (Cn+k \Ω) = ∅, which is absurd. This completes the proof.  Let Mc be the complex affine variety defined by Mc ≡ ϕ−1 (c) =

n+k 6

{z ∈ Cm : Hi (z) = ci } .

[7.10]

i=1

For all c ≡ (c1 , ..., cn+k ) ∈ Cn+k \Ω, the fiber Mc is smooth. D EFINITION 7.1.– The system [7.9] will be called algebraic complete integrable (a.c.i.) in the sense of Adler–van Moerbeke with Abelian functions zi when, for every c ∈ Cn+k \Ω, the fiber Mc [7.10] is the affine part of an Abelian variety (complex >c = T n Cn /Lc , (Lc a lattice in Cn ), and moreover, the flows algebraic torus) M t gX (z), z ∈ M , c t ∈ C, defined by the vector fields XHi , 1 ≤ i ≤ n, are straight line i  t 4   n on T , that is, gX (z) j = fj p + t(k1i , ..., kni ) , where fj (t1 , ..., tn ) are Abelian i functions on T n , fj (p) = zj , 1 ≤ j ≤ m. We will be concerned with algebraically completely integrable systems that are irreducible, that is, when the generic Abelian variety is irreducible (it does not contain a subtorus). The following remark is intended to present several interrelated definitions, all involving algebraically completely integrable systems. For more information and comments on some definitions, see Vanhaecke (2001). R EMARK 7.3.– 1) Let H be a smooth function on a 2n-dimensional symplectic manifold (M, ω). The Hamiltonian system defined by the vector field XH is algebraically completely integrable if there exists a smooth algebraic variety M, a co-symplectic structure ω 2 that restricts to ω along M , that is, ω 2 ∈ Λ2 TM and a morphism h : M −→ U , where U is a Zariski open subset of Cn , all defined over the real field such that (i) h is a proper submersive whose components are in involution (i.e. {Xi oh, Xj oh} ≡ ω 2 (d(Xi oh), d(Xj oh)) = 0, Xi being coordinates on Rn ); (ii) M is a component of MR , the ω 2 on M is the ω 2 on M along M , and H is a C ∞ -function of Xj .oh|M (in such a situation, the fibers of the map h are Abelian varieties or extensions of these by C∗k . This definition also includes the non-compact case and allows fibers that are more general than Abelian varieties). 2) Let (M, {., .}, ϕ) be a complex integrable system where M is a non-singular affine variety and ϕ = (H1 , ..., Hs ) is given by regular algebraic functions Hi . This system is algebraically completely integrable if, for generic c ∈ Cs , the fiber of ϕ−1 (c) is an affine part of an Abelian variety and if the Hamiltonian vector fields XHi are translation invariant when restricted to these fibers.

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3) Let (M, {., .}) be a smooth Poisson variety. An algebraically completely integrable Hamiltonian system consists of a proper flat morphism H : M −→ B, where B is a smooth variety, such that over the complement B\Δ of some proper closed subvariety Λ ⊂ B, the morphism H is a Lagrangian fibration whose fibers are isomorphic to Abelian varieties. 4) Let M be a 2n-dimensional complex manifold with a holomorphic symplectic structure ω, a holomorphic function H : M −→ C and n holomorphic functions which are pairwise in involution and also with H, that is, {Hi , Hj } = {Hi , H} = 0, 1 ≤ i, j ≤ n. Let B be an open dense subset of Cn and F : F −1 (B) ⊂ M −→ B, a submersive map. The Hamiltonian system defined by the vector field XH is an algebraic completely integrable system if there exists a bundle π : A −→ B of Abelian varieties, a divisor D ⊂ A, an isomorphism σ : F −1 (B) −→ A\D and a vector field Y on A\D that restricts to a linear vector field on the fibers of π, so that the following diagram: F −1 (B) −→ A\D F 

σ

*π

B is commutative and such that the vector field XH is σ-related to Y . 5) Note that Hitchin (1994) gave a large class of integrable systems that are almost by construction, algebraic completely integrable and showed that the cotangent bundle of moduli spaces of stable vector bundles on Riemann surfaces carry the structure of integrable systems and are indeed algebraic completely integrable. R EMARK 7.4.– a) The complete algebraic integrability in the sense of Adler– van Moerbeke in the case where M = Cm means that (i) the system [7.9] with polynomial right hand possesses n + k independent polynomial invariants H1 ≡ H, H2 , ..., Hn+k of which k invariants lead to zero vector fields, the n = (m − k)/2 remaining ones are in involution, which give rise to n commuting n+k 3 vector fields. For generic ci , the invariant manifolds {z ∈ Rm : Hi = ci } are i=1

assumed compact, connected and therefore real tori by the Arnold–Liouville’s theorem. (ii) Moreover, the invariant manifolds, thought of as affine varieties in Cm (non-compact), can be completed into complex algebraic tori, that is, ⎧ ⎛ ⎞⎫ n+k one or several ⎨ ⎬ 6 {z ∈ Cm : Hi (z) = ci } = T n \ D ≡ ⎝ codimension one ⎠ , ⎩ ⎭ i=1 subvarieties where the tori T n = Cn /Lattice = complex algebraic torus (Abelian variety) depends on the c’s. In the natural coordinates (t1 , ..., tn ) of these tori, the

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Hamiltonian flows (run with complex time) defined by the vector fields generated by the constants of the motion are straight-line motions, and the coordinates zi = zi (t1 , ..., tn ) are meromorphic in (t1 , ..., tn ). b) The existence of polynomial first integrals for a Hamiltonian system does not necessarily imply the complete integrability of this system. For example, the 2 Hamiltonian system where H(x, y) = x2 + P (y), (P (y) being a polynomial in y), will be algebraically completely integrable with Abelian (here elliptic) functions if and only if P (y) is a polynomial of degree 3 or 4. Following Mumford (1983a, 1983b), the commuting vector fields XH1 , ..., XHn define on the real torus Mc ⊂ R2n defined by the intersection of the constants of the motion H1 = c1 , ..., Hn = cn , an addition law ⊕ : Mc × Mc −→ Mc , (x, y) −→ x ⊕ y = gt+s (p), p ∈ Mc , with x = gt (p), y = gs (p), gt (p) = gtX1 1 ...gtXnn (p), where gtXi i (p) denote the flows generated by XHi . Algebraic complete integrability means that this addition law is rational, that is: (x ⊕ y)j = Rj (xi , yi , c), where Rj (xi , yi , c) is a rational function of all coordinates xi , yi , 1 ≤ i ≤ n. Putting x = p, y = gtXi (p), in this formula, we note that on the real torus Mc , the flows gtXi (p) depend rationally on the initial condition p. Moreover, a Weierstrass theorem on functions admitting an addition theorem states that the coordinates xi are restricted to the real torus: Rn /Lattice −→ Mc , (t1 , ..., tn ) −→ xi (t1 , ..., tn ), are Abelian functions. Geometrically, this means that the real torus Mc Rn /Lattice is the affine part of an algebraic complex torus (Abelian variety) Cn /Lattice and that the real functions xi (t1 , ..., tn ), (ti ∈ R), are the restrictions on this real torus of the meromorphic functions xi (t1 , ..., tn ), (ti ∈ C), of n complex variables, with 2n periods (n periods + n imaginary periods). In degenerate situations, some of these periods may be infinite, as in the case of a harmonic oscillator. c) If the Hamiltonian flow [7.9] is a.c.i., it means that the variables zi are meromorphic on the torus T n and by compactness they must blow up along a codimension one subvariety (a divisor) D ⊂ T n . By the a.c.i. definition, the flow [7.9] is a straight line motion in T n and thus, it must hit the divisor D in at least one place.

Figure 7.1. Divisor and flow trajectory

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Through every point of D, there is a straight line motion and a Laurent expansion around that point of intersection. The differential equations must admit Laurent expansions that depend on the n − 1 parameters defining D and the n + k constants ci defining the torus T n , the total count is m − 1 = dim (phase space) − 1 parameters. The fact that algebraic complete integrable systems possess (m − 1)-dimensional families of Laurent solutions was implicitly used by Kowalewski (1889) in her classification of integrable rigid body motions. Such a necessary condition (see Adler and van Moerbeke (1989)) for algebraic complete integrability can be formulated as follows. T HEOREM 7.2.– If the Hamiltonian system [7.9] (with invariant tori not containing elliptic curves) is algebraic complete integrable, then each zi blows up after a finite (complex) time, and for every zi , there is a family of solutions zi =

∞ 

(j)

zi tj−si ,

si ∈ Z,

some si > 0,

[7.11]

j=0

depending on dim(phase space) − 1 = m − 1, free parameters. The system [7.9] possesses families of Laurent solutions depending on m − 2, m − 3, ..., m − n parameters. The coefficients of each one of these solutions are rational functions on affine algebraic varieties of dimensions m − 1, m − 2, m − 3, ..., m − n. The question raised is whether this criterion is also sufficient. The main problem n+k 3 will be to complete the affine variety Mc = {z ∈ Cm , Hi (z) = ci }, into an i=1

Abelian variety. A naive guess would be to take the natural compactification M c of Mc by projectivizing the equations. Indeed, this can never work for a general reason: >c of dimension bigger or equal than two is never a complete an Abelian variety M intersection, that is, it can never be described in some projective space Pn by n-dim >c global polynomial homogeneous equations. In other words, if Mc is to be the M affine part of an Abelian variety, M c must have a singularity somewhere along the locus at infinity. The trajectories of the vector fields [7.9] hit every point of the singular locus at infinity and ignore the smooth locus at infinity. In fact, the existence of meromorphic solutions to the differential equations [7.9] depending on some free parameters can be used to manufacture the tori, without ever going through the delicate procedure of blowing up and down. Information about the tori can then be gathered from the divisor. More precisely, around the hitting points, the system of differential equations [7.9] admit a Laurent expansion solution depending on m − 1 free parameters, and in order to regularize the flow at infinity, we use these parameters to blow up the variety M c along the singular locus at infinity. The new complex variety obtained in this fashion is compact, smooth and has commuting vector fields on it; it is therefore an Abelian variety. The system [7.9] with k + n polynomial invariants has a coherent tree of Laurent solutions, when it has families of Laurent solutions in t, depending on n − 1, n − 2,...,m − n free parameters. Adler

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and van Moerbeke (1989) have shown that if the system possesses several families of (n − 1)-dimensional Laurent solutions (principal Painlevé solutions) they must fit together in a coherent way, and as we mentioned above, the system must possess (n − 2)-, (n − 3)-,... dimensional Laurent solutions (lower Painlevé solutions), which are the gluing agents of the (n − 1)-dimensional family. The gluing occurs via a rational change of coordinates, in which the lower parameter solutions are seen to be genuine limits of the higher parameter solutions, and which in turn appears due to a remarkable propriety of algebraic complete integrable systems; they can be put into quadratic form in both the original variables and their ratios (to see further). As a whole, the full set of Painlevé solutions glue together to form a fiber bundle with a singular base. A partial converse to theorem 7.8 can be formulated as follows (Abenda and Fedorov 2000): T HEOREM 7.3.– If the Hamiltonian system [7.9] satisfies the condition (a)(i) in the remark 7.7 of algebraic complete integrability and if it possesses a coherent tree of Laurent solutions, then the system is algebraic complete integrable and there are no other m − 1-dimensional Laurent solutions than those provided by the coherent set. We assume that the divisor is very ample and in addition, projectively normal. Consider a point p ∈ D, a chart Uj around p on the torus and a function yj in L(D) having a pole of maximal order at p. Then the vector (1/yj , y1 /yj , . . . , yN /yj ) provides a good system  of coordinates in Uj . Then, taking the derivative with regard to one of the flows

yi yj

˙ =

y˙i yj −yi y˙j , yj2

1 ≤ j ≤ N , are finite on Uj as well.

has a double pole along D, the numerator must also have a Therefore, since double pole (at worst), Hence, when D is projectively  that is, y˙i yj − yi y˙j ∈L(2D). yi yk yl normal, we have yj ˙ = k,l ak,l yj yj , that is, the ratios yi /yj form a closed system of coordinates under differentiation. At the bad points, the concept of projective normality plays an important role: this enables us to show that yi /yj is a bona fide Taylor series starting from every point in a neighborhood of the point in question. Moreover, the Laurent solutions provide an effective tool to find the constants of the motion. To do this, just search polynomials Hi of z, having the property that when evaluated along all of the Laurent solutions z(t), they have no polar part. Indeed, since an invariant function of the flow does not blow up along a Laurent solution, the series obtained by substituting the formal solutions [7.11] into the invariants should, in particular, have no polar part. The polynomial functions Hi (z(t)) being holomorphic and bounded in every direction of a compact space (i.e. bounded along all principle solutions) are thus constant by a Liouville type of argument. In this argument, it is thus important to use all of the generic solutions. To make these informal arguments rigorous is an outstanding question of the subject. As for the system [7.2], we also assume here that Hamiltonian flows are weight-homogeneous with a weight si ∈ N, going with each variable zi , i.e. fi (αs1 z1 , ..., αsm zm ) = αsi +1 fi (z1 , ..., zm ), ∀α ∈ C. Observe that the constants of yj2

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the motion H can then be chosen to be weight-homogeneous: H (αs1 z1 , ..., αsm zm ) = αk H (z1 , ..., zm ), k ∈ Z. The study of the algebraic complete integrability of Hamiltonian systems includes several passages to prove rigorously. Here, we mention the main passages, leaving out the detail when studying the different problems in the following sections. We saw that if the flow is algebraically completely integrable, the differential equations [7.9] must admit Laurent series solutions [7.11] depending on m − 1 free parameters. We must have ki = si and coefficients in the series must satisfy nonlinear equations at the 0th step,  (0) (0) (0) fi z1 , ..., zm + gi zi = 0, 1 ≤ i ≤ m, [7.12] and linear systems of equations at the kth step : % 0 for k = 1 (k) (L − kI)z = some polynomial in z (1) , ..., z (k−1) for k > 1,

[7.13]

where L = Jacobian map of [7.12] = ∂f ∂z + gI |z=z (0) . If m − 1 free parameters are to appear in the Laurent series, they must either come from the nonlinear equations [7.12] or from the eigenvalue problem [7.13], that is, L must have at least m − 1 integer eigenvalues. These conditions are much less than expected, because of the fact that the homogeneity k of the constant H must be an eigenvalue of L. Moreover, the formal series solutions are convergent as a consequence of the majorant method (theorem 7.1). Thus, the first step is to show the existence of the Laurent solutions, which requires an argument precisely every time k is an integer eigenvalue of L and therefore L − kI is not invertible. One shows the existence of the remaining constants of the motion in involution so as to reach the number n + k. Then we have to prove that for given c1 , ..., cm , the set = D≡

 B (0) (1) (2) zi (t) = t−νi zi + zi t + zi t2 + · · · , 1 ≤ i ≤ m Laurent solutions such that : Hj (zi (t)) = cj + Taylor part

defines one or several n − 1 dimensional algebraic varieties ( “Painlevé” divisor) n+k 3 {z ∈ Cm : Hi (z) = ci } ∪ D, is a smooth compact, with the property that i=1

connected variety with n commuting vector fields independent at every point, that is, k+n a complex algebraic torus Cn /Lattice. The flows J ∂H∂zk+i , ..., J ∂H∂z are straight line motions on this torus. Later, we will see in more detail that having computed the space of functions L(D) with simple poles at worst along the expansions, it is often important to compute the space L(kD) of functions having k-fold poles at worst along the expansions. These functions play a crucial role for embedding the invariant tori into projective space. From the divisor D, a lot of information can be obtained with regard to the periods and the action-angle variables. Some others integrable systems appear as coverings of algebraic completely integrable systems. The

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manifolds invariant by the complex flows are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. These systems will be studied in detail in the following chapter. 7.3. The Liouville–Arnold–Adler–van Moerbeke theorem The idea of the Adler–van Moerbeke’s proof (Adler and van Moerbeke 1985) we shall give here is closely related to the geometric spirit of the (real) Arnold–Liouville theorem (Arnold 1989). Namely, a compact complex n-dimensional variety, on which there exist n holomorphic commuting vector fields, which are independent at every point that is analytically isomorphic to a n-dimensional complex torus and the complex flows generated by the vector fields are straight lines on this complex torus. T HEOREM 7.4.– Let A be an irreducible variety defined by an intersection A=

6

{Z = (Z0 , Z1 , ..., Zn ) ∈ PN (C) : Pi (Z) = 0},

i

involving a large number of homogeneous polynomials Pi with a smooth and irreducible affine part A = A ∩ {Z0 = 0}. Put A ≡ A ∪ D, that is, D = A ∩ {Z0 = 0} and consider the map f : A −→ PN (C), Z −→ f (Z). Let A2 = f (A) = f (A), D = D1 ∪ ... ∪ Dr , where Di are codimension 1 subvarieties and also consider S ≡ f (D) = f (D1 ) ∪ ... ∪ f (Dr ) ≡ S1 ∪ ... ∪ Sr . Assume that (i) f maps A smoothly and 1-1 onto f (A). (ii) There exist n holomorphic vector fields X1 , ..., Xn on A, which commute and are independent at every point. One vector field, say Xk (where 1 ≤ k ≤ n), extends holomorphically to a neighborhood of Sk in the projective space PN (C). (iii) For all p ∈ Sk , the integral curve f (t) ∈ PN (C) of the vector field Xk through f (0) = p ∈ Sk has the property that {f (t) : 0 is compact, connected and admits an does not vanish on any point of Sk . Then (a) M embedding into PN (C). (b) A2 is diffeomorphic to a n-dimensional complex torus. 2 The vector fields X1 , ..., Xn extend holomorphically and remain independent on A. 2 > (c) A is a Kähler variety. (d) M is a Hodge variety. In particular, A is the affine part 2 of an Abelian variety A. P ROOF.– a) We will show that the orbits running through Sk form a smooth variety t Σp , p ∈ Sk , such that Σp \Sk ⊆ A. Let p ∈ Sk , ε > 0 small enough, gX the flow k t generated by Xk on A and {gXk : t ∈ C, 0 0, that is, the problem is not expressed in 02 c b c b terms of hyperelliptic integrals, but rather in terms of Abelian integrals associated with the period matrix. As we have seen in the previous section, such Abelian surfaces come up naturally as Prym varieties of double covers of elliptic curves ramified over four points. We look for weight homogeneous Laurent solutions to the system [3.4] in the case of Kowalewski (i.e. [3.12]). The latter is weight

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homogeneous, with M having weight 1 and Γ weight 2. Let M and Γ have the following asymptotic expansions: M=

∞ 

M (k) tk−1 ,

∞ 

Γ=

k=0

Γ(k) tk−2 .

[7.16]

k=0

By substituting [7.16] in the differential equations [3.4] (case of Kowalewski), at the 0th step, the coefficients of t−2 for M and t−3 for Γ yield a nonlinear system     M (0) + M (0) , ΛM (0) + Γ(0) , L = 0,   2Γ(0) + Γ(0) , ΛM (0) = 0,

[7.17]

and at the kth step (k ≥ 1), the coefficients of tk−2 for M and tk−3 for Γ lead to a system of linear equations in M k and Γk : ⎧ 0 for k = 1  (k) ⎪ ⎨ !

k−1  (i) M (k−i) , ΛM M − [7.18] = (Ψ − kI) i=1 Γ(k) ⎪ ⎩ − k−1 Γ(i) , ΛM (k−i)  for k ≥ 2 i=1 where Ψ denotes the linear operator  Ψ

X Y



=

4  4 ΛX + 4 X,ΛM (0) +4[Y, L] + X M (0)  , (0) . Γ , ΛX + Y, ΛM (0) + 2Y

In the basis (e1 , ..., e6 ), Ψ is given by the following matrix ⎛

(0)

1 m3 ⎜ −m(0) 1 ⎜ ⎜ 03 0 ⎜ Ψ=⎜ (0) ⎜ 0 −γ3 ⎜ (0) ⎝ γ3 0 (0) (0) −γ2 γ1

⎞ (0) m2 0 0 0 (0) −m1 0 0 2 ⎟ ⎟ 1 0 −2 0 ⎟ ⎟ (0) (0) (0) ⎟ , 2γ2 2 2m3 −m2 ⎟ (0) (0) (0) ⎟ −2γ1 −2m3 2 m1 ⎠ (0) (0) 2 0 m2 −m1

which is the Jacobian of [7.17]. T HEOREM 7.5.– The nonlinear system [7.17] defines two lines and two points. For the case corresponding to the points, the system [7.18] has two degrees of freedom for k = 2 and one degree of freedom for k = 3 and 4. For the case corresponding to the lines, the system [7.18] has one degree of freedom for k = 1, 2, 3 and 4. The solutions to the system [7.18] are obtained explicitly in a direct way.

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P ROOF.– The proof is a linear algebra problem (a straight forward computation).  The generic solution blows up after a finite time according to a Laurent series within a five-parameter family of Laurent solutions. When using the majorant method (theorem 7.1), any formal Laurent series solution of a system of differential equations with quadratic right-hand  side automatically converges. Now it is easily M (0) checked that (Ψ + I) = 0, and it follows from theorem 7.24 that Γ(0) det(Ψ − kI) = (k + 1)k(k − 1)(k − 2)(k − 3)(k − 4). Consequently, we have the following theorem: T HEOREM 7.6.– The system [3.4] in the case of Kowalewski (i.e.[3.12]) presents ∞ (k) k−1 two distinct families of Laurent series solutions: M = t , k=0 M ∞ (k) k−2 Γ = Γ t , depending on five free parameters such that the coefficients k=0 M (0) and Γ(0) satisfy the nonlinear equations,       M (0) + M (0) , ΛM (0) + Γ(0) , L = 0, 2Γ(0) + Γ(0) , ΛM (0) = 0, and depend on a free variable α, while M (k) and Γ(k) satisfy linear systems ⎧ 0 if k = 1  (k)  ⎪ ⎨ 

 M k−1 (i) (k−i) − i=1 M , ΛM = (M − kI) Γ(k) ⎪ ⎩ − k−1 Γ(i) , ΛM (k−i) if k ≥ 2 i=1 where M is the Jacobian matrix of [7.17]. These systems provide a free variable at each level k = 1, 2, 3, 4. The first meromorphic solution with the Laurent expansion is

α 1 γ1 (t) = 2 + o(t), + i α2 − 2 β + o(t), t 2t iα i m2 (t) = γ2 (t) = 2 + o(t), − α2 β + o(t), t 2t i β m3 (t) = + αβ + o(t), γ3 (t) = + o(t) t t m1 (t) =

and the second meromorphic solution with the Laurent expansion is

α 1 γ1 (t) = 2 + o(t), − i α2 − 2 β + o(t), t 2t iα i m2 (t) = − − α2 β + o(t), γ2 (t) = − 2 + o(t), t 2t β i γ3 (t) = + o(t). m3 (t) = − + αβ + o(t), t t m1 (t) =

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Integrable Systems

The study of convergence of these Laurent series solutions will be carried out via the majorant method (theorem 7.1) around all points where α, β = 0 and the techniques projective normality around bad points. Let $ 34 #of Mc = k=1 x ∈ C6 : Hk (x) = ck , c3 = 1, be the affine variety defined by the four constants of motion. The invariant variety Mc is a smooth affine surface for generic values of c1 , c2 and c3 . Also in this problem, how do we find the compactification of Mc into an Abelian surface? As has been done in the previous two sections, following the methods in Adler and van Moerbeke (1984), the idea of the direct proof as we have seen in theorem 7.4 is closely related to the geometric spirit of the (real) Arnold–Liouville theorem (Arnold 1989). Namely, a compact complex n-dimensional variety on which there exist n holomorphic commuting vector fields that are independent at every point is analytically isomorphic to a n-dimensional complex torus Cn /lattice, and the complex flows generated by the vector fields are straight lines on this complex torus. Now, the main problem will be to complete Mc into a non-singular compact >c = Mc ∪ D in such a way that the vector fields XH complex algebraic variety M 1 and XH4 generated, respectively, by H1 and H4 , extend holomorphically along a >c is an algebraic divisor D and remain independent there. If this is possible, M complex torus (an Abelian variety) and the coordinates restricted to Mc are Abelian functions. As we have already pointed out, a naive guess would be to take the natural compactification M c of Mc by projectivizing the equations in P6 (C). Indeed, this >c of dimension bigger or can never work for a general reason: an Abelian variety M equal than two is never a complete smooth intersection, that is it can never be >c global polynomial described in some projective space Pn (C) by n-dim M homogeneous equations. If Mc is to be the affine part of an Abelian surface, M c must have a singularity somewhere along the locus at infinity M c ∩ {Z = 0}. In fact, m γ what happens in this specific case? Let Z, Mj = Zj , Γj = Zj , be the projective coordinates and consider the transformation M c −→ M c , (Z, M1 , M2 , M3 , Γ1 , Γ2 , Γ3 ) −→ (Z, X1 , X2 , M3 , U1 , U2 , Γ3 ), where 2X1 = M1 + iM2 , 2X2 = M1 − iM2 , U1 = Γ1 + iΓ2 , U2 = Γ1 − iΓ2 . In these new variables, the equations defining M c are written as F1 ≡ 2X1 X2 + M32 + (U1 + U2 )Z − c1 Z 2 = 0, F2 ≡ X1 U2 + X2 U1 + M3 Γ3 − c2 Z 2 = 0, F3 ≡ U1 U2 + Γ23 − Z 2 = 0, F4 ≡ (X12 − U1 Z)(X22 − U2 Z) − c4 Z 4 . T HEOREM 7.7.– The projective variety M c ⊂ P6 (C) defined by the above equations is not an Abelian surface. In addition, M c intersects the hyperplane at infinity Z = 0

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according to the two straight lines d1 = (0, X1 , 0, 0, U1 , 0, 0), d2 = (0, 0, X2 , 0, 0, U2 , 0) and the circle S 1 = (0, 0, 0, 0, U1 , U2 , Γ3 ), where U1 U2 + Γ23 = 0. Moreover, M c is singular along the lines d1 , d2 ; singularity of type: Y 4 + U 3 Z 3 = 0, (Y, Z) small, and along the circle S 1 ; singularity of type: Γ43 X 2 = (Γ43 + Γ23 − 1)Z 2 , (X, Z) small. P ROOF.– Suppose that M c is an Abelian surface. Thus, this surface is isomorphic to a two-dimensional complex algebraic torus C2 /lattice, where the lattice is generated by four independent vectors in C2 . Let (ϕ1 , ϕ2 ) be the natural coordinates of this torus. The canonical divisor of this Abelian surface is empty as, up to scalars, the only 2-form is given by dϕ1 ∧ dϕ2 , which clearly does not vanish anywhere on M c . The canonical bundle KM c of M c is given in terms of the canonical bundle KP6 (C) of P6 (C) by   4  4 the adjunction formula KM c = KP6 (C) ⊗ M c M , where M c denotes the line c bundle associated with the divisor M c in P6 (C). Let x1 = MZ1 , x2 = MZ2 , x3 = MZ3 , y1 = ΓZ1 , y2 = ΓZ2 , y3 = ΓZ3 , be affine coordinates on P6 (C)\ {Z = 0} and consider the form ω = dx1 ∧ dx2 ∧ dx3 ∧ dy1 ∧ dy2 ∧ dy3 . This form has no zeroes or poles M2 M3 Γ1 Γ2 1 in {Z = 0}. Let u = ΓZ3 , u1 = M Γ3 , u2 = Γ3 , u3 = Γ3 , v1 = Γ3 , v2 = Γ3 , be affine x x x coordinates on P6 (C)\ {Z = 0}. Then u = y13 , u1 = y31 , u2 = y32 , u3 = y33 , v1 = yy13 , v2 = yy32 , and so ω = u17 du ∧ du1 ∧ du2 ∧ du3 ∧ dv1 ∧ dv2 . Therefore, the divisor of ω is (ω) = −7H where H is a hyperplane in P6 (C) and consequently KP6 (C) = [−7H]. Since every line bundle on projective space is a multiple of [H], we have KM c = [H]M c and this implies that every holomorphic 2-form on M c would have a zero on M c , a contradiction. The second part of the theorem is straightforward. We will describe the singularities of M c at infinity. First let us analyze the locus at infinity C = M c ∩ {Z = 0}. It is shown that C = d1 ∪ d2 ∪ S 1 , where d1 = (0, X1 , 0, 0, U1 , 0, 0) and d2 = (0, 0, X2 , 0, 0, U2 , 0) are two straight lines tangent to the following circle S 1 = (0, 0, 0, 0, U1 , U2 , Γ3 ), U1 U2 + Γ23 = 0. In theneighborhoods of these lines and  1 ,F2 ,F3 ,F4 ) of this circle, we have used ∂(Z,X∂(F = 3, and the variety M c is  1 ,X2 ,M3 ,U1 ,U2 ,Γ3 ) Z=0

singular along the straight lines d1 , d2 (singularity of type Y 4 + U 3 Z 3 = 0, Y , Z small) and along the circle S 1 (singularity of type Γ43 X 2 = (Γ43 + Γ23 − 1)Z 2 , X, Z small).  The solutions to the system [3.4] in the case of Kowalewski (i.e. [3.12]) intersect d1 and d2 , however the solutions ignore S 1 and the same facts holds for the other vector field [3.14] commuting with the first. To regularize (i.e. make the trajectories parallel) the flow at infinity, it is necessary to solve the singularities of the projective variety M c . We have just seen that these singularities are complicated and we must (probably) blow up M c many times along the lines and blow down M c along the circle. Generally, the method of blowing up and blowing down sub-varieties is considered very delicate and its use in the present case is difficult. In fact, we shall show that the existence of meromorphic solutions to

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the differential equations in question, depending on four free parameters, can be used to manufacture the tori, without ever going through the delicate procedure of blowing up and down. Information about the tori can then be gathered from the divisor.

Figure 7.4. Behavior of solutions around d1 , d2 and S 1

Consider the set of Laurent solutions which remain confined to a fixed affine invariant surface (where c3 = 1 and c1 , c2 , c4 are generic constants), that is, % Dε =

the Laurent solutions mj (t), γj (t), 1 ≤ j ≤ 3, such that : Hk (mj (t), γj (t)) = ck + Taylor part , 1 ≤ k ≤ 4

& ,

= 4 polynomial relations between α, β, γ and μ; c1 = (α2 − 4)β 2 + 18λ εc2 = −(α2 + 2)β 3 + 6βλ − 12θ 8 = (5α2 − 2)β 4 − 6β 2 λ + 84βθ − 240μ   8c4 = (α2 − 1) (11α2 + 10)β 4 + 54β 2 λ + 108βθ + 240μ = algebraic curve ⎫ ⎧ (α, β, ε) such that : ⎬ ⎨ P (α, β, ε) ≡ (α2 − 1)((α2 − 1)β 4 − P (β) + c4 = 0, . = ⎭ ⎩ P (β) ≡ c1 β 2 − 2εc2 β − 1, ε ≡ ±i

[7.19]

The quotient Dε0 = Dε /σε by the involution, σε : Dε −→ Dε ,

(α, β, ε) −→ (−α, β, ε),

[7.20]

is an elliptic curve defined by u2 = P 2 (β) − 4c4 β 4 .

[7.21]

The curve Dε is a two-sheeted ramified covering of Dε0 , ϕε : Dε −→ Dε0 ,

(α, u, β, ε) −→ (u, β, ε),

[7.22]

Algebraically Integrable Systems

= Dε :

181

4

(β)+u α2 = 2β +P 2β 4 u2 = P 2 (β) − 4c4 β 4 .

[7.23]

Let us look more closely at certain points of interest on the non-singular version √ 2 4 2β +P (β)+ P (β)−4c β4 4 = of the curve Dε . For β sufficiently small, α2 = 2β 4 √ 2β 4 +P (β)− P 2 (β)−4c4 β 4 1 2 1 + c4 + o(β) and α = = 4 (−1 + o(β). At β = ∞, the 2β 4 "β 2 2 curve Dε behaves as follows: 2(α − 1)β = c1 ± c21 − 4c4 + o(β). The curve Dε has four points at infinity pj (1 ≤ j ≤ 4) and four branch points qj ≡ (α = 0, u = −2β 4 − P (β), β 4 + P (β) + c4 = 0) (1 ≤ j ≤ 4) on the elliptic

4

4 qj − j=1 pj , curve Dε0 . The divisor structure of α and β on Dε is (α) =  0j=1 

4 (β) = 4 zeroes − j=1 pj . Let g (Dε ) = genus of Dε , g Dε = genus of Dε0 , n = number of sheets and v = number   of branch  points . According to the Riemann–Hurwitz formula, g (Dε ) = n g Dε0 − 1 + 1 + v2 = 3. The map (α, u, β, ε) −→ (α, u, −β, −ε) is an isomorphism between Dε=i and Dε=−i and so we have the following commutative diagram ∼

Dε=i −→ ↓ϕε=i ∼ 0 Dε=i −→

Dε=−i ↓ϕε=−i . 0 Dε=−i

Thus, we have proved the following theorem. T HEOREM 7.8.– The divisor on which the solutions of [3.12] blow up consists of two isomorphic irreducible components Dε=i and Dε=−i , which are both non-singular curves of genus 3. Each of these curves Dε [7.19], [7.23] is a double cover of an elliptic curve Dε0 [7.21] ramified at four points. Let D ≡ Dε=i + Dε=−i . For positive integers k, define the space L(kD) of meromorphic functions having a k-fold pole at worst along D. T HEOREM 7.9.– The space L(D) is spanned by the following functions: f0 = 1, f1 = m1 , f2 = m2 , f3 = m3 , f4 = γ3 , f5 = f12 + f22 , f6 = 4f1 f4 − f3 f5 , f7 = (f2 γ1 − f1 γ2 ) f3 + 2f4 γ2 . In addition, Dis embedded into P7 (C) according to (0)

(0)

p = (α, u, β) −→ lim t(1, f1 (p), ..., f7 (p)) = 0, f1 (p), ..., f7 (p) , in a way that t→0

intersect each other D transversally in four points at infinity  ε=i and Dε=−i " 2 2 c1 − 4c4 , β = ∞ and that the geometric genus of D is 9. α = ±1, u = ±β P ROOF.– Let L(r) be the set of polynomial functions f = f (x(t)) of degree less than or equal to r, modulo the constants Hk = ck , 1 ≤ k ≤ 4 and such that: f (x(t)) = t−1 (x(0) + x(1) t + · · · ), x(0) = 0 on D, with x(t) = (m1 , m2 , m3 , γ1 , γ2 , γ3 ) given explicitly above. We look for r such that the geometric genus of D(r) = Nr + 2,

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D(r) ⊂ PNr (C), and we shall show that it is unnecessary to go beyond r = 4. Indeed, using the Laurent series obtained previously, one obtains that the spaces L(r) , nested according to weighted degree, are generated as follows: L(1) = {f0 = 1, f1 = m1 , f2 = m2 , f3 = m3 }, % & 1 L(2) = L(1) ⊕ f4 = γ3 , f5 = − (f12 + f22 ) , 4 L(3) = L(2) ⊕ {f6 = f3 f5 + f1 f4 } , L(4) = L(3) ⊕ {f7 = (f2 γ1 − f1 γ2 )f3 + 2f4 γ2 } . Note that the spaces L(1) , L(2) and L(3) do not provide the embedding because g(D(r) ) = dim L(r) + 1, r = 1, 2, 3. More precisely, since the embedding in P3 (C) via L(1) does not separate the sheets, we pass to a L(2) andwe find that the  then 6 (2) embedding in P (C) is invalid because g D embedded in P5 (C) − 2 > 5, which (2) (3) contradicts the  fact that Nr + 1 = g(D  ) − 1. In the same way, using space L we obtain g D(3) embedded in P6 (C) − 2 > 6, and the contradiction persists. We must therefore consider L(4) and study the embedding of D into P7 (C) using the (0) (1) functions of L(4) . Let ff04 , ..., ff74 ≡ Fk = Fk + Fk t + · · · , 0 ≤ k ≤ 7, and  " pj ≡ α = ±1, u = ±β 2 c21 − 4c4 , β = ∞ , 1 ≤ j ≤ 4, be the four points at ∞ of Dε . Therefore α εα ε c2 ε(1 − u) (pj ), , , , 1, εα, ε(α2 − 1)β, − + β β β β β2  2 = 0, 0, 0, 0, 1, ±ε, 0, ∓ε c1 − 4c4 = 4 distinct points. 

(0)

Fk (pj ) =

0,

The four points at ∞ are separated on each curve Dε , but using the transformation Dε=i −→ Dε=−i , (α, u, β) −→ (−α, −u, β), one shows    (0) (0) Fk (pj ) = Fk (pj ) , that is, the four points at ∞ are identified pairwise Dε=i

Dε=−i

with the four points at ∞ on the other curve. Let s = Since

1 β

be a local parameter for pj .

 (0) (0)  ∂Fk ∂Fk  (pj ) = (0, ±1, ±ε, ε, 0, −c2 ) =⇒ (pj )  ∂s ∂s

Dε=i

 (0)  ∂Fk  = (pj )  ∂s

, Dε=−i

then the curve Dε=i intersects the curve Dε=−i transversely in four points at infinity b1 , ..., b4 = (α = ∞, β = 0), p1 , ..., p4 = (α = ±1, u = ±β 2

c21 − 4c4 , β = ∞).

Algebraically Integrable Systems

183

Figure 7.5. Curves Dε=±i

In a neighborhood  of the points bj = (α = ∞, β = 0), 1 ≤ j ≤ 4, one divides by f1 . By setting

f0 f7 f1 , ..., f1

(0)

(1)

≡ Gk = Gk + Gk t + · · · , 0 ≤ k ≤ 7, one obtains

(0)

Gk (pj ) = (0, 1, ε, 0, 0,  0, ∓1, 0), which are thecoordinates of four different  points in P7 (C). Therefore, g D(4) embedded in P7 (C) − 2 = 7, that is, g D(4) = 9.  To avoid overburdening the text, we will use the notation D instead of D(4) when embedding in P7 (C). At all points where α, β = 0, ∞, the Laurent solutions are nicely convergent (theorem 7.1). Therefore, at most points of D there is a transversal fiber to the curve. Hence, this defines a smooth surface strip around D except at the bad points. Now we need to construct a surface strip around D at the bad points as well. To do so, we must use the concept of a normally generated line bundle or projective normality. Ultimately, we wish to prove that in the various charts  .  fj fj , 1 ≤ j ≤ 7, k fixed [7.24] = polynomial fk fk  f This enables us to show that fkj is a bona fide Taylor series starting from every point in a neighborhood of the point in question ⊆ P7 (C). As mentioned above, let L ≡ L(4) and D ≡ D(4) . T HEOREM 7.10.– The orbits of the vector field [3.12] going through the curve D form >c = Mc ∪Σ a smooth surface Σ near D such that Σ\D ⊆ Mc . Moreover, the variety M is smooth, compact and connected. P ROOF.– Let φ(t, p) = {z(t) = (m1 (t, p), m2 (t, p), m3 (t, p), γ1 (t, p), γ2 (t, p), γ3 (t, p), )}, t ∈ C, 0 < |t| < ε, be the orbit of the vector field [3.12] going through the point p ∈ D. Consider the surface element Σp ⊂ P7 (C) formed by the divisor D and the orbits going through p, and set Σ ≡ ∪p∈DΣp . Let D = H ∩ Σ be the curve where H ⊂ P7 (C) is a hyperplane transversal to the direction of the flow. If D is smooth, then using the implicit function theorem the surface Σ is smooth. But if D is singular at 0, then Σ would be singular along the trajectory (t−axis), which immediately goes into the affine part Mc . Hence, Mc would be singular, which is a

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contradiction because Mc is the fiber of a morphism from C6 to C4 and so smooth for almost all of the three constants of the motion ck . Let M c be the projective closure of Mc into P6 (C), let Z = [Z0 , M1 , M2 , M3 , Γ1 , Γ2 , Γ3 ] ∈ P6 (C) where Z0 , m γ Mj = Z0j , Γj = Zj0 , are the projective coordinates and let C = M c ∩ {Z0 = 0} be the locus at infinity. By theorem 7.7, we have C = d1 ∪ d2 ∪ S 1 where d1 , d2 are straight lines and S 1 is a circle with d1 ∩ d2 = ∅ and dj ∩ S 1 = point, j = 1, 2. Consider the application M c ⊆ P6 (C) −→ P7 (C), Z −→ f (Z), where >c = f (M c ). In a f = (f0 , f1 , ..., f7 ) ∈ L(D) (see theorem 7.9) and let M 7 > neighborhood V (p) ⊆ P (C) of p, we have Σp = Mc and Σp \S ⊆ Mc . Otherwise >c such that: Σp ∩ Σp = (t − axis), there would exist an element of surface Σp ⊆ M orbit φ(t, p) = (t − axis)\ p ⊆ Mc , and hence Mc would be singular along the t−axis, which is impossible. Since the variety M c ∩ {Z0 = 0} is irreducible and the generic hyperplane section Hgen. of M c is also irreducible, all hyperplane sections are connected and hence C is also connected. Now, consider the graph Γf ⊆ P6 (C) × P7 (C) of the map f , which is irreducible together with M c . It follows from the irreducibility of C that a generic hyperplane section Γf ∩ {Hgen. × P7 (C)} is irreducible, hence the special hyperplane section Γf ∩ {{Z0 = 0} × P7 (C)} is connected and the projection map projP7 (C) {Γf ∩ {{Z0 = 0} × P7 }} = f (C) ≡ D >c is compact, connected and embeds is connected. Hence, the variety Mc ∪ Σ = M 7 smoothly into P (C) via f .  We shall prove a somewhat stronger statement than [7.24], namely that [7.24] is satisfied with quadratic polynomials. By we see that the f0 , ..., f7 do  inspection, .  not fj 2 γ1 satisfy that property. For example, ff40 , = f1 γ2f−f = polynomial 2 f 4 4 1 ≤ j ≤ 7. The problem is that D, although very ample, will not be projectively normal (the line bundle [D] is not normally generated), which forces us to look at the divisor 2D or maybe simply 2Dε=i + Dε=−i . Hence, we must take functions with higher order poles. For instance, let us consider the space L(2Dε=i + Dε=−i ) of functions, which have double poles at worst along Dε=i and simple ones at worst along Dε=−i . When using the Riemann–Roch theorem, if 2Dε=i + Dε=−i is a divisor on an Abelian variety, this space must be 18-dimensional. Indeed, let L(2Dε=i + Dε=−i ) = L(4) ⊕ {g8 , ..., g17 } ≡ {g0 , ..., g17 }, where g0 = 1, g1 = 12 (m1 + im2 ), g2 = 12 (m1 − im2 ), g3 = f3 , g4 = f4 , g5 = f5 , g6 = f6 , g7 = f7 , g8 = g22 , g9 = g2 g3 , g10 = g2 g4 , g11 = g22 − γ1 − iγ2 , g12 = g1 g11 , g13 = g12 g11 , g14 = g13 g11 , g15 = g2 g5 , g16 = g2 g6 , g17 = g2 g7 , where one reads off the behavior of the gk along Dε from the Laurent series solutions; here (gk ) denote the divisor restricted to Dε : (g1 ) = −Dε=−i , (g2 ) = −Dε=i , (gk ) = −Dε=i − Dε=−i where k = 3, ..., 7, (g8 ) = −2Dε=i , (gk ) = −2Dε=i − Dε=−i , where k = 9, 10, 15, 16, 17, (g11 ) = −2Dε=i + 2Dε=−i , (g12 ) = −2Dε=i + Dε=−i , (gk ) = −2Dε=−i , where k = 13, 14. Note that all gk have the property (gk ) ≥ −2Dε=i − Dε=−i . Now consider the embedding of −→ P17 (C), 2Dε=i + Dε=−i via these functions, 2Dε=i + Dε=−i

Algebraically Integrable Systems

185

(0) (0) 0, g1 (p), ..., g17 (p) . About the points  bj = (α = ∞, β = 0), it is appropriate to divide by g8 , which implies that ggk8 (bj ), 0 ≤ j ≤ 17, is finite. In a neighborhood of the points at infinity " pj = α = ±1, u = ±β 2 c21 − 4c4 , β = ∞ , 1 ≤ j ≤ 4, one divides g0 , ..., g17 by  k g10 , which makes gg10 (pj ), 0 ≤ j ≤ 17, finite. Using [3.12], we show that in a neighborhood of the points pj and modulo linear combination of the constants of   · · · k g 10 0 k . Also, in a = gk g10g−g = quadratic polynomial of gg10 , ..., gg17 motion gg10 2 10 10  · neighborhood of the points bj we have ggk8 = quadratic polynomial of    g0 g17 gk gk g8 , ..., g8 . It follows that at the bad points pj , bj the series g10 (pj ), g8 (bj ), where 0 ≤ k ≤ 17, converge as a consequence  of Picard’s theorem applied to the p −→ limt→0 t (g0 (p), ..., g17 (p)) =

system of ordinary differential equations following theorem:



gk gl

·

, l = 8, 10. Therefore, we have the

T HEOREM 7.11.– The divisor 2Dε=i + Dε=−i is projectively normal and has a smooth embedding in P17 (C), which shows that in particular the Laurent series solutions converge everywhere. T HEOREM 7.12.– The two commuting vector fields [3.12] and [3.14] extend >c . holomorphically and remain independent on M P ROOF.– Let ϕt1 , ϕt2 be the flows generated, respectively, by vector fields [3.12] >c \Mc = D. For δ sufficiently small, ϕτ1 , −δ < and [3.14]. Consider a point p ∈ M >c by g t2 (q) = τ1 < δ, is well defined and ϕτ1 ∈ Mc . We may define ϕt2 on M g −t1 g t2 g t1 (q), q ∈ U (p) = g −t1 (U (g t1 (p))), where U (p) is a neighborhood of p. By commutativity one can see that g t2 is independent of t1 ; g −t1 −ε1 g t2 g t1 +ε1 (q) = g −t1 g −ε1 g t2 g t1 g ε1 (q) = g −t1 g t2 g t1 (q). Note that g t2 (q) is holomorphic away from D. This is because g t2 g t1 (q) is holomorphic away from D and g t1 is holomorphic in U (p) and maps bi-holomorphically U (p) onto U (g t1 (p)).  The flows ϕt1 and ϕt2 are holomorphic, independent on D and we can show along >c is a torus. And that will be done by the same lines as in the theorem 7.10 that M 2 >c , (t1 , t2 ) −→ ζ(t1 , t2 ) = ϕt1 ϕt2 (p), considering the holomorphic map ζ : C −→ M for a base point p ∈ Mc . Then L = {(t1 , t2 ) ∈ C2 : ζ(t1 , t2 ) = p}, is a lattice of >c is a biholomorphic diffeomorphism. Thus, M >c ⊆ P7 C2 , hence ζ : C2 /L −→ M 2 is conformal to a complex torus C /L and an Abelian surface as a consequence of Chow. We have the following theorem: >c is an Abelian surface on which the Hamiltonian flows [3.12] T HEOREM 7.13.– M and [3.14] are straight line motions.

186

Integrable Systems

T HEOREM 7.14.– The three holomorphic differentials on D are   k1 α2 − 1 β 2 dβ dβ k2 dβ ω0 = , ω1 = , ω2 = , u αu αu

[7.25]

>c there are two where u is given by [7.21] and k1 , k2 ∈ C. In addition, on M holomorphic differentials dt1 and dt2 , such that: dt1 |Dε = ω1 , dt2 |Dε = ω2 , where ω1 and ω2 are the two holomorphic differentials [7.25] on Dε . P ROOF.– From the Poincaré residue formula, we know that the three holomorphic differentials on Dε are of the form g(α,β,ε)dβ = g(α,β,ε)dβ , where g(α, β, ε) is a ∂P αu (α,β,ε) ∂α

polynomial of at most degree five in α and β and P (α, β, ε) is given by [7.19]. It is easy to verify that ω0 , ω1 , ω2 [7.25] effectively form a basis of holomorphic differentials on Dε . Let p ∈ Dε ∩ {αu = 0}. Around the point p, we consider two >c , coordinates on M % 1 m1 + im2 = −iβ + o(t) along Dε=i τ= = −εt + o(t2 ), x= m1 − im2 = iβ + o(t) along Dε=−i m3 We denote by ∂t∂1 (resp. ∂t∂2 ) the derivative according to the vector field [3.12] (respectively, [3.14]). Obviously, we have 1 dt1 = Δ(τ, x)



∂x ∂τ dτ − dx , ∂t1 ∂t1

∂τ ∂x . − where Δ(τ, x) = ∂t 1 ∂t2 expansions, we find that

∂τ ∂x ∂t2 . ∂t1 .

1 dt2 = Δ(τ, x)



∂x ∂τ − dτ + dx , ∂t1 ∂t1

By direct computation using the asymptotic

∂τ = −ε + o(t), ∂t1

∂x = −2αβ 2 + o(t), ∂t1

∂τ = −4ε(α2 − 1)β 2 + o(t), ∂t2

∂x = 8α(α2 − 1)β 4 − P (β) + o(t), ∂t2

where P (β) ≡ c1 β 2 − 2εc2 β − 1, from which one can deduce the two differentials dt1 and dt2 . The restrictions of dt1 and dt2 to the curve Dε are given by dt1 |D

  k1 α2 − 1 β 2 dβ = , αu

dt2 |D =

k2 dβ , αu

(k1 , k2 ∈ C),

and are the two holomorphic differentials ω1 , ω2 [7.25] on Dε .  T HEOREM 7.15.– The vector field [3.12] (respectively, [3.14]) is regular along D, transversal to D at every point β = 0 (respectively, β = ∞) and (doubly) tangent at β = 0 (respectively, β = ∞).

Algebraically Integrable Systems

187

P ROOF.– Using the same notation  as in the proof of theorem 7.9, one can see that ∂ ∂ 0 0 F (p ) F (p ) j j 0 1 ∂s l = 0, and consequently there exist Fk at Fl such that det ∂s 1k Fk (pj ) Fl1 (pj ) the vector field [3.12] is transversal to D at the four points pj of Dε=i ∩ Dε=−i . k1 1 2 2 1 From theorem 7.14, the function ω ω2 = k2 (α − 1)β ∼ β 2 is meromorphic along a neighborhood of bj = (α = ∞, β = 0), (1 ≤ j ≤ 4) and provides the tangent to the 1 curve D in the coordinates t1 and t2 . The function ω ω2 vanishes whenever the vector field [3.14] is tangent to D and has a pole whenever [3.12] is tangent to D. Hence, the zeroes bj of ω2 provide the four points of tangency of the vector field [3.12] to D.  ∂ 0 ∂ Gk (bj ) ∂β G0l (bj ) 0 1 ∂β = 0, where the We find that for all Gk , Gl , we have det G1l (bj ) G1k (bj ) notation G0k , G1l has been introduced in the proof of theorem 7.9. Consequently, [3.12] is (doubly) tangent to D at four points bj , which concludes the proof of the theorem.  T HEOREM 7.16.– The space 0 1 of holomorphic differentials on the divisor D is (0) (0)) (0) (0) (0) (0) f1 ω2 f2 ω2 , ..., f7 ω2 ⊕ {ω1 , ω2 }, where f1 , f2 , ..., f7 are the first coefficients (the residues) of the functions f1 , f2 , ..., f7 ∈ L(D) (theorem 7.9) and the embedding of D into P7 (C) is the canonical embedding 1 0 (0) (0) (0) p = (α, u, β) ∈ D −→ ω2 , f1 ω2 , f2 ω2 , ..., f7 ω2 ∈ P7 (C). P ROOF.– The adjunction formula gives us a map, the Poincaré residue map, between >c , with a pole along D and holomorphic 1-forms on D. meromorphic 2-forms on M Applied to the 2-form ω = fj dt1 ∧ dt2 with fj ∈ L(D),

ω=

dt1 ∧ dt2 1 fj

−→ Res ω|D = −

  dt1    = 1  f

∂ ∂t2

j

D

  dt2    , 1  f

∂ ∂t1

j

D

hence,

Res ω|D = ∂ ∂t1

where

     dt2   = fj(0) dt2  = fj(0) ω2 , D t1  2 (0) + o(t1 )  f j

D

∂ ∂t1

is the derivative according to the vector field [3.12]. Using the second vector (0) (0) field [3.14], we find ω = f2j ω1 , with f2j the residue. The differentials ω1 , ω2 , (0)

(0)

f1 ω2 , ..., f7 ω2 , form a basis for the space of holomorphic differential forms on D,

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Integrable Systems

0 1 (0) that is, H 0 (D, Ω1D ) = {ω1 , ω2 } ⊕ fj ω2 , 1 ≤ j ≤ 7 = {ω1 , ω2 } ⊕ W ω2 , where the space W consists of all the residues of the functions of L(D). The embedding into P7 (C) via the residues of fj is the canonical embedding of the curve D via its holomorphic differentials, except for the 0 1 two differentials ω1 and ω2 , p = (α, u, β) ∈ (0)

(0)

(0)

D −→ ω2 , f1 ω2 , f2 ω2 , ..., f7 ω2 ∈ P7 (C). This proves theorem 7.16. 

As we have seen (theorem 3.4 (a)), the involution σ [3.17] has eight fixed points on the affine variety Mc . In fact (which corresponds to involution σε [7.20] on Dε ), it has eight other fixed points at infinity given by the branch points of Dε on Dε0 . So the >c , thus confirming the number of fixed involution in question has 16 fixed points on M points that it should have on an Abelian variety. >c coming from T HEOREM 7.17.– The involution σ [3.17] on the Abelian surface M the one defined on the affine variety Mc has eight fixed points at infinity. P ROOF.– From the asymptotic expansions (see theorem 7.6), the functions m1 , m2 , γ1 , γ2 remain invariable by the transformation (t, α, β) −→ (−t, −α, β), whereas m3 , γ3 change into −m3 , −γ3 . Then the involution σ [3.17] is transformed at infinity into an involution σε [7.20] on Dε . Now the fixed points of σε are given by the branch points of Dε on Dε0 .  > T HEOREM 7.18.– The Abelian  surface  Mc that completes the affine surface Mc is the dual Prym variety P rym∗ Dε /Dε0 of the genus 3 curve Dε [7.19] or [7.23] for the involution σε [7.20] interchanging the sheets of the double covering ϕε [7.2]. P ROOF.– Let (a1 , a2 , a3 , b1 , b2 , b3 ) be a canonical homology basis of D , such that σ (a1 ) = a3 , σ (b1 ) = b3 , σ (a2 ) = −a2 , σ (b2 ) = −b2 for the involution σ .

Figure 7.6. Genus 3curve with a canonical basis of cycles

Algebraically Integrable Systems

189

As a basis of holomorphic differentials ω1 , ω2 , ω3 on the curve D , we take,   k1 α2 − 1 β 2 dβ k2 dβ dβ ω1 = , ω2 = , ω3 = , [7.26] αu αu u and obviously σ ∗(ω1 ) =−ω1 , σ ∗ (ω2 ) = −ω2 , σ ∗ (ω3 ) = ω3 . Recall that the Prym variety P rym Dε /Dε0 is a subabelian variety of the Jacobi variety Jac(Dε ) = P ic0 (Dε ) = H 1 (ODε )/H 1 (Dε , Z), constructed from the double cover ϕε (7.22), the involution σε on Dε interchanging sheets, extends by linearity to a map σε : Jac(Dε ) → Jac(Dε ) and up to some points of order two, Jac(Dε ) splits into an even part and an odd part: the even part is an elliptic curve (the quotient of Dε by the 0 involution  σ , i.e.  Dε [7.21]) and the odd part is a two-dimensional Abelian surface P rym Dε /Dε0 . We consider the period matrix Ω of Jac(Dε )      ⎞ ⎛ a1 ω1 a2 ω1 a3 ω1 b1 ω1 b2 ω1 b3 ω1 Ω = ⎝ a1 ω2 a2 ω2 a3 ω2 b1 ω2 b2 ω2 b3 ω2 ⎠ . ω ω ω ω ω ω a1 3 a2 3 a3 3 b1 3 b2 3 b3 3 Then,      ⎞ ⎛ a1 ω1 a2 ω1 − a1 ω1 b1 ω1 b2 ω1 − b1 ω1 Ω = ⎝ a1 ω2 a2 ω2 − a1 ω2 b1 ω2 b2 ω2 − b1 ω2 ⎠ , ω 0 ω ω 0 ω a1 3 a1 3 b1 3 b1 3 Dε0 ), P rym(Dε /Dε0 ) and and therefore the period matrices of Jac(Dε0 ) (i.e.   P rym∗ (Dε /Dε0 ) are, respectively, Δ = a1 ω3 ω , b1 3 Γ=

     2 a1 ω1 a2 ω1 2 b1 ω1 b2 ω1 , 2 a1 ω2 a2 ω2 2 b1 ω2 b2 ω2

    ω ω ω ω a1 1 a2 1 b1 1 b2 1  Γ = . ω ω ω ω a1 2 a2 2 b1 2 b2 2 ∗

Let

LΩ =

⎧ 3 ⎨ ⎩

i=1

+ mi

⎫ ⎞ ⎛ ⎞ + ω1 ω1 ⎬ ⎝ ω 2 ⎠ + ni ⎝ ω 2 ⎠ : m i , ni ∈ Z , ⎭ ai bi ω3 ω3 ⎛

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Integrable Systems

be the period lattice associated with Ω. Let us also denote by LΔ , the period lattice associated Δ. We have the following diagram (Nρε is the norm mapping, surjective), 0 ↓ Dε0 ↓ ρ∗ε

* Nρ

ε 0 −→ ker Nρε −→ P rym(Dε /Dε0 ) ⊕ Dε0 = Jac(Dε ) −→  τε ↓ >c ∪ 2Dε C2 /Lattice M ↓ 0

Dε ↓ ρε Dε0

−→ 0

>c = P rym∗ (Dε /Dε0 ) has kernel The polarization map τε : P rym(Dε /Dε0 ) −→ M 0 (ρ Dε ) Z2 × Z2 and the induced polarization on P rym(Dε /Dε0 ) is of type (1,2).    >c −→ C2 /LΛ : p −→ p dt1 be the uniformizing map where dt1 , dt2 are Let M p0 dt2 >c corresponding to the flows generated, respectively, by H1 , H2 two differentials on M  % &

4 dt1 νk  : nk ∈ Z , such that: dt1 |Dε = ω1 and dt2 |Dε = ω2 , LΛ = k=1 nk dt2 νk is the lattice associated with the period matrix     dt1 ν2 dt1 ν3 dt1 ν4 dt1 ν 1  , Λ= dt2 ν2 dt2 ν3 dt2 ν4 dt2 ν1 ∗

>c , Z). By the Lefschetz theorem on hyperplane and (ν1 , ν2 , ν3 , ν4 ) is a basis of H1 (M >c , Z) induced by the inclusion Dε → M >c is section, the map H1 (Dε , Z) −→ H1 (M surjective and we can find four cycles ν1 , ν2 , ν3 , ν4 on the curve Dε such that     ω1 ν2 ω1 ν3 ω1 ν4 ω1 Λ = ν1 , ω ω ω ω ν 1 2 ν 2 2 ν 3 2 ν4 2 and LΛ =

 & ω νk 1  : nk ∈ Z . k=1 nk ω νk 2

%

4

The cycles ν1 , ν2 , ν3 , ν4 that we look for in Dε are a1 , a2 , b1 , b2 and they generate >c , Z), such that H1 ( M     ω ω ω ω a1 1 a2 1 b1 1 b2 1  Λ= , ω ω ω ω a1 2 a2 2 b1 2 b2 2 is a Riemann matrix. So Λ = Γ∗ , that is, the period matrix of P rym∗ (Dε /Dε0 ) dual >c and P rym∗ (Dε /Dε0 ) are two Abelian of P rym(Dε /Dε0 ). Consequently, M

Algebraically Integrable Systems

191

varieties analytically isomorphic to the same complex torus C2 /LΛ . By Chow’s >c and P rym∗ (Dε /D0 ) are then algebraically theorem (Griffiths and Harris 1978), M ε isomorphic. 

Figure 7.7. Sheet +and sheet −

In summary, the use of the Laurent series (and the study of convergence of various series intervening in the problem via the majorant method and the concept of normally generated line bundle) show, that the orbits of the vector field [3.12] going through the >c and the variety curve D form a smooth surface Σ near D, such that Σ\Mc ⊆ M >c = Mc ∪ Σ is smooth, compact and connected. More precisely, for generic c, M >c of polarization (1, 2), the affine surface Mc completes into an Abelian surface M by adding a singular divisor D consisting of two isomorphic genus 3 curves Dε=i and Dε=−i intersecting in four points. Each Dε=±i is a double cover of an elliptic curve Dε0 ramified at four points. It defines a line bundle and a polarization (1, 2) on >c . The divisors 2Dε=i , 2Dε=−i or Dε=i + Dε=−i (of geometric genus 9) are all M very ample and define a polarization (2, 4); moreover, the eight-dimensional space of sections of the corresponding line bundle embeds the Abelian surface into P7 (C). >c and In other words, the line bundle [D] defines a polarization of type (2, 4) on M 7 > leads to an embedding of Mc in P (C); it is not normally generated, but the line >c is equipped with two everywhere bundle [2Dε=i + Dε=−i ] is. The Abelian surface M > independent commuting vector fields. Otherwise,  the Abelian surface Mc can also be identified as the dual of the Prym variety Dε /Dε0 and the problem linearizes on this variety. 7.6. The Hénon–Heiles system Consider the system x1 = y 1 ,

.

y 1 = −Ax1 − 2x1 x2 ,

.

.

x2 = y 2 ,

.

[7.27]

y 2 = −Bx2 − x21 − εx22 ,

corresponding to a generalized Hénon–Heiles Hamiltonian 1 ε 1 H = (y12 + y22 ) + (Ax21 + Bx22 ) + x21 x2 + x32 , 2 2 3

[7.28]

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Integrable Systems

where A, B, ε are constant parameters and x1 , x2 , y1 , y2 are canonical coordinates and momenta, respectively. First studied as a mathematical model to describe the chaotic motion of a test star in an axisymmetric galactic mean gravitational field (Hénon and Heiles 1964), this system is widely explored in other branches of physics. It is well known from applications in stellar dynamics, statistical mechanics and quantum mechanics. It provides a model for the oscillations of atoms in a three-atomic molecule (Berry 1978). Usually, the Hénon–Heiles system is not integrable and represents a classical example of chaotic behavior. Nevertheless, at some special values of the parameters, it is integrable; to be precise, there are three known integrable cases: i) ε = 6, A and B arbitrary. The second integral of motion is H2 = x41 +4x21 x22 −4y12 x2 +4y1 y2 y1 +4Ax21 x2 +(4A−B)y12 +A(4A−B)x21 . [7.29] ii) ε = 1, A = B. The second integral of motion is 1 H2 = y1 y2 + x31 + x1 x22 + Ax1 x2 . 3 iii) ε = 16, B = 16A. The second integral of motion is H2 = 3y14 + 6Ay12 x21 + 12y12 x21 x2 − 4y1 y2 x31 − 4Ax41 x2 − 4x41 x22 2 +3A2 x41 − x61 . 3 In the two cases (i) and (ii), the system [7.27] has been integrated by making use of genus one and genus two theta functions. For case (i), it was shown (Ankiewicz and Pask 1983) that this case separates in translated parabolic coordinates. Solving the problem in case (ii) is not difficult (this case trivially separates in Cartesian coordinates). In case (iii), the system can also be integrated (Ravoson et al. 1993) by making use of elliptic functions. The general solutions of the equations of motion for Hamiltonian [7.28], for cases (i) and (ii), have the Painlevé propriety, that is, they only admit poles in the complex time variable. This section deals with case (i) (case (iii) will be studied in section 8.3). The system [7.27] can be written in the form .

z = f (z) = J

∂H , ∂z

z = (x1 , x2 , y1 , y2 ) ,

[7.30]

O I . The second −I O . 2 flow commuting with the first is regulated by the equations: z = J ∂H ∂z ,  z = (x1 , x2 , y1 , y2 ) , with H2 defined by [7.29]. The invariant (or level) variety where H = H1 [7.28],

Mc =

2 6 i=1

∂H ∂z



=

∂H ∂H ∂H ∂H ∂x1 , ∂x2 , ∂y1 , ∂y2

{z : Hi (z) = ci } ⊂ C4 ,





,J =

[7.31]

Algebraically Integrable Systems

193

is a smooth affine surface for generic c = (c1 , c2 ) ∈ C2 . As before, the question is: how do we find the compactification of Mc into an Abelian surface? Now the system [7.30] is integrable in the sense of Liouville, but how does one effectively integrate the problem? We check the existence near each movable singularity of a Laurent series that represents the general solution. In fact, the Laurent decomposition of such asymptotic solutions have the following form z(t) ≡ (x1 (t), x2 (t), y1 (t), y2 (t)), where x1 =

∞ 

(k)

x1 tk−1 ,

x2 =

k=0

∞ 

(k)

x2 tk−2 ,

.

y1 = x 1 ,

.

y2 = x 2 ,

k=0

which depend on three free parameters. Putting these expressions into [7.30], solving inductively for the z (k) , one finds a nonlinear equation at the 0th step, z (0) + f (z (0) ) = 0,

[7.32]

and a linear system of equations at the kth step : % 0 for k = 1, (L − kI)z (k) = quadratic polynomial in z (1) , ..., z (k−1) for k > 1,

[7.33]

where L denotes the Jacobian map of [7.32]. One parameter appears at the 0th step, that is, in the resolution of [7.32] and the two remaining ones at the kth step, k = 3, k = 6. The resolution of these systems is a linear algebra problem, which is a straight forward computation. Using the majorant method (see theorem 7.1), we can show that the formal Laurent series solutions are convergent. Consequently, we have the following theorem: T HEOREM 7.19.– The non-identically zero solutions of [7.32] define a line. The system [7.33] has 1 degree of freedom for k = 3 and k = 6. The flow XH1 [7.27] possesses Laurent series solutions that depend on three free parameters α, β and γ. The first free parameter α appears in the resolution of [7.32] and the two remaining ones β, γ at the kth step, k = 3, k = 6 of [7.33]. These Laurent series solutions are explicitly given by (0)

x1 =

x1 (2) (3) (4) (5) (6) + x1 t + x1 t2 + x1 t3 + x1 t4 + x1 t5 + · · · t

x2 =

x2 (2) (4) (5) (6) + x2 + x2 t2 + x2 t3 + x2 t4 + · · · t2

(0)

.

y1 = x1 ,

[7.34]

.

y2 = x 2 (0)

(0)

(1)

with leading terms given explicitly by x1 = α = free parameter, x2 = −1, x1 = 3 (2) (3) (3) (1) (2) αB α2 B x2 = 0, x1 = α12 + αA 2 − 12 , x2 = 12 − 12 , x1 = β = free parameter, x2 = 0,

194

Integrable Systems

(4) αAB α5 11α3 B 11α3 A αB 2 αA2 α4 α2 A α2 B B2 24 − 72 + 720 − 120 − 720 − 8 , x2 = 48 + 10 − 60 − 240 , 2 (5) (5) (6) βB Aβ αβ αγ α7 α5 A α5 B α3 B 2 x1 = − βα 12 + 60 − 10 , x2 = 3 , x1 = − 9 − 15552 − 2160 + 12960 + 25920 + (6) α3 A2 α3 AB αAB 2 αB 3 αA2 B αA3 1440 − 4320 + 1440 − 19440 − 288 + 144 and x2 = γ = free parameter. (4)

x1 =

Consider points at infinity that are limit points of trajectories of the flow. We search for the set of Laurent solutions that remain confined to the fixed affine invariant surface Mc [7.31] related to specific values of c1 and c2 . T HEOREM 7.20.– The pole solutions [7.34] restricted to the surface Mc [7.31] is a smooth genus 3 hyperelliptic curve D[7.35], which is a double ramified cover of an elliptic curve E[7.37]. P ROOF.– By substituting [7.34] in the constants of the motion H1 = c1 and H2 = c2 , we eliminate the parameter γ linearly, leading to algebraic relation between the two remaining parameters, which is the equation of the divisor D along which the z(t) ≡ (x1 (t), x2 (t), y1 (t), y2 (t)) blow up. So, D is the closure of the continuous components of {Laurent series solutions z(t) such that Hi (z(t))$= ci , 1 ≤ i ≤ 2}, 32 # that is, D = t0 − coefficient of k=1 z ∈ C4 : Hk (z(t)) = ck . Thus, we find an algebraic curve defined by D = {(β, α) : β 2 = P8 (α)},

[7.35]

where

 13 7 1 8 P8 (α) = − 5A − B α6 α − 15, 552 432 18  671 1 17 943 − B 2 + A2 − BA α4 36 15, 120 7 1, 260  1 1 13 2 2 10 1 3 3 2 4A − − B − A B + AB − c1 α2 + c2 . 36 2, 520 6 9 7 36

The curve D determined by an eight-order equation is smooth, hyperelliptic and its genus is 3. Moreover, the map σ : D −→ D, (β, α) −→ (β, −α),

[7.36]

is an involution on D and the quotient E = D/σ is an elliptic curve defined by E = {(β, ζ) : β 2 = P4 (ζ)},

[7.37]

where P4 (ζ) is the degree 4 polynomial in ζ = α2 obtained from the polynomial P8 (α) above. The hyperelliptic curve D is thus a two-sheeted ramified covering of the elliptic curve E [7.37], ρ : D −→ E, (β, α) −→ (β, ζ), ramified at the four points covering ζ = 0 and ∞. 

[7.38]

Algebraically Integrable Systems

195

Let T be a smooth surface compactifying Mc [7.31]. Consider a basis 1, f1 , ..., fN of the vector space L(D) ≡ {f : f meromorphic on T, (f ) ≥ −D}, of meromorphic functions on T with at worst a simple pole along D and the map T → PN (C), p  [1, f1 (p), ..., fN (p)], considered projectively, because if at p some fi (p) = ∞, we divide by fi having the highest order pole near p, which makes every element finite. The Kodaira embedding theorem tells us that if the line bundle associated with the divisor is positive, then for k ∈ N, the functions of L(kD) embed smoothly T into PN (C) and 3 #then by Chow’s theorem,$T can be realized as an algebraic variety, that is, T = i Z ∈ PN (C) : Pi (Z) = 0 , where Pi (Z) are homogeneous polynomials. In fact in our case, k = 2 and N = 7 suffice, that is, the divisor 2D provides a smooth embedding into P7 (C), via the meromorphic section of L(2D). As in previous section, it is easy to find a set of polynomial functions {1, f1 , ..., fN } in L(2D) such that the embedding of 2D with those functions into PN yields a curve of genus N + 2. Straightforward calculation, using asymptotic expansions, shows that the space L(2D) is spanned by the following functions: L(2D) = {f0 , f1 , ..., f7 }, $ # = 1, x1 , x21 , x2 , y1 , y12 + x21 x2 , y2 x1 − 2y1 x2 , y1 y2 + 2Ax1 x2 + 2x1 x22 , & % 1 α α2 (α2 − 8A − B) α(α2 + 4A − B) 6β α α2 , , 2 = 1, , 2 , − 2 , − 2 , − t t t t t2 2t2 t +higher order terms in t. The application   (0) (0) 2D −→ P7 , p −→ lim t2 [f0 (p), f1 (p), ..., f7 (p)] = 0, 0, f2 (p), ..., f7 (p) , t→0

> ⊆ P7 and the genus of 2D is 9. maps the curve 2D into 2D T HEOREM 7.21.– The orbits of the vector field [7.7] running through 2D form a smooth surface Σ near 2D such that Σ\2D ⊆ T . Moreover, the variety T = Mc ∪ Σ, is smooth, compact, connected and it comes equipped with two everywhere independent commuting vector fields, which extend holomorphically on T . The system [7.7] is algebraic complete integrable and the corresponding flow evolves on an Abelian surface T C2 /Lattice. P ROOF.– Let ψ(t, p) = {z(t) = (x1 (t), x2 (t), y1 (t), y2 (t)) : t ∈ C, 0 < |t| < ε} be the orbit of the vector field [7.7] going through the point p ∈ 2D and consider the surface element Σp ⊂ P7 (C) formed by the divisor 2D and the orbits going through p. Consider the curve S = H ∩ Σ where H ⊂ P7 (C) is a hyperplane transversal to

196

Integrable Systems

 the direction of the flow and Σ ≡ p∈2D Σp . If S is smooth, then using the implicit function theorem, the surface Σ is smooth. But if S is singular at 0, then Σ would be singular along the trajectory (t−axis) which immediately goes into the affine part Mc . Hence, Mc would be singular, which is a contradiction (Mc is the fiber of a morphism from C4 to C2 and so smooth for almost all of the two constants of the motion ci ). Let M c be the projective closure of Mc into P4 (C), Z = [Z0 , Z1 = z1 Z0 , ..., Z4 = z4 Z0 ] ∈ P4 (C), and let M∞ = M c ∩ {Z0 = 0} be the locus at infinity. Consider the map M c ⊆ P4 −→ P7 (C), Z −→ f (Z), where f = (f0 , f1 , ..., f7 ) ∈ L(2D) and let T = f (M c ). In a neighborhood V (p) ⊆ P7 (C) of p, we have Σp = T and Σp \2D ⊆ Mc . Otherwise, there would exist an element of surface Σp ⊆ T such that Σp ∩ Σp = t − axis, orbit ψ(t, p) = t − axis\p ⊆ Mc , and hence Mc would be singular along the t−axis, which is impossible. Since the variety M c ∩ {Z0 = 0} is irreducible and the generic hyperplane section Hgen. of M c is also irreducible, all hyperplane sections are connected and hence M∞ is also connected. Consider the graph Γf ⊆ P4 × P7 (C) of the map f , which is irreducible together with M c . It follows from the irreducibility of M∞ that a generic hyperplane section Γf ∩ {Hgen. × P7 (C)} is irreducible, hence the special hyperplane section Γf ∩ {{Z0 = 0} × P7 (C)} is connected and therefore the projection map projP7 [Γf ∩ {{Z0 = 0} × P7 (C)] = f (M∞ ) ≡ 2D is connected. Hence, the variety Mc ∪ Σ = T is compact, connected and embeds smoothly into P7 (C) via f . Let g t1 and g t2 be the flows generated, respectively, by vector fields XH1 and XH2 . For p ∈ 2D and for small ε > 0, g t1 (p), ∀t1 , 0 < |t1 | < ε, is well defined and g t1 (p) ∈ Mc . We define g t2 on Mc by g t2 (q) = g −t1 g t2 g t1 (q), q ∈ U (p) = g −t1 (U (g t1 (p))), where U (p) is a neighborhood of p. By commutativity one can see that g t2 is independent of t1 ; g −t1 −ε1 g t2 g t1 +ε1 (q) = g −t1 g −ε1 g t2 g t1 g ε1 = g −t1 g t2 g t1 (q). We affirm that g t2 (q) is holomorphic away from 2D. This is because g t2 g t1 (q) is holomorphic away from 2D and that g t1 is holomorphic in U (p) and maps bi-holomorphically U (p) onto U (g t1 (p)). Since the flows g t1 and g t2 are holomorphic and independent on 2D, we can show along the same lines as in theorem 7.10 that T is a torus by considering the holomorphic map C2 −→ T , (t1 , t2 ) −→ g t1 g t2 (p), for a fixed origin p ∈ Mc . The additive subgroup Lattice = {(t1 , t2 ) ∈ C2 : g t1 g t2 (p) = p} is a lattice of C2 and hence C2 /lattice −→ T is a biholomorphic diffeomorphism. Therefore, T ⊆ P7 is conformal to a complex torus C2 /lattice and an Abelian surface as a consequence of the Chow theorem.  R EMARK 7.5.– Recall that a Kähler variety is a variety with a Kähler metric, that is, a Hermitian metric whose associated differential 2-form

of type (1, 1) is closed. The complex torus C2 /lattice with the Euclidean metric dzi ⊗ dz i is a Kähler variety and any compact complex variety that can be embedded in projective space is also a Kähler variety. A compact complex Kähler variety having as many independent meromorphic functions as its dimension is a projective variety. We have shown that T is a complex torus C2 /lattice, and so, in particular, T is a Kähler variety with Kähler metric given by dt1 ⊗ dt1 + dt2 ⊗ dt2 . As mentioned above, a compact complex Kähler variety having the required number as (its dimension) of

Algebraically Integrable Systems

197

independent meromorphic functions is a projective variety. Thus, T is both a projective variety and a complex torus C2 /Lattice and hence an Abelian surface as a consequence of the Chow theorem. 7 T HEOREM The flow  7.22.–  [7.27] evolves on Abelian surface T ⊆ P (C) of period 2 0a c ac matrix , Im > 0, (a, b, c ∈ C), and is expressed in terms of 0 4c b c b Abelian integrals (involving the differentials [7.40]).

P ROOF.– Note that the affine invariant surface Mc [7.31] has the following involution σ : (x1 , x2 , y1 , y2 ) −→ (x1 , x2 , −y1 , −y2 ), which maps XHj into −XHj , j = 1, 2, where XH1 is the flow [7.27] and XH2 the other flow commuting with the first, thus σ amounts to a reflection about some appropriately chosen origin on T . This map acts on the parameters of the Laurent solution [7.34] as follows: (t, β, α) −→ (−t, β, −α). Since L is symmetric (σ ∗ L L), σ can be lifted to L as an involution σ 2 in two ways, differing in sign and for each section (theta-function) s ∈ H 0 (L), we therefore have σ 2s = 2s = ±s. Recall that a section s ∈ H 0 (L) is called even (respectively, odd) if σ +s (respectively, σ 2s = −s). Under σ 2, the vector space H 0 (L) splits into an even and odd subspace H 0 (L) = H 0 (L)even ⊕ H 0 (L)odd , with H 0 (L)even containing all of the even sections and H 0 (L)odd all of the odd ones. Using the inverse formula (Mumford 1967a, p. 331), we see after a small computation that h0 (L)

δ1 δ2 + 2−1+ even δk , k = 1, 2 2 δ1 δ2 = − 2−1+ even δk , k = 1, 2 2

even

≡ dim H 0 (L)

even

odd

≡ dim H 0 (L)

odd

h0 (L)

=

[7.39]

By the classification theory of ample line bundles on Abelian varieties and theorem7.21, T C2 /LΩwith period lattice given by the columns of the following   a c δ1 0 a c , Im > 0, (a, b, c ∈ C), with δ1 δ2 = dim H 0 L⊗2 = matrix 0 δ2 c b c b dim L(2D) = g(2D) − 1 = 8. We have two possibilities: (i) δ1 = 1, δ2 = 8 and (ii) δ1 = 2, δ2 = 4. From [7.39], the corresponding line bundle L⊗2 has five even sections and three odd sections in case (i) and six even sections and two odd sections   even odd and L⊗2 of in case (ii). The space L⊗2 splits into two subspaces L⊗2 even and odd functions for the σ-involution  even  ⊗2 odd L⊗2 = L⊗2 ⊕ L = {f0 , f1 , f2 , f3 , f5 , f7 } ⊕ {f4 , f6 }, = {1, x1 , x21 , x2 , y12 + x21 x2 , y1 y2 + 2Ax1 x2 + 2x1 x22 } ⊕ {y1 , y2 x1 − 2y1 x2 }. Among the functions of L⊗2 , there are six even and two odd functions for the involution that  σ, showing  case (ii) is the only alternative and the period matrix has the 20 ac ac form , Im > 0, (a, b, c ∈ C). On T , let the holomorphic 1-forms 02 c b c b

198

Integrable Systems

  2 dt1 and dt2 defined by dti XHj = δij . The 1-forms ω0 = αdα , ω1 = α βdα , ω2 = β   dα 0 D, Ω1D . Using the Laurent solutions, the differentials dt1 β , yield a basis of H and dt2 corresponding to the flows generated, respectively, by H1 and H2 , restricted to the curve D[7.35], descend to two differentials on D: dt1 |D =

α2 dα = ω1 , β

dt2 |D =

dα = ω2 . β

[7.40]

This concludes the proof.  R EMARK 7.6.– The divisor D defines on the Abelian surface T a polarization (1, 2). The eight-dimensional space of sections of the line bundle L⊗2 splits into two   even odd and L⊗2 of even and odd sections (theta functions). In subspaces L⊗2  ⊗2 even  ⊗2 odd ⊗2 = L ⊕ L , f7 }⊕ other words, we have L 0 , f1 , f2 , f3 , f5  = {f  ⊗2 odd  ⊗2 even {f4 , f6 }. The remarkable property is that W L , L ⊂  ⊗2 even ⊗2 , where W (, ) is the Wronskian W (fi , fj ) ≡ fi X (fj ) − fj X (fi ) L between two theta functions, with respect to an arbitrary holomorphic vector field X on T . The Abelian surface T , as embedded in P7 (C) can be described by six quadratic relations between the theta functions, three of which only involve even sections and another three involving even and odd sections. From the divisor D and the line bundle L, a lot of information can be obtained. For example, Adler and van Moerbeke (1988) have obtained a one-dimensional family of birationally maps between the Hénon-Heiles system (case i), Kowalewski’s top and the geodesic flow on SO(4) for the Manakov metric. Such birationally maps are given by identifying the three eight-dimensional spaces L of Hénon–Heiles , Kowalewski and Manakov. T HEOREM 7.23.– The Abelian surface T which completes the affine surface Mc is the dual Prym variety P rym∗ (D/E) of the genus 3 hyperelliptic curve D [7.35] for the involution σ [7.36] interchanging the sheets of the double covering ρ [7.38] and the problem linearizes on this variety. P ROOF.– Simply follow the same reasoning that was provided in theorem 7.18, while taking account of the notations used here.  We will now show that T can also be seen as a double unramified cover of the Jacobian variety Jac(C) of the 2-genus hyperelliptic curve C [7.41]. The remarkable property of this curve is that the flow of solutions of the equations of motion is linearized on its Jacobian Jac(C) and so, the solutions can be expressed in terms of theta-functions of two variables. We also discuss the Kummer surface associated with T ; it will be constructed explicitly. T HEOREM 7.24.– The torus T can be regarded as a double unramified cover of the Jacobian variety Jac(C) of the 2-genus hyperelliptic curve C [7.41] and the system

Algebraically Integrable Systems

199

[7.27] can be integrated in terms of genus 2 hyperelliptic functions of time. The involution σ [7.36] on the Jacobian Jac(C) leads to a singular surface KC , which >C . after desingularization, defines a K3 surface K P ROOF.– Remember that the Laurent solutions [7.34] restricted to the surface Mc [7.31] are parameterized by an hyperelliptic curve D [7.35] of genus 3. The latter is a double ramified cover of an elliptic curve E [7.37] and can also be seen as a two-sheeted unramified cover π : D −→ C, (β, α) −→ (η, ζ) of the following hyperelliptic curve C of genus 2: C : η 2 = ζP4 (ζ).

[7.41]

We show, using the results obtained above and applying the method explained in Vanhaecke (2001), that the linearized flow can be realized on the Jacobian variety Jac(C) of the 2-genus hyperelliptic curve C. The torus T can be regarded as a double unramified cover of the Jacobian variety Jac(C) and the system [7.27] can be integrated in terms of genus 2 hyperelliptic functions of time. The differentials dt1 and dt2 , corresponding to the flows XH1 and XH2 , restricted to the curve D, 2 go down to C. Indeed, using [7.40], dt1 |D = ω1 = α βdα = ζdζ η , dζ dt2 |D = ω2 = dα β = η , yielding the two hyperelliptic differentials on C. The map ρ [7.38] extends to a map ρ2 : T → Jac(C). We denote by |C| the linear system, that is, the set of all effective divisors linearly equivalent to C. We have |C| = P(L(D)); associating to each non-zero function f ∈ L(C). An Abelian variety T Cn /lattice has a natural involution σ, induced by the sign flip (z1 , . . . , zn ) −→ (−z1 , . . . , −zn ), in Cn . Its fixed points are exactly the 22n half-periods of T . The quotient T /σ is called the Kummer surface. In our case, n = 2 and the involution σ [7.36] on the Jacobian Jac(C) leads to a singular surface KC , which after desingularization, at the : 16 fixed points of the involution σ, defines a K3 surface K C . It has no holomorphic 1-forms and it has a trivial canonical divisor. The Kummer surface KC of Jac(C) can be considered as the image of ψρ∗ |2C| : T → P3 (C), with ρ2∗ |2C| ⊂ |2D|. Since C is hyperelliptic of genus 2, it has six distinct Weierstrass points (indeed, let Γ a smooth hyperelliptic curve of genus g ≥ 2 with φ : Γ → P1 (C) the two-sheeted map, then all of the branch points of φ are the only Weierstrass points of Γ. By the Riemann–Hurwitz formula, the number of these points is equal to 2g + 2). Choose a Weierstrass point P on the curve C and coordinates [Z0 , Z1 , Z2 , Z3 ] for P3 such that ψϕ∗ |2C| (P ) = [0, 0, 0, 1]. Then this point will be a singular point for the Kummer surface K of equation a (Z0 , Z1 , Z2 ) Z32 + b (Z0 , Z1 , Z2 ) Z3 + c (Z0 , Z1 , Z2 ) = 0, where a, b and c are polynomials of degree, 2, 3 and 4, respectively. After a projective transformation that fixes [0, 0, 0, 1] we may assume that a (Z0 , Z1 , Z2 ) = Z12 − 4Z0 Z2 . We can construct an algebraic map from Mc to the Jacobi variety Jac (C): Mc −→ Jac(C), p ∈ Mc −→ (ζ1 + ζ2 ) ∈ Jac(C), and the

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Integrable Systems

flows generated by the constants of the motion are straight lines on Jac(Γ), that is, the linearizing equations are given by +

+

ζ1 (t)

ζ1 (0)

+

ζ2 (t)

ω1 + ζ2 (0)

+

ζ1 (t)

ω1 = c1 t,

ζ2 (t)

ω2 + ζ1 (0)

ω2 = c2 t, ζ2 (0)

where ω1 , ω2 [7.40] span the 2-dimensional space of holomorphic differentials on the curve C and ζ1 , ζ2 , two appropriate variables given by " " −Z1 + a (Z0 , Z1 , Z2 ) −Z1 − a (Z0 , Z1 , Z2 ) ζ1 = , ζ2 = , 2Z0 2Z0 algebraically related to the ones originally given, for which the Hamilton–Jacobi equation could be solved by the separation of variables.  7.7. The Manakov geodesic flow on the group SO(4) Let x ∈ C6 , t ∈ C and U ⊂ C6 a non-empty Zariski open set. The map Ψ: (H1 , ..., H4 ) :C6 −→ C4 (see section 4.5) is submersive on U, that is, dH are linearly independent on U. $ Let  1 (x),..., dH # 4 (x) I = Ψ C6 \U = c = (ci ) ∈ C4 : ∃x ∈ Ψ−1 (c), dH1 (x) ∧ ... ∧ dH4 (x) = 0 , be 4 the set of critical values of Ψ and I the Zariski closure # of I in C . The non-empty $ Zariski open set U can be chosen as the set U = x ∈ C6 : Ψ(x) ∈ C4 \I . The $ 34 # invariant variety defined by Mc = Ψ−1 (c) = i=1 x ∈ C6 : Hi (x) = ci , is the fiber of a morphism from C6 to C4 , thus Mc is a smooth affine surface for generic c = (c1 , ..., c4 ) ∈ C4 and the main problem will be to complete Mc into an Abelian surface. Now, how do we find the compactification of Mc into an Abelian surface? This compactification is not $ trivial and the simplest one obtained as a closure: 34 # M c = i=1 Hi (x) = ci x20 ⊂ P6 (C), that is, x1 x4 + x2 x5 + x3 x6 = c1 x20 ,

λ1 x21 + λ2 x22 + · · · + λ6 x26 = c3 x20 ,

x21 + x22 + · · · + x26 = c2 x20 ,

μ1 x21 + μ2 x22 + · · · + μ6 x26 = c4 x20 ,

where [x0 : x1 : ... : x6 ] are homogeneous coordinates on P6 (C), does not lead to this result (in the following we will not distinguish between x1 as a homogeneous coordinates [x0 : x1 ] and as an affine coordinate x1 /x0 ). An Abelian surface is not simply connected and therefore cannot be a projective complete intersection. In other Mc must have a singularity words, if Mc is to be the affine part of an Abelian surface, 3 somewhere along the locus at infinity C = M c {x0 = 0}. A direct calculation shows that C is an ordinary double curve of M c except at 16 ordinary pinch points of M c ; the variety M c has a local analytic equation x2 = yz 2 . The reduced curve Cr is a smooth elliptic curve. Now, it is only after blowing up M c along the curve Cr that one gets the desired Abelian surface.

Algebraically Integrable Systems

201

T HEOREM 7.25.– The divisor of poles of the functions x1 , x2 , ..., x6 is a Riemann surface D of genus 9. For generic constants, the surface Mc is the affine part of an >c obtained by gluing to Mc the divisor D. Abelian surface M P ROOF.– Consider points at infinity which are limit points of trajectories of the flow. There is a Laurent decomposition of such asymptotic solutions,  X(t) = t−1 X (0) + X (1) t + X (2) t2 + · · · , [7.42] which depend on dim(phase space) − 1 = 5 free parameters. Putting [7.42] into [4.8], solving inductively for the X (k) , one finds a nonlinear equation at the 0th step,   X (0) + X (0) , Λ.X (0) = 0, [7.43] and a linear system of equations at the kth step : 

(L − kI) X

(k)

%

=

quadratic polynomial in X

(1)

, ..., X

0 for k = 1 for k ≥ 2

(k−1)

[7.44] where L denotes the linear map     L(Y ) = Y, Λ.X (0) + X (0) , Λ.Y + Y = Jacobian map of [7.43]. One parameter appears at the 0th step, that is, in the resolution of [7.43] and the four remaining ones at the kth step, k = 1, ..., 4. By only taking into account solution trajectories lying on the surface Mc , we obtain one-parameter families that are parameterized by a Riemann surface. To be precise, we search for the set D of Laurent solutions [7.42] restricted to the affine invariant surface Mc , that is D = closure of the continuous components of {Laurent solutions X(t) such that Hi (X(t)) = ci , 1 ≤ i ≤ 4} , =

4 6 #

$ t0 − coefficient of Hi (X(t)) = ci ,

i=1

= a Riemann surface (algebraic curve) whose affine equation is  2  2  2 (0) (0) (0) (0) (0) (0) (0) (0) (0) w 2 + c1 x 5 x 6 + c 2 x4 x6 + c 3 x4 x5 + c4 x 4 x 5 x 6  (0) (0) (0) ≡ w 2 + F x4 , x5 , x6 = 0, [7.45]

202

Integrable Systems (0)

(0)

(0)

where w is an arbitrary parameter and x4 , x5 , x6 parameterizes the elliptic curve ⎧  2  2  2 (0) (0) (0) ⎨ x4 + x5 + x6 = 0,  E :  (0) [7.46] ⎩ βx + αx(0) βx(0) − αx(0) = 1, 5

6

5

6

with (α, β) such that: α2 + β 2 + 1 = 0. The Riemann surface D is a two-sheeted ramified covering of the elliptic curve E and it easy to check that the elliptic curve E is  exactly the reduced curve Cr . The branch points are defined by the 16 zeroes of (0) (0) (0) F x4 , x 5 , x 6 on E. The Riemann surface D is unramified at infinity and by Riemann–Hurwitz’s formula, 2g (D) − 2 = N (2g (E) − 2) + R, where N is the number of sheets and R the ramification index, the genus g (D) of D is 9. To show >c with M >c \ Mc = D, we can use that Mc is the affine part of an Abelian surface M the same method of Laurent’s developments used previously (see Haine (1983)). Here, by following Mumford (see appendix of Adler and van Moerbeke (1982)), we give an abstract algebro-geometrical proof that the four quadrics in this problem intersect in the affine part of an Abelian surface using Enriques classification of >c and use Enriques algebraic surfaces. For this, we will compute the invariants of M classification of algebraic surfaces (Griffiths and Harris 1978, p. 590). Let KM c be >c ) the irregularity of the canonical bundle, χ(O  ) the Euler characteristic and q(M Mc

>c . Now if φ : M >c −→ M c ⊂ P6 (C) is the normalization of M c , then the pullback M    >c , O  , is an isomorphism and map on sections φ∗ : H 0 M c , OM c −→ H 0 M Mc   ∗   KM = K − D, where K = φ K and so for H a hyperplane in P6 (C), c Mc Mc Mc 

4 KM = φ∗ M c .KP6 (C) + ( i=1 deg Hi ).H − D = 0. Also c       χ OM = χ φ∗ OM = χ (φ∗ OD /OE ) + χ OM c . The c c /OM c + χ OM c Riemann surface D[7.45] of genus 9 is a double cover ramified over  16points of the elliptic curve E[7.46]. We use the Koszul complex to compute χ OM c . In the local ring at each point of P6 (C) the localizations of the 4 homogeneous polynomials Hi give a regular sequence, and the Koszul complex gives a canonical resolution 0 → OP6 (C) (−8) → OP6 (C) (−6)4 → OP6 (C) (−4)6 → OP6 (C) (−2)4 → OP6 (C) → OM c → 0

    >c = 2. By Enriques–Kodaira’s = 0, q M Thus χ OM c = 8, hence χ OM c >c is an Abelian classification theorem (Griffiths and Harris 1978), it follows that M surface.  >c ∼ T HEOREM 7.26.– The flow on an Abelian surface M = C2 /lattice of  [4.8]  evolves 20 ac a c polarization , Im > 0. 04 c b c b

Algebraically Integrable Systems

203

0 1 >c , (f ) + D ≥ 0 be the vector space of P ROOF.– Let L ≡ f : f meromorphic on M >c with at worst a simple pole along D, and let meromorphic functions on M   0 > >c , O(D) be the Euler characteristic of χ(D) = dim H Mc , O(D) − dim H 1 M D. The adjunction formula and the Riemann–Roch theorem for divisors  on Abelian KM .D+D.D  c >c + 1 + surfaces imply that g(D) = + 1, and χ(D) = pa M 2   1 > c ) , where g(D) is the geometric genus of D and pa Mc is the 2 D.(D − KM > >c arithmetic genus Since M is an Abelian surface   of Mc . D.D > KM = −1 , g(D) − 1 = M = 0, p = χ(D). Using Kodaira–Serre a c c 2 duality (Griffiths and Harris 1978, p. 153), the Kodaira–Nakano vanishing theorem (Griffiths and Harris 1978, p. 154) and a theorem on theta-functions (Griffiths and Harris 1978, p. 317), it easy to see that   g(D) − 1 = dim L(D) ≡ h0 (L) = δ1 δ2 , [7.47] >c . Note where δ1 , δ2 ∈ N, are the elementary divisors of the polarization c1 (L) of M that σ ≡ −id : (x0 , x1 , . . . , x6 ) −→ (−x0 , x1 , . . . , x6 ), is the reflection about the >c , given by the 16 branch points on D origin of C2 and has 16 fixed points on M covering the 16 roots of the polynomial F (x04 , x05 , x06 ) [7.45]. Since L is symmetric (σ ∗ L L), σ can be lifted to L as an involution σ 2 in two ways differing in sign and for each section (theta-function) s ∈ H 0 (L), we therefore have σ 2s = ±s. A section s ∈ H 0 (L) is called even (respectively, odd) if σ 2s = +s (respectively, σ 2s = −s). Under σ 2 the vector space H 0 (L) splits into an even and odd subspace H 0 (L) = H 0 (L)even ⊕ H 0 (L)odd , with H 0 (L)even containing all of the even sections and H 0 (L)odd all of the odd ones. Note that c1 (L) = φ∗ (H) and (c1 (L)2 ) = 16 (since the degree of M c >c is 16). By the classification theory of ample line bundles on Abelian varieties, M  δ1 0 a c C2 /LΩ with period lattice given by the columns of the matrix Ω = , 0 δ2 c b  a c Im > 0, according to [7.47], with δ1 δ2 = h0 (L) = g(D) − 1 = 8, δ1 | δ2 , c b δi ∈ N∗ . Hence, we have two possibilities: (i) δ1 = 1, δ2 = 8 and (ii) δ1 = 2, δ2 = 4. From formula [7.39], the corresponding line bundle L has five even sections and three odd sections in case (i), and six even sections and two odd sections in case (ii). Now x1 , . . . , x6 are six even sections, showing  that  case (ii) is the only alternative and the 20 ac a c period matrix has the form , Im > 0.  04 c b c b >c that completes the affine surface Mc is the T HEOREM 7.27.– The Abelian surface M Prym variety P rymα (Γ) of the genus 3 Riemann surface Γ:

204

Integrable Systems

% Γ:

 0 2  2  0 0 2 w2 = −c z − c3 x05 z + c4 y, 1 x5 x6 − c2 x6  y 2 = z α2 z − 1 (β 2 z + 1),

[7.48]

for the involution σ : Γ −→ Γ, (w, y, z) −→ (−w, y, z), interchanging the two sheets of the double covering Γ −→ Γ0 , (w, y, z) −→ (y, z), Γ0 is the elliptic curve,   [7.49] Γ0 : y 2 = z α2 z − 1 (β 2 z + 1).  2 P ROOF.– After substitution z ≡ x04 , the curve D can also be seen as a four-sheeted unramified covering of another curve Γ, determined by the equation   2  2  2 2  2 Γ : G(w, z) ≡ w2 + c1 x05 x06 + c2 x06 z + c3 x05 z − c24 x05 x06 z = 0.  2  2 Equations [7.46] are equivalent to x05 = β 2 z + 1 and x06 = α2 z − 1. The curve Γ is invariant under an involution σ : Γ −→ Γ,

(w, z) −→ (−w, z).

[7.50]

Consider a map ρ : Γ −→ Γ0 ≡ Γ/σ, (w, y, z) −→ (y, z), of the curve Γ onto an elliptic curve Γ0 ≡ Γ/σ, that is given by the equation [7.49]. The genus of Γ [7.48] is calculated easily by means of the map ρ. The latter is a two-sheeted ramified covering of Γ0 and it has four branch points. Using the Riemann–Hurwitz formula, >c we obtain g(Γ) = 3. We will now proceed to show that the Abelian surface M can be identified as Prym variety P rymσ (Γ). Let (a1 , a2 , a3 , b1 , b2 , b3 ) be a basis of cycles in Γ with the intersection indices ai oaj = bi obj = 0, ai obj = δij , such that σ (a1 ) = a3 , σ (b1 ) = b3 , σ (a2 ) = −a2 , σ (b2 ) = −b2 for the involution σ [7.50]. By the Poincaré residue formula, the three holomorphic 1-forms ω0 , ω1 , ω2 in Γ are   dz dz the differentials P (w, z) (∂G/∂w)(w,z)  = P (w, z) 4wy , for P a polynomial G(w,z)=0

zdz dz of degree ≤ deg G − 3 = 1. Therefore, ω0 = dz y , ω1 = wy , ω2 = wy form a basis of holomorphic differentials on Γ and obviously σ ∗ (ω0 ) = ω0 , σ ∗ (ωk ) = −ωk , (k = 1, 2), for the involution σ [7.50]. It is well known that the period matrix Ω of P rymσ (Γ) can be written as follows:      2 a1 ω1 a2 ω1 2 b1 ω1 b2 ω1 . Ω= 2 a1 ω2 a2 ω2 2 b1 ω2 b2 ω2

>c , such Let (dt1 , dt2 ) be a basis of holomorphic M 0 1-forms onthe Abelian surface dt1  2 dt1 that dtj |D = ωj , (j = 1, 2), LΩ = : mk , k=1 mk ak dt2 + nk bk dt2 1 nk ∈ Z , is the lattice associated with the period matrix 

    dt1 a dt1 b dt1 b dt1 a1 2 2 1     , Ω = dt2 a dt2 b dt2 b dt2 a 

1

2

1

2

Algebraically Integrable Systems

205

   >c , Z) and A2 −→ C2 /LΩ : p −→ p dt1 , where (a1 , a2 , b1 , b2 ) is a basis of H1 (M p0 dt2 is the uniformizing map. By the Lefschetz theorem on hyperplane section (Griffiths >c , Z) induced by the inclusion and Harris 1978, p. 156), the map H1 (D, Z) −→ H1 (M > D → Mc is surjective and consequently we can find four cycles a1 , a2 , b1 , b2 on the Riemann surface D such that 

     ω1  ω1  ω1  ω1 a a b b   2 3 4 Ω =  1 , ω ω ω ω a 2 a 2 b  2 b  2 1

and LΩ

=

2

0

2 k=1

3

mk

4

 ak

ω1  ω2

+ nk

 bk

ω1  ω2

1 : mk , nk ∈ Z . Recall that

F (x04 , x05 , x06 )[7.45] has four zeroes on Γ0 [7.49] and 16 zeroes on E[7.46], and it follows that the four cycles a1 , a2 , b1 , b2 on D that we are looking for are >c , Z), such that 2a1 , a2 , 2b1 , b2 and they form a basis of H1 (M      2 a1 ω1 a2 ω1 2 b1 ω1 b2 ω1 Ω = =Ω 2 a1 ω 2 a 2 ω2 2 b 1 ω 2 b 2 ω2 

>c and P rymσ (Γ) are two Abelian varieties analytically is a Riemann matrix. Thus, M >c and isomorphic to the same complex torus C2 /LΩ . By Chow’s theorem, M P rymσ (Γ) are then algebraically isomorphic.  R EMARK 7.7.– Strange as it may seem, the use of the Lax spectral curve technique may not give the tori correctly, but perhaps with period doubling, in contrast with the statement that the correct tori would be obtained by the Kowalewski–Painlevé analysis. This indicated a need for caution in the interpretation of the result for tori calculated from the Lax spectral curve technique. A striking example of this phenomenon appears in the Euler equations associated with a class of geodesic flow on SO(4) for a left-invariant diagonal metric. We know from section 4.5 that the linearization of the Euler–Arnold equations [4.8] takes place on the Prym variety P rymσ (C) of the genus 3 Riemann surface C [4.16]; the latter is a double ramified cover of an elliptic curve C0 . Also, from the asymptotic analysis (section 7.7) of >c upon equations [4.8], the affine variety Mc completes into an Abelian surface M adding a Riemann surface D [7.45] of genus 9, which is a fourfold unramified cover of a Riemann surface Γ [7.48] of genus 3; the latter is a double ramified cover of an >c can also be identified as the Prym variety elliptic curve Γ0 . The Abelian surface M P rymσ (Γ) and the problem linearizes on P rymσ (Γ). From the fundamental exp . ∗ → OM exponential sequence 0 → Z → OM  c → 0, we get the map   c 1 > ∗ 2 > · · · → H Mc , OM → H Mc , Z → · · · , that is, the first Chern class of a line  c

>c . Any line bundle with Chern class zero can be realized by constant bundle on M  >c of holomorphic line bundles on M >c multipliers. Therefore, the group P ico M

206

Integrable Systems

   >c = H 1 M >c , O  /H 1 M >c , Z and with Chern class zero is given by P ico M Mc ∗

>c (∗ means the dual >c of M is naturally isomorphic to the dual Abelian surface M ∗ >c is symmetric like the >c and M Abelian surface). The relationship between M relationship between two vectors spaces set up a bilinear pairing. It is interesting to >c = P rymσ (Γ) obtained from the asymptotic observe that the Abelian surfaces M analysis of the differential equations and P rymσ (C) obtained from the orbits in the Kac–Moody Lie algebra are not identical but only isogenous, that is, one can be obtained from the other by doubling some periods and leaving other unchanged. The >c = (P rymσ (C))∗ , that is, precise relation between these two Abelian surfaces is M they are dual of each other. The functions x1 , ..., x6 are themselves meromorphic on >c , while only their squares are on P rymσ (C). The relationship between the M Riemann surfaces Γ and C is quite intricate. As usual we let Θ the theta divisor on Jac(Γ), we have P rymσ (C)\Π = Θ ∩ P rymσ (C) = Γ, with Π a Zariski open set of >c = C, where Θ is a translate of the theta divisor of Jac(C) P rymσ (C). Also Θ ∩ M invariant under the involution σ. Moser (1980) was aware of a similar situation in the context of the Jacobi’s geodesic flow problem on ellipsoids. 7.8. Geodesic flow on SO(4) with a quartic invariant Often, when studying the geodesic flow on SO(4), it is more convenient to use the coordinates u = (x1 , x2 , x3 ) and v = (x4 , x5 , x6 ), they correspond to the decomposition u ⊕ v ∈ so(4) so(3) ⊕ so(3). In these coordinates, the geodesic flow on the group SO(4) can be written as XH : u˙ = u ×

∂H , ∂u

v˙ = v ×

∂H , ∂v

for invariant metric defined by the quadratic form  1  3(3ci + di )x2i + (ci + 3di )x2i+3 + 6(di − ci )xi xi+3 , 24 i=1 3

H=

[7.51]

3

3 b −b with coefficients ci = abii , di = ajj −akk , i=1 ai = 0, i=1 bi = 0, and ijk permutations of 123. This geodesic flow has three quadratic invariants, namely, the Casimir functions "u"2 and "v"2 , and the metric [7.51], and one quartic invariant will be given later. The invariants "u"2 and "v"2 define the four-dimensional non-degenerate symplectic leaves of Hamiltonian structure, which are therefore parameterized by the values of "u"2 and "v"2 . After the following linear change of coordinates (motivated by the equations of a curve of rank three quadrics, see Adler and van Moerbeke (1987)), which is meaningful insofar a = 0, −1, 1, −1/3, 1/3,

Algebraically Integrable Systems



 a − 1 −1 (a − 1)z1 , (3a − 1)(a + 1)z4 3a + 1 1    √ a + 1 −1 (a + 1)z2 x2 = − −1 , x5 (3a + 1)(a − 1)z5 3a − 1 1    √ a−1 a+1 (a − 1)z3 x3 = −1 , x6 (a + 1)z6 3a + 1 3a − 1 x1 x4



207

=

√



−1

the geodesic flow takes (after rescaling time) on the simple form: dz1 dt dz2 dt dz3 dt dz6 dt

dz4 2a = z5 z6 + dt 3a − 1 dz5 2a = z4 z6 , = z5 z6 + dt 3a + 1 1+a 1−a = z4 z5 + z1 z5 + z1 z2 , 2 2 1−a 1+a = z4 z5 + z2 z4 + z1 z2 , 2 2 = z3 z5 ,

a−1 z2 z3 , 3a − 1 a+1 z2 z3 , 3a + 1

[7.52]

with three quadratic invariants (in z): 1−a 2 (z − z32 + z1 z4 ) = A1 , 3a + 1 1 a+1 Q2 ≡ −a(z42 − z2 z5 ) − = A2 , 3a − 1

Q1 ≡ a(z52 − z1 z4 ) +

Q3 ≡

2(z1 z4 + z2 z5 − z3 z6 ) z42 − z2 z5 z 2 − z1 z4 − + 5 = A3 , (3a − 1)(3a + 1) 3a + 1 3a − 1

and a quartics invariant (in z):  1−a 2 2 2 2 (z4 − z2 z5 ) + ( Q4 ≡ − (z2 z3 − z5 z6 )) 3a + 1 3a − 1  −2 1+a 2 2 2 (z5 − z1 z4 ) + ( (z1 z6 − z3 z4 )) + 3a − 1 3a + 1 +

  2 3(1 − a2 ) 2(z4 − z2 z5 )(z52 − z1 z4 ) − (z1 z2 − z4 z5 )2 (3a − 1)(3a + 1)

+

4(1 + a) (z 2 − z1 z4 )(z1 z4 + z2 z5 − z3 z6 + z22 − z62 + z2 z5 ) (3a − 1)(3a + 1) 5

+

4(1 − a) (z 2 − z2 z5 )(z1 z4 + z2 z5 − z3 z6 + z12 − z32 + z1 z4 ) = A4 . (3a − 1)(3a + 1) 4

[7.53]

208

Integrable Systems

The geodesic flow in question admits one family of Laurent solutions, ζ z= t

1 + UY t +

1

1

γZ 2 + δ

U

2

Y02

+

3 

 Ai Yi2

 2

3

t + o(t ) ,

i=1

where 1 = (1, 1, ..., 1) , Y 1 , Y02 , Yi2 are appropriatevectors depending on Y, Z and a only, γ ≡ 4a, δ ≡ (a − 1)(3a + 1) and ζ = diag

Y 2 Z2 Y Z Z , Y , − Z , Z, Y, − Y

, with

Y, Z ∈ C such that Y + Z = 1. The five-dimensional family of Laurent solutions depend on the parameters Z, U, A1 , A2 and A3 . The vectors Yi2 can be chosen such that Qi (z(t)) = Ai for i = 1, 2, 3. Confining the five-dimensional family of Laurent solutions to the invariant manifold A = {z : Qi (z) = Ai } yields a relation between the free parameters, defining a curve 2

2

⎧ 2 2 ⎨ (U, V, Y, Z)such hat Z = V, Y = 1 − V and 2 D: P (U, V ) = U 2 (1 − V )V (αV + β) ⎩ −2U 2 (1 − V )V P (V ) + Q(V ) = 0, where α = 16a3 , β = (a − 1)3 (3a + 1), 4  P (V ) = (αV + β) ((3a2 + 1)A3 − A1 − A2 )(V − 1) + A1 V − A2 (V − 1)  −2V (V − 1) A1 (1 − a)3 (1 + 3a) + A2 (1 + a3 (1 − 3a) 4 −A3 (1 − a2 )(1 − 9a2 ) ,  42 Q(V ) = ((3a2 + 1)A3 − A1 − A2 )V (V − 1) − A1 V + A2 (V − 1) +V (V − 1) [(4aV + (a − 1)(3a + 1))A4 + 4A1 A2 4 −(a − 1)(3a + 1)(a + 1)(3a − 1)A23 . Note that P 2 (V ) − (αV + β)2 Q(V ) = V (1 − V )R(V ),

[7.54]

with R(V ) being a cubic polynomial. The curve D is an unramified 4 − 1 cover of the curve C : P (U, V ) = 0. In view of [7.54], the curve C itself is a double cover of the hyperelliptic curve H : W 2 = V (1 − V )R(V ), of genus 2, ramified over four points where Q(V ) = 0. Therefore, C has genus 5 and D has genus 17. The 2 curve D must be thought of as being a very ample divisor on some Abelian surface A, to be constructed according to the method described in the preceding problems (for 2 more details, see Adler and van Moerbeke (2004)). The curve D, wrapped around A, intersects itself transversally in eight points, adding 8 to the genus 17. Therefore, the torus A2 C2 /LΩ on which the geodesic flow linearizes is defined by a period lattice

Algebraically Integrable Systems

209

 ac δ1 0 a c , Im > 0, 0 δ2 c b c b according to [7.47], with δ1 δ2 = g(D) − 1 = 24, δ1 | δ2 , δi ∈ N∗ . We have two possibilities: (i) δ1 = 1, δ2 = 24 and (ii) δ1 = 2, δ2 = 12. The line bundle L(D) = {1, z1 , ..., z6 , F1 , ..., F5 , G1 , ..., G8 , H1 , ..., H4 } is specified as follows F1 = z42 − 2 −2 z2 z5 , F2 = z52 −z1 z4 , F3 = z1 z2 −z4 z5 , F4 = 3a−1 (z2 z3 −z5 z6 ), F5 = 3a+1 (z1 z6 − 2 2 2 z3 z4 ), F6 = z1 z4 + z2 z5 − z3 z6 , F7 = z1 − z3 + z1 z4 , F8 = z2 − z62 + z2 z5 , G1 = −2az2 F2 − (1 − a)z4 F3 , G2 = −2az1 F1 − (1 + a)z5 F3 , G3 = (1 − a)z5 F4 + (1 + a)z4 F5 , G4 = (1 + a)z5 F5 + (1 − a)z1 F4 , G5 = (1 − a)z4 F4 + (1 + a)z2 F5 , G6 = −(1 − a)z3 F1 − (1 + a)z6 F2 , G7 = 2az5 F2 − (1 − a)z1 F3 , G8 = −2az4 F1 − (1 + a)z2 F3 , H1 = 4a2 F1 F2 + (1 − a2 )F3 , H2 = 2a2 F1 F5 − (1 − a)F3 F4 , H3 = −2a2 F2 F4 − (1 + a)F3 F5 , H4 = 2a2 F4 F5 + F3 ((1 + a)F2 − (1 − a)F1 ). The reflection about the origin on the Abelian surface amounts to flipping the time for dz each linear flow on it, but since the flow dt given by [7.52] is quadratic and since 1 dz the other flow dt2 (commuting with the first) is quartic (as it derives from the quartic Hamiltonian [7.53]), flipping the signs of t1 and t2 for each of the flows amounts to the flip (z1 , ..., z6 ) −→ (−z1 , ..., −z6 ). From formula [7.39], the above line bundle L(D) has 11 even sections and 13 odd sections in case (i) and 10 even sections and 14 odd sections in case  (ii), showing that case  (ii) is the only alternative and the period 2 0 a c a c matrix has the form , Im > 0. Differentiating z11 and zz21 with 0 12 c b c b respect to t1 (corresponding to the flow [7.52]) and t2 (corresponding to the quartic flow generated by the invariant Q4 [7.53]) yields two differentials ω1 and ω2 defined on the curve C: 

Ω given by the columns of the following matrix Ω =

ω1 =

ϕ(V )dV " U V (1 − V )R(V )

ω2 =

dV " , U V (1 − V )R(V )

where ϕ(V ) is a rational function in V having the form ϕ(V ) =

4aV + (a − 1)(3a + 1)  (αV + β)U 2 V (1 − V ) V (1 − V )  +(A3 (3a2 + 1) − A1 − A2 )(1 − V )V − A1 V − A2 (1 − V ) ].

The restriction of the differentials dt1 and dt2 to the curve D is ω1 = dt1 |D = ϕ(Z 2 )ω2 ,

ω2 = dt2 |D =

dZ √ . UY R

Recall that C is a double ramified cover of a hyperelliptic curve H of genus 2, whose sheets are interchanged by the involution (V, U ) −→ (V, −U ). Hence Jac(C) = P rym(C/H) ⊕ Jac(H). Since ω1 and ω2 are both odd differentials for

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that involution, the flows evolve on the three-dimensional P rym(C/H) and therefore A2 ⊂ P rym(C/H). This shows that P rym(C/H) splits further, up to isogenies, into 2 an elliptic curve E and the two-dimensional invariant torus A, 2 A ⊕ E = P rym(C/H). In summary, the affine invariant surface A for the Adler-van Moerbeke geodesic flow completes into a generic Abelian  surface A2 of 1 0 ac polarization (1, 6), that is, defined by a period matrix of the form , 0 6 c b  a c Im > 0, by adjoining at infinity a curve of genus 25, with eight normal c b crossings and smooth version D. There exists an elliptic curve E such that A2 satisfies A2 ⊕ E = P rym(C/H). More precisely 4 6 # j=1

$ 2 x ∈ C6 : Qj (x) = cj = A\

%

a curve of genus 25 with 8 singular points

& .

2 contain a very ample and Put in a more geometrical language, the tori A projectively normal curve of geometric genus 25, with eight normal crossings whose smooth version D is a 4 − 1 unramified cover of a curve C of genus 5. The curve C itself is a double cover ramified over four points of a genus 2 hyperelliptic curve H. The linearization takes place on a two-dimensional subtorus of the three-dimensional 2 ⊕ E, where E is an elliptic curve Prym variety P rym (C/H) with P rym (C/H) = A (for more details, see Adler and van Moerbeke (2004)). This situation provides a full description of the moduli for the Abelian surfaces of polarization (1, 6). 7.9. The geodesic flow on SO(n) for a left invariant metric Consider the group SO(n) and its Lie algebra so(n) paired with itself, via the customary inner product X, Y  = − 12 tr (X.Y ) , where X, Y ∈ so(n). A left invariant metric on SO(n) is defined by a non-singular symmetric linear map Λ : so(n) −→ so(n), X −→ Λ.X, and by the following inner product; given two vectors gX and gY in the tangent space SO(n) at the point g ∈ SO(n), gX, gY  = X, Λ−1 .Y . The question of classifying the metrics for which geodesic flow on SO(n) is algebraically completely integrable is difficult. Case n = 3: The Euler problem of a rigid body (section 7.4) is always algebraically completely integrable and can be regarded as geodesic flow on SO(3). Case n = 4: In the classification Adler and van Moerbeke (1984) of algebraic integrable geodesic flow on SO(4), three cases come up; two are linearly equivalent to cases of rigid body motion in a perfect fluid studied last century, respectively, by Clebsch and Lyapunov–Steklov, and there is a third new case, namely the Kostant–Kirillov Hamiltonian flow on the dual of so(4). As mentioned in the

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previous section, it is more convenient to use the coordinates u = (x1 , x2 , x3 ) and v = (x4 , x5 , x6 ), they correspond to the decomposition u ⊕ v ∈ so(4) so(3) ⊕ so(3). In these coordinates, the geodesic flow on the group SO(4) can be written as u˙ = u × ∂H , v˙ = v × ∂H for the metric defined by ∂u

∂v ,

6 3 1 2 λ x + where the quadratic form H = j=1 j j j=1 μj xj xj+3 , 2 λ1 , ..., λ6 , μ1 , μ2 , μ3 ∈ C and λ12 λ23 λ31 λ45 λ56 λ64 μ1 μ2 μ3 = 0 with λjk ≡ λj − λk . Explicitly, the equations above are written as dx4 dx1 = λ32 x2 x3 + μ3 x2 x6 − μ2 x3 x5 , = λ65 x5 x6 + μ3 x3 x5 − μ2 x2 x6 , dt dt dx2 dx5 = λ13 x3 x1 + μ1 x3 x4 − μ3 x1 x5 , = λ46 x6 x4 + μ1 x1 x6 − μ3 x3 x4 , dt dt dx3 dx6 = λ21 x1 x2 + μ2 x1 x5 − μ1 x2 x4 , = λ54 x4 x5 + μ2 x2 x4 − μ1 x1 x5 . dt dt Besides the energy Q1 = H, the equations have two trivial constants of the motion Q2 = x21 + x22 + x23 , Q3 = x24 + x25 + x26 . Adler and van Moerbeke (2004) have shown that the geodesic flow on SO(4) for the metric defined by the above quadratic form is algebraically completely integrable if and only if: (a) The quadratic form H is diagonal with regard to the customary so(4) coordinates (Manakov metric), that

4 β −β 2 is, 2H = j,k=1 Λjk Xjk , (Xjk )1≤j,k≤4 ) ∈ so(4), with Λjk = αjj −αkk , (αj , βj ∈ j 0, (a, b, c ∈ C), and also the linearization takes place on a Prym variety c b as discussed earlier in section 7.8. The periods of this Prym variety provide the exact periods of the motion in terms of Abelian integrals. The problem of the solid body in a fluid in the case of Clebsch is a particular case of this metric. See sections 3.3.1 and 4.5 where the problem has been studied via the spectral method and section 7.7 via the Kowalewski-Painlevé analysis. (b) The quadratic form H satisfies the conditions 

  λ12 λ23 λ31 λ45 λ56 λ64 (λ23 − λ56 )2 (λ31 − λ64 )2 (λ12 − λ45 )2 μ21 , μ22 , μ23 = , , , 2 λ23 λ56 λ31 λ64 λ12 λ45 (λ46 λ32 − λ65 λ13 )

with the product μ1 μ2 μ3 being rational in λ1 , ..., λ6 and with the following sign 12 λ23 λ31 λ45 λ56 λ64 specification μ1 μ2 μ3 = λ(λ 3 (λ12 − λ45 )(λ23 − λ56 )(λ31 − λ64 ). The 46 λ32 −λ65 λ13 ) extra invariant Q4 is quadratic and the flow linearizes on two-dimensional hyperelliptic Jacobians. More precisely 4 6 # j=1

$ x ∈ C6 : Qj (x) = cj = Jac (hyperelliptic curve C of genus 2)\D,

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Integrable Systems

where D is a divisor of genus 17, which contains four translates of the Θ-divisor in Jac(C), each of which is isomorphic to C. The hyperelliptic curve # $

C is a double cover of the curve C0 defined as t1 : t2 : t3 : t4 ] ∈ P3 (C) such that tj Qj has rank 3 , isomorphic to P1 (C). The periods of the motion are given by the periods of the hyperelliptic curve C. When studying the differential systems in this case, as well as the invariants via Kowalewski’s analysis, it is advantageous to rewrite them in a simpler form in order to reduce the notations and thus avoid too much calculation (see exercise 7.6 for explicit notation and Adler and van Moerbeke (2004) for more details). The problem of the solid body in a fluid in the case of Lyapunov–Steklov is a particular case of this metric (see section 3.3.2). (c) The form H satisfies 

 μ41 , μ42 , μ43 = λ13 λ46 λ21 λ54 λ32 λ65



1 1 1 , , λ32 λ65 λ13 λ46 λ21 λ54

.

The quantities ζ, ξ and η defined by ζ 2 ≡ λλ46 , ξ 2 ≡ λλ54 , η 2 ≡ λλ65 , satisfy the 13 21 32 quadratic relations ζξ + ξη + ηζ + 1 = 0, 3ξη + η − ξ + 1 = 0. The geodesic 23 2 flow has  a quartic invariant,  evolves on Abelian surfaces A ⊆ P (C) having period 2 0 ac ac matrix , Im > 0, (a, b, c ∈ C), and it will be expressed in terms 0 12 c b c b of Abelian integrals. Case n ≥ 5: We have seen that if a system is algebraically completely integrable, then it has a family of meromorphic Laurent series depending on “dim (phase space) − 1” free parameters. Now, trying to generalize the result to the geodesic flow on SO(n) for n ≥ 5 using the same method leads to insurmountable calculations. As shown by Haine (1984), for n ≥ 5 Manakov’s metrics are the only left invariant diagonal metrics on SO(n), for which the geodesic flow is algebraically completely integrable. Note that it turns out that the geodesic flow on SO(n) admits a lot of invariant manifolds on which they reduce to geodesic flow on SO(3) and the solutions of the differential equation with initial conditions on these manifolds are elliptic functions and this without any condition on the metric. Haine (1984) has shown that looking at solutions near these special a priori known solutions and imposing these solutions to be single-valued functions of t ∈ C, suffices to single out the left invariant diagonal metrics for which the geodesic flow is algebraically completely integrable. This criterion was first used, without proof by Lyapunov (1893) (the proof is due to Haine (1984)), who showed that the only integrable tops whose solutions have analytic properties belong to the classical known cases: Euler top, Lagrange top and Kowalewski top. 7.10. The periodic five-particle Kac–van Moerbeke lattice The periodic five-particle Kac–van Moerbeke lattice (Kac and van Moerbeke . 1975) is given by the quadratic vector field xj = xj (xj−1 − xj+1 ), j = 1, ..., 5, 5 where (x1 , ..., x5 ) ∈ C and xj = xj+5 . This system forms a Hamiltonian vector

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field for the Poisson structure {xj , xk } = xj xk (δj,k+1 − δj+1,k ), 1 ≤ j, k ≤ 5, and admits three independent first integrals H 1 = x1 x 3 + x 2 x 4 + x 3 x 5 + x 4 x 1 + x 5 x 2 , H2 = x1 + x 2 + x 3 + x 4 + x 5 , H3 = x 1 x 2 x 3 x 4 x 5 . Let us show that this system is algebraically completely integrable. We easily check that H1 and H2 are in involution while H3 is a Casimir. The system in question is therefore integrable in the Liouville sense.3 In addition, it is shown (Adler # $ 3 and van Moerbeke 2004) that the affine variety j=1 x ∈ C5 : Hj (x) = cj , (c1 , c2 , c3 ) ∈ C3 , c3 = 0, defined by # the intersection of the constants of motion $is isomorphic to Jac(C)\D where C = (z, w) ∈ C2 : w2 = (z 3 − c1 z 2 + c2 z)2 − 4z , is a smooth curve of genus 2 and D consists of five copies of C in the Jacobian variety Jac(C). The flows generated by H1 and H2 are linearized on Jac(C) and the system in question is algebraically completely integrable. The reader interested in the study of this system via various methods can find further information with more detail in Adler and van Moerbeke (2004), as well as in Teschil (2000). 7.11. Generalized periodic Toda systems Let e0 , ..., el be linearly dependent vectors in the following Euclidean vector space (Rl+1 , .|.), l ≥ 1 , such that they are l to l linearly independent (i.e. for all j, the vectors e0 , ..., eCj , ..., el are linearly independent). Suppose that the non-zero reals

l

l ξ0 , ..., ξl satisfying j=0 ξj ej = 0 are non-zero sum; that is, j=0 ξj = 0. Let e |e 

A = (aij )0≤i,j≤l be the matrix whose elements are defined by aij = 2 eji |ejj  , %. x = x.y . 0 ≤ i, j ≤ l. We consider the vector field XA on C2(l+1) , XA : , where y = Ax x, y ∈ Cl+1 and x.y = (x0 y0 , ..., xl yl ). Using the method described in this chapter, we show (Adler and van Moerbeke 2004) that if XA is an integrable vector field of an irreducibly algebraically completely integrable system, then A is the Cartan matrix of a possibly twisted affine Lie algebra. This system was studied by many authors (see Adler and van Moerbeke (2004) and the references therein). Specific detailed results can be found in the technical paper (Adler and van Moerbeke 1991) (and also in Adler and van Moerbeke (2004), concerning the link between the Toda lattice, Dynkin diagrams, singularities and Abelian varieties). The periodic l + 1 particle Toda lattices are associated with extended Dynkin diagrams. They have l + 1 polynomial invariants, as many as there are dots in the Dynkin diagram and are integrable Hamiltonian systems. The complex invariant manifold defined by putting these invariants equal to generic constants completes into an Abelian variety by gluing on a specific divisor D. The latter is entirely described by the extended

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Integrable Systems

Dynkin diagram: each point of the diagram corresponds to a component of the divisor and each subdiagram determines the intersection of the corresponding divisors. The global geometry of the complex invariant tori (Abelian varieties), such as polarization, divisor equivalences, dimension of certain linear systems, etc., is also entirely given by the extended Dynkin diagram and the linear equivalence between them is expressed in Lie-theoretic terms. More precisely, the divisor D consists of l + 1 irreducible components Dj each associated with a root αj of the Dynkin diagram Δ. The intersection of k components Dj1 , ..., Djk satisfies the following relation: the intersection multiplicity of the intersection of k components of the (W ) divisor equals order where W and A are the Weyl group and the Cartan matrix det(A) going with the sub-Dynkin diagram αj1 , ..., αjk associated with the k components. The intersection of all the divisors is empty and the intersection of all divisors but one is a discrete set of points whose number is explicitly determined. we have the following expression for the set-theoretical number of points in terms of the Dynkin diagram ⎛ ⎝

6 β=α

⎞ pα Dβ ⎠ = p0



order(Weyl group of the Dynkin diagram Δ\α0 ) order(Weyl group of the Dynkin diagram Δ\α)

,

where the integers pα are given by the null vector of the Cartan matrix going with Δ. The singularities of the divisor are canonically associated with semi-simple Dynkin diagrams. The singularities of each component only occur at the intersections with other components, and their multiplicities at the intersection with other divisors are expressed in terms of how a corresponding root is located in the sub-Dynkin diagram determined by this root and those of the members of the above divisor intersection. The following inclusion holds for the singular locus sing(Dk ) of Dk : sing(Dk ) ⊆ Dk ∩

0≤j≤l Dj , k = 0, ..., l. The multiplicity of the singularity of a particular component j=k

Dk , at its intersection with m other divisors, that is, sing(Dk ) ∩ (Dj1 , ..., Djm ), all j1 , ..., jm = k, is entirely specified by the way the corresponding root αk sits in the sub-Dynkin diagram αk , αj1 , ..., αjm (for proof of these results, as well as other information, see Adler and van Moerbeke (1991)). 7.12. The Gross–Neveu system The Gross–Neveu Hamiltonian system plays an important role in particle physics . ∂H . ∂H and written asxj = ∂y , y j = − ∂x , with j = 1, ..., n and whose energy of the j j



n 1 2 icα,x , where c is a constant, α = (α1 , ..., αn ) and form H = 2 j=1 yj + α∈R e

n α, x = j=1 αj xj . The sum on α above extends over the root system of a simple Lie algebra L. We show (Adler and van Moerbeke 1982) that (a) the Hamiltonian system above for j = 1, 2, 3, L = sl(3) and

Algebraically Integrable Systems

H=

215

3 3  1 2 yj + ei(xj −xk ) , 2 j=1 j,k=1

with Abelian functions yj , eixj , 1 ≤ j ≤ 3, is not algebraically completely integrable. (b) The same conclusion is obtained for the Hamiltonian system in (a) where j = 1, 2, 3, 4 and L = sl(4) o(6). 7.13. The Kolossof potential Let us study the integrability of the following Kolossof Hamiltonian system . ∂H . ∂H . ∂H . ∂H (Gavrilov et al. 1992): q 1 = ∂p , p1 = − ∂q , q 2 = ∂p , p2 = − ∂q , where 1 1 2 2 1 2 1 2 H = 2 (p1 + p2 ) + r + r − a cos θ, a ∈ R, is the Hamiltonian and q1 = r cos θ, q2 = r sin θ. This system describes the motion of a particle of mass unit in the plane (q1 , q2 ) under the effect of the Kolossof potential V (q1 , q2 ) = r + 1r − a cos θ, a ∈ R. The Kolossof system admits a second first integral F = −(a2 + q22 )p21 + (q1 − a)(2q2 p1 − q1 p2 + ap2 )p2 −2a(q1 − a)(aq1 − 1)(q12 + q22 )−1/2 , and it is Liouville integrable (but it is not algebraically completely integrable). Let Mc = {(q1 , q2 , p1 , p2 , z) ∈ C5 : H = c1 , F = c2 , q12 + q22 = z 2 , z = 0} (where c = (c1 , c2 ) is not a critical value) be the invariant affine variety by the Kolossof flow. Let us consider P (u) = −2u3 + 2c1 u2 − 2(1 − a2 )u + c2 . We show (see Gavrilov et al. (1992)) that if the polynomial (u2 − a2 )P (u) does not have a double root, then >c \D where the variety Mc is smooth and biholomorphic to the complex manifold M >c is an Abelian variety and D a divisor on M >c . In addition, M >c is an unbranched M double cover of the Jacobian variety Jac(C) where C is a 2-genus curve defined by C : w2 = (u2 − a2 )P (u). The trajectories of the Hamiltonian flow generated by H in Mc are straight lines, but the motion is nonlinear. The trajectories of the flow generated by H + sF , s = 0 in Mc are not linear. As a result, the flows generated by >c . H and F do not linearize on M 7.14. Exercises E XERCISE 7.1.– We consider the system described by the Hamiltonian (Roekaerts 1987): H=

 1 2 y1 + y22 + y1 x1 x2 + y2 2



1 2 x1 + 2x22 . 32

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Is this system algebraically completely integrable? A second first integral is     F = 256y14 y2 + 32y12 4y12 x21 + 8x22 + y22 x21     2 +x21 64y13 x1 x2 + 8y12 y2 x21 + 16x22 + y23 x21 + x41 (y1 x1 + 2y2 x2 ) . E XERCISE 7.2.– Consider a system of differential equations (Haine 1984): z˙ = f (z),

z ∈ Cm ,

[7.55]

where f is holomorphic on Cm . Assume that all solutions of [7.55] with initial conditions in a dense set A ⊂ Cm are (analytic) single-valued functions of t ∈ C. a) Show that all solutions of [7.55] are single-valued functions of t ∈ C. b) Let ϕ(t) be a particular solution of [7.55] holomorphic along some closed path l in the complex t-plane. Show that the analytic continuation along l of any solution of the variational (linearized) equations: δ˙ = ∂f ∂x (ϕ(t))δ, has to be single-valued. c) We use the notation from definition 7.1 (section 7.2). Show that if the system [7.9] is algebraically completely integrable with Abelian functions zi , then the conclusions of questions (a) and (b) hold. E XERCISE 7.3.– Suppose that the Hamiltonian system [7.9] is algebraically completely integrable with Abelian functions zi . Can we conclude that the analytic continuation of any solution of this system can at worst lead to pole singularities? (In other words, all of its solutions are meromorphic functions of t ∈ C). Justify your answer. E XERCISE 7.4.– Consider the following system of differential equations: X˙ = [X, λX], [7.56]

where X = (xij ) ≡ ic \Mc = Hi + H−i consists of two smooth isomorphic genus 2 curves Hε . M e) Deduce that the system of differential equations [7.57] is algebraic complete >c . The system [7.57] and the integrable and the corresponding flows evolve on M invariants [7.58] will play a crucial role in exercise 8.8 (hint: see Lesfari (2020)).

8 Generalized Algebraic Completely Integrable Systems

Some interesting cases of integrable systems, to be discussed in this chapter, appear as coverings of algebraic completely integrable systems. The manifolds in variant by the complex flows are coverings of Abelian varieties and these systems are called generalized algebraic completely integrable. The latter are completely integrable in the sense of Arnold–Liouville and so generically, the compact connected manifolds invariant by the real flows are tori, the real parts of complex affine coverings of Abelian varieties. Also, we will see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. We will see that a large class of algebraic completely integrable systems in the generalized sense are part of new algebraic completely integrable systems. We consider (as examples of applications) the Hénon–Heiles problem, the Ramani–Dorizzi–Grammaticos (RDG) potential, the Yang–Mills system, Goryachev–Chaplygin and Lagrange tops, as well as other interesting systems related to these examples. 8.1. Generalities There are many examples of differential equations z˙ = f (z), z ∈ Cm , which have the weak Painlevé property that all movable singularities of the general solution have only a finite number of branches, and some integrable systems appear as coverings of algebraic completely integrable systems. The manifold invariants by the complex flows are coverings of Abelian varieties and these systems are called generalized algebraic completely integrable. They are Liouville integrable and the compact connected manifolds invariant by the real flows are tori, the real parts of complex affine coverings of Abelian varieties. Most of these systems possess solutions that are Laurent series of t1/n (t ∈ C), and whose coefficients depend Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

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rationally on certain algebraic parameters. These parameters come from an (n − 1)-dimensional family of affine varieties (where n is the dimension of the invariant manifolds) and a number of independent constants of the motion. D EFINITION 8.1.– Let (M, {., .}, ϕ) be a complex integrable system where M is a non-singular affine variety and ϕ = (H1 , ..., Hs ) where each Hi in ϕ is a regular function. We say that this system is a generalized algebraic completely integrable system if each generic fiber of H is a Zariski open subset of a commutative algebraic group, on which the Hamiltonian vector fields generated by Hi are translation invariant. The integrable systems we are going to deal with here are complex integrable systems where M = Cm . Therefore, for the generalized algebraic completely integrable systems, it suffices to replace the condition (ii) in remark 7.4 (a), by the n+k 3 {z ∈ Cm : Hi (z) = ci } are related to an following: (iii) the invariant manifolds i=1

l-fold cover T2n of the complex algebraic torus T n ramified along a divisor D in T n n+k 3 as follows: {z ∈ Cm : Hi (z) = ci } = T2n \D. i=1

E XAMPLE 8.1.– The following differential equations x˙ = y 3 ,

y˙ = −x3 ,

[8.1]

are written in the form of a Hamiltonian vector field z˙ = J ∂H z = (x, y) , ∂z , where 0 −1 H = 14 (x4 + y 4 ) = a is the Hamiltonian and J = . This system is 1 0 completely integrable and can be solved in terms of Abelian integrals. We deduce  from the equations x˙ = y 3 , 14 (x4 + y 4 ) = a, the integral form t = (a−xdx4 )3/4 + t0 . √ The system [8.1] admits four one-dimensional families of Laurent solutions in t, depending on one free parameter, x = √1t (x0 + x1 t + x2 t2 + · · · ), y = √1t (y0 + y1 t + y2 t2 + · · · ), where x0 + 2y03 = 0, −y0 + 2x30 = 0, x1 = y1 = 0,  4 −x2 + 2y02 y2 = 0, y2 + 2x20 x2 = 0. Hence, (2x0 y0 )2 = −1, xy00 = −1, x1 = y1 = 0 and x2 , y2 depend on one free parameter. We have seen √ that it is possible for the variables x, y to contain square root terms of the type t, which are strictly not allowed by the Painlevé test. However, these terms are trivially removed by introducing some new variables z1 , z2 , z3 , which restores the Painlevé property to the system. A simple inspection of the Laurent series mentioned above suggests choosing z1 = x2 , z2 = y 2 , z3 = xy. By using the first integrals H = a, and the differential equations [8.1], we obtain a new system of differential equations in three unknowns z1 , z2 , z3 , having two quadrics invariants F1 , F2 : z˙1 = 2z2 z3 , z˙2 = −2z1 z3 , z˙3 = # z22 − z12 , and F1 = z12 + z22 = 4a, F2 = z12 − z22$+ z32 = b. The intersection A = z ≡ (z1 , z2 , z3 ) ∈ C3 : F1 (z) = 4a, F2 (z) = b is an elliptic

Generalized Algebraic Completely Integrable Systems

223

curve E : {z22 = −z12 + 4a, z32 = −2z12 + 4a + b}. Note that the equation x4 + y 4 = 4a defines a Riemann surface#of genus 3, but is not a torus. An equivalent $ description of x4 + y 4 = 4a$is given by z22 = −z12 + 4a, z32 = −2z12 + 4a + b and # x2 = z1 , y 2 = z2 , xy = z3 ; as a double cover of E ramified at the four points where zi = ∞. Consequently, the invariant surface completes into a double cover of an elliptic curve ramified at the points where the variables blow up. This example corresponds to definition (i), (iii) and we shall see more complicated examples, but very interesting problems, later. Consider the change of variable: z1 = 12 (m2 − m1 ), z2 = 12 (m1 + m2 ), z3 = m3 . Taking the derivative and using the differential equations above for z1 , z2 , z3 leads to the following system of differential equations: m ˙ 1 = −2m2 m3 , m ˙ 2 = 2m1 m3 , m ˙ 3 = m1 m2 . We see the resemblance with the equations of the Euler rigid body motion. It was shown in series of publications (Abenda and Fedorov 2000; Vanhaecke 2001) that the θ-divisor can serve as a carrier of integrability. Let H be a hyperelliptic curve of genus g and Jac(H) = Cg /Λ its Jacobian variety, where Λ is a lattice of maximal rank in Cg . Let Ak : Sym (H) −→ Jac(H), (P1 , ..., Pk ) −→ k

k +  j=1

Pj ∞

(ω1 , ..., ωg ) mod.Λ,

(0 ≤ k ≤ g), be the Abel map where (ω1 , ..., ωg ) is a canonical basis of the space of differentials of the first kind on 4H. The theta divisor Θ is a subvariety of Jac(H) defined as Θ ≡ A Symg−1 (H) /Λ. By Θk , we will  denote the subvariety (called

strata) of Jac(H) defined by Θk ≡ Ak Symk (H) /Λ and we have the following stratification: {O} ⊂ Θ0 ⊂ Θ1 ⊂ Θ2 ⊂ ... ⊂ Θg−1 ⊂ Θg = Jac(H), where O is the origin of Jac(H). It was shown in Vanhaecke (2001) that these stratifications of the Jacobian are connected with stratifications of the Sato Grassmannian via an extension of Krichever’s map and some remarks on the relation between Laurent solutions for the Master systems and stratifications of the Jacobian of a hyperelliptic curve. In Vanhaecke (2001) there is a study about Lie–Poisson structure in the Jacobian, which indicates that invariant manifolds associated with Poisson brackets can be identified with these strata. Some problems were considered in Vanhaecke (2001) and Abenda and Fedorov (2000), where a connection was established with the flows on these strata. Such varieties or their open subsets often appear as coverings of complex invariants manifolds of finite dimensional integrable systems (Hénon–Heiles and Neumann systems). Let us consider the RDG series of integrable potentials (Ramani 1982; Hietarinta 1987): 



[m/2]

V (x, y) =

k=0

2m−2i

m−i i

x2i y m−2i ,

m = 1, 2, ...

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It can be straightforwardly proven that a Hamiltonian H: H=

1 2 (p + p2y ) + αm Vm , 2 x

m = 1, 2, ...

containing V is Liouville integrable, with an additional first integral: F = px (xpy − ypx ) + αm x2 Vm−1 ,

m = 1, 2, ...

The study of cases m = 1 and m = 2 is easy. The study of other cases is not obvious. For the case m = 3, we obtain the Hénon–Heiles system that we will see in section 8.3. The case m = 4 corresponds to the system that will be studied in section 8.2. However, the case m = 5 corresponds to a system with an Hamiltonian 3 4 of the form H = 21 (p2x + p2y ) + y 5 + x2 y 3 + 16 x y. The corresponding Hamiltonian 1 6 2 system admits a second first integral: F = −px y + px py x − 12 x2 y 4 + 38 x4 y 2 + 32 x , and admits three-dimensional family solutions x, y, which are Laurent series of t1/3 : 1 2 x = at− 3 , x = bt− 3 , b3 = − 29 , but for which there are no polynomials P , such that P (x(t), y(t), x(t), ˙ y(t)) ˙ is Laurent series in t. We introduce a method for generating new integrable systems from known ones. For the algebraic integrable systems in the generalized sense, the Laurent series solutions contain square root terms of the type t−1/n that are strictly not allowed by the Painlevé test (i.e. the general solutions should have no movable singularities other than poles in the complex plane). However, for some problems these terms are trivially removed by introducing new variables, which restores the Painlevé property to the system. By inspection of the Laurent solutions of the algebraic integrable systems in the generalized sense, we look for polynomials in the variables defining these systems, without fractional exponents. For many problems, obtaining these new variables is not a problem, just use (by simple inspection) the first terms of the Laurent solutions. These new variables belong to the space L(D), where D is a divisor on an Abelian variety T n , which completes the affine defined by the intersection of the invariants of the new algebraically completely integrable system. In all of the problems we have studied, we find that the known algebraically integrable systems in the generalized sense are part of new algebraically integrable m systems. Let x˙ = J ∂H ∂x , x ∈ C , be an algebraically integrable system in the generalized sense. The Laurent series solutions of this system contain fractional exponents and the manifolds invariant by the complex flows are coverings of Abelian varieties. We might conjecture (with some additional conditions to be determined) from the problems discussed in this chapter that this system is part of a new algebraically integrable system in m + 1 variables. In other words, there is a new m+1 algebraically integrable system z˙ = J ∂H , that is, whose solutions ∂z , z ∈ C expressible in terms of theta functions are associated with an Abelian variety with a divisor on it, and the Hamiltonian flows are linear on this Abelian variety.

Generalized Algebraic Completely Integrable Systems

225

8.2. The RDG potential and a five-dimensional system Consider the RDG system (Ramani 1982),     q¨2 − q2 3q12 + 8q22 = 0, q¨1 − q1 q12 + 3q22 = 0,

[8.2]

corresponding to the Hamiltonian H1 =

1 2 3 1 (p + p22 ) − q12 q22 − q14 − 2q24 , 2 1 2 4

where p1 and p2 are the momenta conjugate to q1 and q2 , respectively. This system is integrable, with the second first integral (of degree 8) being 1 H2 = p41 − 6q12 q22 p21 + q14 q24 − q14 p21 + q16 q22 + 4q13 q2 p1 p2 − q14 p22 + q18 . 4 The first integrals H1 and H2 are in involution, that is, {H1 , H2 } = 0. Recall that a system z˙ = f (z) is weight-homogeneous with a weight νk going with each variable zk if fk (λνi z1 , . . . , λνn zn ) = λνk +1 fk (z1 , . . . , zn ), for all λ ∈ C. The system [8.2] is weight-homogeneous, with q1 , q2 having weight 1 and p1 , p2 having weight 2, so that H1 and H2 have weight 4 and 8, respectively. When we examine all of the possible singularities, √we find that it is possible for the variable q1 to contain square root terms of the type t, which are strictly not allowed by the Painlevé test. However, we will see later that these terms are trivially removed by introducing new variables z1 , . . . , z5 , which restores the Painlevé property to the system. Let B be the affine variety defined by 2 6 B= {z ∈ C4 : Hk (z) = bk }, [8.3] k=1

for generic (b1 , b2 ) ∈ C2 . T HEOREM 8.1.– a) The system [8.2] admits Laurent solutions depending on three free parameters: u, v and w. These solutions restricted to the surface B [8.3] are parameterized by two copies Γ1 and Γ−1 of the Riemann surface Γ [8.4] of genus 16. b) The system [8.2] is algebraic complete integrable in the generalized sense and extends to a new system [8.5] of five differential equations algebraically completely integrable with three quartics invariants. Generically, the invariant manifold A [8.6] defined by the intersection of thesequartics form the affine part of an Abelian surface 2 The reduced divisor at infinity A2 \ A = C1 + C−1 is very ample and consists A. of two components C1 and C−1 of a genus 7 curve C [8.7]. In addition, the invariant 2 surface B can be completed as a cyclic double cover B of the Abelian surface A, ramified along the divisor C1 + C−1 . Moreover, B is smooth except at the point lying 2 of B is a surface of over the singularity (of type A3 ) of C1 + C−1 and the resolution B 2 2 general type with invariants: X (B) = 1 and pg (B) = 2.

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Integrable Systems

P ROOF.– a) The system [8.2] possesses a three-dimensional family of Laurent solutions (principal balances) depending on three free parameters u, v and w. There are precisely two such families, labeled by ε = ±1, and they are given as follows: (q1 , q2 , p1 , p2 ) = (t−1/2 , t−1 , t−3/2 , t−2 ) × a Taylor series in t, and it is not difficult to determine them explicitly. These formal series solutions are convergent as a consequence of the majorant method. By substituting these series in the constants of the motion H1 = b1 and H2 = b2 , we eliminate the parameter w linearly, leading to an equation connecting the two remaining parameters u and v: Γ:

 65 3 93 6 2 3  uv + u v + −9829u8 + 26112H1 u3 v 4 64 8192 10299 16 123 1536298731 − u − H1 u 8 + H2 + = 0. 65536 256 52

[8.4]

According to Hurwitz’s formula, this defines a Riemann surface Γ of genus 16. The Laurent solutions restricted to the affine surface B [8.3] are thus parameterized by two copies Γ−1 and Γ1 of the same Riemann surface Γ. b) Let ϕ : B −→ C5 , (q1 , q2 , p1 , p2 ) −→ (z1 , z2 , z3 , z4 , z5 ), be the morphism defined on the affine variety B (8.3) by z1 = q12 , z2 = q2 , z3 = p2 , z4 = q1 p1 , z5 = p21 − q12 q22 . These variables are easily obtained by simple inspection of the above Laurent series. By using the variables z1 , ..., z5 and differential equations [8.2], we obtain z˙1 = 2z4 ,

z˙3 = z2 (3z1 + 8z22 ),

z˙2 = z3 ,

z˙4 = z12 + 4z1 z22 + z5 ,

[8.5]

z˙5 = 2z1 z4 + 4z22 z4 − 2z1 z2 z3 . This new system on C5 admits the following three first integrals: F1 =

1 1 1 z5 − z1 z22 + z32 − z12 − 2z24 , 2 2 4

1 F2 = z52 − z12 z5 + 4z1 z2 z3 z4 − z12 z32 + z14 − 4z22 z42 , 4 F3 = z1 z5 + z12 z22 − z42 . The first integrals F1 and F2 are in involution, while F3 is trivial (Casimir function). The invariant variety A defined by A=

3 6

{z : Fk (z) = ck } ⊂ C5 ,

k=1

[8.6]

Generalized Algebraic Completely Integrable Systems

227

is a smooth affine surface for generic values of (c1 , c2 , c3 ) ∈ C3 . The system [8.5] is completely integrable and possesses Laurent series solutions, (z1 , z2 , z3 , z4 , z5 ) = (t−1 , t−1 , t−2 , t−2 , t−2 ) × a Taylor series in t, labeled by ε = ±1, depending on four free parameters α, β, γ, θ. The convergence of these series is guaranteed by the majorant method. By substituting these developments in equations Fk (z) = ck , k = 1, 2, 3, we obtain three polynomial relations between α, β, γ, θ. Eliminating γ and θ from these equations leads to an equation connecting the two remaining parameters α and β:   C : 64β 3 − 16α3 β 2 − 4 α6 − 32α2 c1 − 16c3 β   +α 32c2 − 32α4 c1 + α8 − 16α2 c3 = 0.

[8.7]

The Laurent solutions restricted to the surface A [8.6] are parameterized by two copies C−1 and C1 of the same Riemann surface C [8.7]. According to the Riemann– Hurwitz formula, the genus of C is 7. Applying the method used in Chapter 7, we embed these curves in a hyperplane of P15 (C) using the 16 functions: 1, z1 , z2 , 5 − z12 , z3 + 2εz22 , z4 + εz1 z2 , W (f1 , f2 ), f1 (f1 + 2εf4 ), f2 (f1 + 2εf4 ), z4 (f3 + 2εf6 ), z5 (f3 + 2εf6 ), f5 (f1 + 2εf4 ), f1 f2 (f3 + 2εf6 ), f4 f5 + W (f1 , f4 ), W (f1 , f3 ) + 2εW (f1 , f6 ), f3 − 2z5 + 4f42 , where W (sj , sk ) ≡ s˙ j sk − sj s˙ k (Wronskian) and we show that these curves have two points in common in which C1 is tangent to C−1 . The system [8.2] is algebraic complete integrable in the generalized sense. The invariant surface B [8.3] can be 2 ramified along the completed as a cyclic double cover B of the Abelian surface A, divisor C1 + C−1 . Moreover, B is smooth except at the point lying over the singularity (of type A3 ) of C1 + C−1 (double points of intersection of the curves C1 and C−1 ) and the resolution B2 of B is a surface of general type. We shall resume the proof of these results (already used previously in other similar problems) with more detail. Observe that the morphism ϕ is an unramified cover. The Riemann surface Γ [8.4] plays an important role in the construction of a compactification B of B. Let us denote by G a cyclic group of two elements {−1, 1} on Vεj = Uεj × {τ ∈ C : 0 < |τ | < δ}, where τ = t1/2 and Uεj is an affine chart of Γε for which the Laurent solutions are defined. The action of G is defined by (−1) ◦ (u, v, τ ) = (−u, −v, −τ ) and is without fixed points in Vεj . So we can identify the quotient Vεj /G with the image of the smooth map hjε : Vεj −→ B defined by the Laurent expansions. We have (−1, 1).(u, v, τ ) = (−u, −v, τ ), (1, −1).(u, v, τ ) = (u, v, −τ ), that is, G × G acts separately on each coordinate. Thus, identifying V εj /G2 with the image of ϕ ◦ hjε in

228

Integrable Systems

A. Note that Bεj = Vεj /G is smooth (except for a finite number of points) and the coherence of the Bεj follows from the coherence of Vεj and the action of G. By taking B and by gluing on various varieties Bεj \{some points}, we obtain a smooth complex C which is a double cover of the Abelian variety A2 ramified along manifold B, 2 To see C1 + C−1 , and can therefore be completed to an algebraic cyclic cover of A. what happens to the missing points, we investigate the image of Γ × {0} in ∪Bεj . The quotient Γ × {0}/G is birationally equivalent to the Riemann surface Υ of genus 7:  65 3 93 3 2 3  y + x y + −9, 829x4 + 26, 112b1 x2 y 4 64 8192  15, 362 98, 731 10, 299 8 123 4 = 0, y ≡ uv, x ≡ u2 . +x − x − b1 x + b2 + 65, 536 256 52

Υ:

The Riemann surface Υ is birationally equivalent to C. The only points of Υ fixed under (u, v) −→ (−u, −v) are the points at ∞, which correspond to the ramification 2−1 points of the map Γ × {0} −→ Υ : (u, v) −→ (x, y), and coincide with the points at ∞ of the Riemann surface C. Then, the variety BC constructed above is birationally equivalent to the compactification B of the generic invariant surface B. So B is a cyclic double cover of the Abelian surface A2 ramified along the divisor C1 + C−1 , where C1 and C−1 have two points in common at which they are tangent to each other. It follows that the system [8.2] is algebraic complete integrable in the generalized sense. Moreover, B is smooth except at the point lying over the singularity (of type A3 ) of C1 + C−1 . In term of an appropriate local holomorphic coordinate system (X, Y, Z), the local analytic equation about this singularity is X 4 + Y 2 + Z 2 = 0. Now, let 2 be the Euler characteristic of B2 and B2 be the resolution of singularities of B, X (B) 2 the geometric genus of B. 2 Then B2 is a surface of general type with invariants: pg (B) 2 2 X (B) = 1 and pg (B) = 2. This completes the proof.  2 consider the holomorphic 1-forms dt1 and dt2 defined On the Abelian variety A, by dti (XFj ) = δij , where XF1 and XF2 are the vector fields generated, respectively, by F1 and F2 . Taking the differentials of ζ = 1/z2 and ξ = zz12 viewed as functions of t1 and t2 , using the vector fields and the Laurent series and solving linearly for dt1 ∂ζ ∂ξ ∂ζ ∂ξ and dt2 , we obtain the holomorphic differentials (where Δ ≡ ∂t − ∂t ): 1 ∂t2 2 ∂t1 ω1 = dt1 |Cε =

1 ∂ξ ∂ζ 8 dζ − dξ)|Cε = ( dα, Δ ∂t2 ∂t2 α (−4β + α3 )

ω2 = dt2 |Cε =

1 −∂ξ ∂ζ 2 dζ − dξ)|Cε = ( 2 dα. Δ ∂t1 ∂t1 (−4β + α3 )

The zeroes of ω2 provide the points of tangency of the vector field XF1 to Cε . 4 3 1 Wehave ω and XF1 is tangent to Hε at the point covering α = ∞. ω2 = α −4β + α

Generalized Algebraic Completely Integrable Systems

229

Note that the reflection σ on the affine variety A amounts to the flip σ : (z1 , z2 , z3 , z4 , z5 ) −→ (z1 , −z2 , z3 , −z4 , z5 ), changing the direction of the commuting vector fields. It can be extended to the (-Id)-involution about the origin of 2 where t1 and t2 C2 to the time flip (t1 , t2 ) −→ (−t1 , −t2 ) on the Abelian variety A, are the time coordinates of each of the flows XF1 and XF2 . The involution σ acts on the parameters of the Laurent solution as follows: σ : (t, α, β, γ, θ) −→ (−t, −α, −β, −γ, θ), interchanges the Riemann surfaces Cε and the linear space L can be split into a direct sum of even and odd functions. Geometrically, this involution interchanges C1 and C−1 , that is, C−1 = σC1 . 8.3. The Hénon–Heiles problem and a five-dimensional system As discussed in section 7.6 of the Hénon–Heiles study, the present section deals with the case (iii) (see section 7.6). We will briefly study this problem knowing that the method is the same as the one described previously. When we examine all of the possible singularities,√we find that it is possible for the variable y1 to contain square root terms of the type t, which are strictly not allowed by the Painlevé test. However, these terms are trivially removed by introducing some new variables z1 , . . . , z5 , which restores the Painlevé property to the system. And as mentioned above, we obtain a new algebraically completely integrable system. The system [7.27] for case (iii), that is, x1 = y 1 ,

.

y 1 = −Ax1 − 2x1 x2 ,

.

y 2 = −16Ax2 − x21 − 16x22 ,

x2 = y 2 ,

. .

[8.8] 

can be written in the form u˙ = J ∂H ∂u , u = (x1 , x2 , y1 , y2 ) , J =

H ≡ H1 =

O I , where −I O

1 2 A 16 (y + y22 ) + (x21 + 16x22 ) + x21 x2 + x32 . 2 1 2 3

The functions H1 and 2 H2 = 3y14 + 6Ay12 x21 + 12y12 x21 x2 − 4y1 y2 x31 − 4Ax41 x2 − 4x41 x22 + 3A2 x41 − x61 , 3 commute, that is, {H1 , H2 } = 0. The second flow commuting with the first is 2 regulated by the equations u˙ = J ∂H ∂u . The system [8.8] admits Laurent solutions in √ t, (x1 , x2 , y1 , y2 ) = (t−1/2 , t−2 , t−3/2 , t−3 ) × a Taylor series in t,

230

Integrable Systems

depending on three free parameters : α, β, γ. These formal series solutions are convergent as a consequence of the majorant method. By substituting these series in the constants of the motion H1 = b1 and H2 = b2 , one eliminates the parameter γ linearly, leading to an equation connecting the two remaining parameters α and β: 144αβ 3 −

 294A2 3 143 8 4 44  3 α β+ α − α6 + 4A − 3b1 α4 + b2 = 0 5 504 21 21

which is the equation of an algebraic curve D, along which the u(t) blow up. To be more precise, D is the closure of the continuous components of {Laurent series solutions u(t) # such that Hk (u(t)) = $bk , 1# ≤ k ≤ 2}, that is to say, $ D = t0 − coefficient of u ∈ C4 : H1 (u(t)) = b1 ∩ u ∈ C4 : H2 (u(t)) = b2 . 32 The invariant variety A = k=1 {z ∈ C4 : Hk (z) = bk } is a smooth affine surface for generic (b1 , b2 ) ∈ C2 . The Laurent solutions restricted to the surface A are parameterized by the curve D. We show that the system [8.8] is part of a new system of differential equations in five unknowns having one quartic and two cubic invariants (constants of motion). By inspection of the above Laurent expansions, we look for polynomials in (x1 , x2 , y1 , y2 ) without fractional exponents. Let ϕ : A −→ C5 , (x1 , x2 , y1 , y2 ) −→ (z1 , z2 , z3 , z4 , z5 ), be a morphism on the affine variety A, where z1 , . . . , z5 are defined as z1 = x21 , z2 = x2 , z3 = y2 , z4 = x1 y1 , z5 = 3y12 + 2x21 x2 . Using the two first integrals H1 , H2 and differential equations [8.8], we obtain a system of differential equations in five unknowns, z˙1 = 2z4 ,

z˙3 = −z1 − 16A1 z2 − 16z22 ,

1 8 z˙4 = −A1 z1 + z5 − z1 z2 , 3 3 z˙5 = −6A1 z4 + 2z1 z3 − 8z2 z4 , z˙2 = z3 ,

[8.9]

having one quartic and two cubic invariants (constants of motion), F1 =

1 1 1 2 16 A1 z1 + z5 + 8A1 z22 + z32 + z1 z2 + z23 , 2 6 2 3 3

F2 = 9A21 z12 + z52 + 6A1 z1 z5 − 2z13 − 24A1 z12 z2 − 12z1 z3 z4 + 24z2 z42 − 16z12 z22 , F3 = z1 z5 − 3z42 − 2z12 z2 . This new system is completelyintegrableand

the Hamiltonian structure is defined 5 ∂H ∂F ∂H by the Poisson bracket {F, H} = ∂F , J = k,l=1 Jkl ∂zk ∂zl , and ∂z ∂z ⎛

0 ⎜ 0 ⎜ J =⎜ ⎜ 0 ⎝ −2z1 −12z4

0 0 −1 0 0

⎞ 0 2z1 12z4 ⎟ 1 0 0 ⎟ ⎟, 0 0 −2z1 ⎟ 0 0 −8z1 z2 + 2z5 ⎠ 2z1 8z1 z2 − 2z5 0

Generalized Algebraic Completely Integrable Systems

231

is a skew-symmetric matrix for which the corresponding Poisson bracket satisfies the Jacobi identities. The system [8.9] can be written as z˙ = J ∂H ∂z , z = (z1 , z2 , z3 , z4 , z5 ) , where H = F1 . The second flow commuting with the first 2 is regulated by the equations z˙ = J ∂F ∂z ,z = (z1 , z2, z3 , z4 , z5 ) . These vector fields ∂F1 ∂F2 are in involution, that is, {F1 , F2 } = ∂z , J ∂z = 0 and the remaining one is 3 The system [8.9] is integrable in the sense of Liouville. Casimir, that is, J ∂F ∂z = 0. 33 The invariant variety B = k=1 {z ∈ C5 : Fk (z) = ck } is a smooth affine surface for generic values of c1 , c2 , c3 . The system [8.9] possesses Laurent series solutions that depend on four free parameters. These meromorphic solutions restricted to the surface B can be read off from the above expansions and the change of variable ϕ. Following the methods previously used, we find the compactification of B into an 2 the system [8.9] is algebraic complete integrable and the Abelian surface B, 2 Also, we show (as in the proof of theorem 8.1b)) corresponding flows evolve on B. that the invariant surface A can be completed as a cyclic double cover A of an 2 The system [8.8] is algebraic complete integrable in the Abelian surface B. generalized sense. Moreover, A is smooth except at the point lying over the singularity of type A3 and the resolution A2 of A is a surface of general type. We have shown that the morphism ϕ maps the vector field [8.8] into an algebraic completely integrable system [8.9] in five unknowns, and the affine variety A onto the affine part 2 This explains (among other) why the asymptotic solutions B of an Abelian variety B. to the differential equations [8.8] contain fractional powers. All of this is summarized as follows: T HEOREM 8.2.– The √ system [8.8] admits Laurent solutions containing square root terms of the type t, depending on three free parameters, and is algebraic complete integrable in the generalized sense. The morphism ϕ maps this system into a new algebraic completely integrable system [8.9] in five unknowns. 8.4. The Goryachev–Chaplygin top and a seven-dimensional system The Goryachev–Chaplygin top mentioned in section 3.2.4 is a rigid body rotating about a fixed point, for which the principal moments of inertia I1 , I2 , I3 satisfy the relation: I1 = I2 = 4I3 , the center of mass lies in the equatorial plane through the fixed point and the principal angular momentum is perpendicular to the direction of gravity. The equations of the motion can be written in the form .

m1 = 3m2 m3 ,

.

γ 1 = 4m3 γ2 − m2 γ3 ,

.

γ 2 = m1 γ3 − 4m3 γ1 ,

.

.

m2 = −3m1 m3 − 4γ3 , m3 = 4γ2 ,

.

γ 3 = m2 γ1 − m1 γ2 ,

[8.10]

232

Integrable Systems

where m1 , m2 , m3 , γ1 , γ2 , γ3 are the coordinates of the phase space. The following four quadrics are constants of motion for this system: H1 = m21 + m22 + 4m23 − 8γ1 = 6b1 , H2 = (m21 + m22 )m3 + 4m1 γ3 = 2b2 , H3 = γ12 + γ22 + γ32 = b3 ,

H4 = m1 γ1 + m2 γ2 + m3 γ3 = 0,

for generic b1 , b2 , b3 ∈ C. This system is completely integrable, and H1 (energy) and H4 are in involution, while H2 , H3 are Casimir invariants.√The Goryachev–Chaplygin system has asymptotic solutions that are meromorphic in t, (m1 , m2 , m3 , γ1 , γ2 , γ3 ) = (t−3/2 , t−3/2 , t−1 , t−2 , t2 , t−1/2 ) × a Taylor series in t, depending on four free parameters b1 , b2 , b3 and u, v. Let A be the affine variety defined by A = {x : H1 (x) = 6b1 , H2 (x) = 2b2 , H3 (x) = b3 , H4 (x) = 0},

[8.11]

where x = (m1 , m2 , m3 , γ1 , γ2 , γ3 ). These solutions restricted to A are parameterized by two copies Cε=+i and Cε=−i of the curve C of genus 4: C : 16b3 u4 + εu2 (b2 + 6b1 v − 16v 3 ) − v 2 = 0. The asymptotic solutions √ of the system [8.10] contain fractional powers, that is, square root terms of the type t, which are strictly not allowed by the Painlevé test, but the new variables (z1 , z2 , z3 , z4 , z5 , z6 , z7 ) defined by z1 = m21 + m22 , z2 = m3 , z3 = γ32 , z4 = γ1 , z5 = γ2 , z6 = m1 γ3 , z7 = m2 γ3 restore the Painlevé property to the system. These variables are easily obtained by a simple inspection of the Laurent series above; we use the first terms of the Laurent series and it is generally sufficient to choose combinations (often obvious) of the initial variables so as not to have terms √ of the type t. Now let ϕ : A −→ C7 , (m1 , m2 , m3 , γ1 , γ2 , γ3 ) −→ (z1 , z2 , z3 , z4 , z5 , z6 , z7 ), be a morphism on the affine variety A. These affine variables were originally used in Bechlivanidis and van Moerbeke (1987) without any discussion of their origin and algebraic properties. The morphism ϕ maps the vector field [8.10] into the system in seven unknowns (z1 , z2 , z3 , z4 , z5 , z6 , z7 ) ∈ C7 (Bechlivanidis and van Moerbeke 1987), .

z 1 = −8z7 ,

.

.

z 3 = 2(z4 z7 − z5 z6 ),

z 2 = 4z5 ,

.

z 4 = 4z2 z5 − z7 ,

.

z 5 = z6 − 4z2 z4 ,

.

z 6 = −z1 z5 + 2z2 z7 ,

[8.12]

.

z 7 = z1 z4 − 2z2 z6 − 4z3 ,

having five quadrics invariants F1 = z1 − 8z4 + 4z22 = 6c1 ,

F2 = z1 z2 + 4z6 = 2c2 ,

F3 = z3 + z42 + z52 = c3 ,

Generalized Algebraic Completely Integrable Systems

F4 = z2 z3 + z4 z6 + z5 z7 = c4 ,

233

F5 = z62 + z72 − z1 z3 = c5 ,

where c1 , c2 , c3 , c4 , c5 are generic constants. To obtain these invariants, we used the first integrals H1 , H2 , H3 , H4 and differential equations [8.10]. This system is completelyintegrableand the symplectic structure is defined by the Poisson bracket ∂H {F, H} = ∂F ∂z , J ∂z , where ⎛

⎞ 0 0 −A z7 −z6 B −C ⎜ 0 0 0 − 12 z5 41 z4 − 12 z7 12 z6 ⎟ ⎜ ⎟ ⎜ A 0 0 0 0 −z3 z5 z3 z4 ⎟ ⎜ ⎟ 1 0 0 0 0 − 12 z3 ⎟ J =⎜ ⎜ −z7 21z5 ⎟, 1 ⎜ z6 − z4 ⎟ 0 0 0 z 0 2 2 3 ⎜ ⎟ 1 ⎝ −B 1 z7 z3 z5 ⎠ 0 − z 0 −z z 2 3 2 2 3 C − 12 z6 −z3 z4 21 z3 0 z2 z3 0 A ≡ 2z4 z7 − 2z5 z6 ,

B ≡ z1 z5 + 2z2 z7 ,

C ≡ z1 z4 + 2z2 z6 ,

is a skew-symmetric matrix whose elements polynomial satisfy the Jacobi identity. . The system [8.12] is written in the form z = J ∂H ∂z , where H = F1 and x = (x1 , x2 , x3 , x4 , x5 , x6 , x7 ) . The two first integrals F1 and F2 are in involution, k that is, {F1 , F2 } = 0, while F3 , F4 and F5 are Casimir invariants, that is, J ∂F ∂x = 0, where k = 3, 4, 5. The first fact to observe is that if the system [8.12] is to have Laurent solutions depending on six free parameters, the Laurent decomposition of

∞ (k) k−1 such asymptotic solutions must have the following form: zi = for k=0 zi t

∞ (k) k−2 i = 1, 2, 3 and zi = k=0 zi t for i = 4, 5, 6, 7. By putting these expansions (k) into the five quadrics invariants above, solving inductively for the zi , we explicitly find the following Laurent solutions depending on six free parameters: 2εα + 2α2 − 2ε(α(α2 − 2c1 ) + ζ)t − (2ξ + ζ)αt2 + o(t3 ), t ε 1 α ε − − − (α2 − 2c1 )t − (2ξ + ζ)αt2 + o(t3 ), 2t 2 2 4 ε 3α ε (ξ + ζ) + (ξ + ζ) − ((5α2 − c1 )(ξ + ζ) − 8(2c3 α + c4 ))t + o(t2 ), 8t 8 8 1 1 2 ε − 2 + (α − 2c1 ) + (2ξ + ζ)t + o(t2 ), [8.13] 8t 8 8 ε ε 1 − (α2 − 2c1 ) − (2ξ + 3ζ)t + o(t2 ), 8t2 8 8 α 1 εα + (2ξ − (α2 − 2c1 )α + ζ) − (2ξ + 3ζ)t + o(t2 ), 2 4t 4 4 εα ε α − 2 + (α(α2 − 2c1 ) + ζ) + (2ξ + ζ)t + o(t2 ), 4t 4 4

z1 = − z2 = z3 = z4 = z5 = z6 = z7 =

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where ε = ±i, ξ(α) = 2α3 − 3c1 α + c2 and the parameters α, ζ belong to a genus 2 hyperelliptic curve, H : ζ 2 = (2α3 − 3c1 α + c2 )2 − 4(4c3 α2 + 4c4 α + c5 ).

[8.14]

Using the majorant method, we can show that the formal Laurent series solutions are convergent. In fact, the Laurent solutions are parameterized by two copies H+i and H−i of the genus 2 hyperelliptic curve H for ε = ±i. In order to embed H into some projective space, we search for functions z0 , z1 , ..., zN of increasing degree in the original variables, having the property that the embedding D of Hi + H−i into PN (C) via those functions satisfies the relation geometric genus(2D) = N + 2. As in the previous chapter, we show that this occurs for the first time for N = 15, that is, the embedding D of Hi + H−i into P15 (C) is done via the original functions z0 , z1 , ..., z7 enlarged by adjoining the following eight other functions: z8 ≡ z2 z3 , z9 ≡ z1 z3 , z10 ≡ z1 z4 + 2z2 z6 , z11 ≡ z1 z5 + 2z2 z7 , z12 ≡ −z5 z6 + z4 z7 , z13 ≡ z2 z12 − 2z3 z5 , z14 ≡ z32 , z15 ≡ z1 z12 + 4z3 z7 . Using these functions, we embed the curves Hi and H−i into a hyperplane of P15 (C). Thus embedded, these curves have one point in common at which they are tangent to each other.

Figure 8.1. Curves H±i

In the neighborhood of α = ∞, the curve H has two points at which ξ + ζ behaves 2 4 α+c5 + as follows: ξ +ζ = 4α3 +o(α), picking the + sign for ζ and ξ +ζ = 4c3 α +4c α3 lower order terms, and picking the − sign for ζ. So when choosing the + sign for ζ and dividing the vector (1, z1 , ..., z15 ) by z14 = z32 , the corresponding point is sent to the point [0 : · · · : 1 : 0] ∈ P15 (C), which is independent of ε. The choice of the sign - for ζ conducts to two different points, taking into account the sign of ε. Therefore, the divisor D obtained in this way has genus 5 and 2D has genus 17, satisfying the relation: geometric genus of 2D = N + 2, that is, 2D ⊂ P15 (C) = Pg−2 (C). Following the method explained and used in detail in the previous chapter, we show that the affine surface B=

5 6

{z = (z1 , z2 , z3 , z4 , z5 ) : Fk (z) = ck } ⊂ C7 ,

[8.15]

k=1

can be completed into an Abelian surface B2 by adjoining the divisor D = Hi + H−i at infinity. The variety B2 is equipped with two commuting, linearly independent vector

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fields. Let dt1 , dt2 be two holomorphic 1-forms on B2 corresponding, respectively, to the vectors fields XF1 , XF2 . By letting y1 = xx12 , y2 = x12 , we obtain the forms

ω2 = dt2 |Hε



∂y1 aα ∂y2 dy2 − dy1 = dα, ∂t2 ∂t2 ζ  b 1 ∂y1 ∂y2 =− dy2 − dy1 = dα, Δ ∂t1 ∂t1 ζ

ω1 = dt1 |Hε =

1 Δ

where a and b are constants and Δ =

∂y2 ∂y1 ∂t1 ∂t2

−

∂y2 ∂y1 ∂t2 ∂t1 .

The points where the vector 2 field XF1 is tangent to the curves Hi and H−i on B are provided by the zeros of the form ω2 . Note that XF1 is tangent to Hi and H−i at the point where both curves touch; this point corresponds to α = ∞. The involution σ : (z1 , z2 , z3 , z4 , z5 , z6 , z7 ) −→ (z1 , z2 , z3 , z4 , −z5 , z6 , −z7 ) on the variety B acts on the free parameters (keeping the same notation) as follows: σ : (t, α, ζ, ε, c1 , c2 , c3 , c4 , c5 ) −→ (−t, α, ζ, −ε, c1 , c2 , c3 , c4 , c5 ). Therefore, we have Hi = σH−i and geometrically, this means that Hi and H−i are deduced from 2 Following the same reasoning one another by a translation in the Abelian variety B. used in the proof of theorem 8.1b), we show that the invariant variety A [8.11] can be completed as a cyclic double cover A of the Jacobian of a genus two curve, and A is smooth except at the point (tacnode) lying over the singularity of type A3 . Therefore, we have the following result (Bechlivanidis and van Moerbeke 1987; Piovan 1992): T HEOREM 8.3.– a) The system [8.12] is algebraically completely integrable. The Laurent solution [8.13] depends on six free parameters. The affine surface B [8.15] completes into an Abelian variety B2 by adjoining a divisor Hi + H−i , where H+i and H−i are two copies of the same genus 2 hyperelliptic curve H [8.14] for ε = ±1 that intersect each other in a tacnode belonging to Hi + H−i . b) The invariant variety A[8.11] of the Goryachev–Chaplygin top can be compactified as a cyclic double cover A of the Jacobian of a genus 2 curve, ramified along the divisor Hi + H−i . Moreover, A is smooth except at the point (tacnode) lying over the singularity (of type A3 ) of Hi + H−i , and the resolution A2 of A is a 2 = 1 and surface of general type with invariants: Euler characteristic of A2 = X (A) 2 = 2. The system [8.10] is algebraic completely geometric genus of A2 = pg (A) integrable in the generalized sense. The extended system [8.12] includes some other known integrable systems. It is shown (Bechlivanidis and van Moerbeke 1987) that the system [8.10] is rationally related to the three-body Toda system.

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8.5. The Lagrange top We show that the equations governing the motion of the Lagrange top (see sections 3.2.2 and 4.7) form an algebraic completely integrable system in the generalized sense. These equations are explicitly written in the form .

γ 1 = λ3 m3 γ2 − λ1 m2 γ3 ,

.

γ 2 = λ1 m1 γ3 − λ3 m3 γ1 ,

.

λ1 m1 = λ1 (λ3 − λ1 )m2 m3 − γ2 ,

.

λ1 m2 = λ1 (λ1 − λ3 )m1 m3 + γ1 , .

.

γ 3 = λ1 (m2 γ1 − m1 γ2 ).

m3 = 0,

This system admits the following four first integrals: λ21 2 λ1 λ3 2 (m1 + m22 ) + m3 − γ 3 , 2 2 H3 = λ1 (m1 γ1 + m2 γ2 + m3 γ3 ),

H1 =

H2 = γ12 + γ22 + γ32 , H4 = λ3 m3

and is integrable in the sense of Liouville. The Poisson structure is given by {mi , mj } = −ijk mk , {mi , γj } = −ijk γk , {γi , γj } = 0, where 1 ≤ i, j, k ≤ 3 and ijk is the total antisymmetric tensor for which ijk = 1. Let # $ Mc = (m1 , m2 , m3 , γ1 , γ2 , γ3 ) ∈ C6 : H1 = c1 , H2 = 1, H3 = c3 , H4 = c4 , be the affine variety defined by the intersection of the constants of the motion and let C∗ ∼ C/2πiZ be the group of rotations defined by the flow of the vector field . . . . . generated by H4 , that is, m1 = m2 , m2 = −m1 , m3 = 0, γ 1 = γ2 , γ 2 = −γ1 , . ∗ γ 3 = 0. The quotient Mc /C is an elliptic curve. We show that the algebraic variety Mc is not isomorphic to the direct product of the curve Mc /C∗ and C∗ . For generic constants cj , the complex invariant manifold Mc is biholomorphic to an affine subset of C2 /Λ where Λ ⊂ C2 is a lattice of rank 3,    2πi 0 τ1 , Re(τ1 ) < 0. Λ=Z ⊕Z ⊕Z τ2 0 2πi Hence, C2 /Λ is an non-compact algebraic group and can be considered as a non-trivial extension of the elliptic curve C/{2πiZ ⊕ τ1 Z} by C∗ ∼ C/2πiZ, ϕ

0 −→ C/2πiZ −→ C2 /Λ −→ C/{2πiZ ⊕ τ1 Z} −→ 0,

ϕ(z1 , z2 ) = z1 .

The algebraic group C/2πiZ is the generalized Jacobian of an elliptic curve with two points identified at infinity. We have the following result (Gavrilov and Angel Zhivkov 1998): T HEOREM 8.4.– The differential system governing the Lagrange top form an algebraic completely integrable system in the generalized sense.

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8.6. Exercises E XERCISE 8.1.– Let A be the complexified invariant surface of a generalized algebraic completely integrable system, with A being an unramified m-covering of the affine part of an Abelian variety B. Assume that the system possesses Laurent 1 solutions z(τ, p), depending on the parameter τ = t m , (p ∈ D, where D is a generalized “Painlevé” divisor going with such solutions), and a (r − 1)-dimensional family of divisors. Moreover, assume that there is a cyclic automorphism of A exchanging sheets and that there is a properly discontinuous action of the cyclic group μm of m elements on the space Uε = {|τ | < ε} × D, which is free on Uε \D and leaves the Laurent expansions invariant. a) Show (Piovan 1992) that the degree m holomorphic map ψε = (τ, p) ∈ {τ ∈ C, 0 < |τ | < ε} × D = Uε \D −→ z(τ, p) ∈ A, is full rank and induces an isomorphism ψε : (Uε \D)|με −→ ψε (Uε \D) ⊂ A. b) Show (Piovan 1992) that A is birationally equivalent to an m-cyclic covering of B and the coordinate functions extend meromorphically. E XERCISE 8.2.– Let A be a m-cyclic cover of an Abelian surface B ramified along the smooth reduced divisor D, such that D is linearly equivalent to mS, for some ample divisor S. Show (Piovan 1992) that A is a surface of general type. E XERCISE 8.3.– Let A be a m-cyclic cover of an Abelian surface B ramified along the reduced divisor D. Assume D is linearly equivalent to mS, for some ample divisor S. Show (Piovan 1992) that A has the following invariants: a) The genus pg (A) of A is pg (A) = 1 +

θ2 12 (m

− 1)m(2m − 1).

2

θ (m − 1)m(2m − 1), where X (A) is the holomorphic Euler b) X (A) = 12 characteristic X (A) of A.

E XERCISE 8.4.– Let X be a double cover of an Abelian surface Y ramified along the divisor B. We assume that B only has simple singularities, that is, double or triple points of type A − D − E, and B is linearly equivalent to 2θ, for some ample divisor θ. Then, a resolution of singularities of X is a surface of general type with invariants: θ2 θ2 X (X) = 12 , pg (X) = 1 + 12 . E XERCISE 8.5.– We consider the differential system described by the Hamiltonian (Roekaerts 1987):   1 2 1 2 2 2 H= y + y2 + y1 x1 x2 + y2 x + 2x2 . 2 1 8 1

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Is this system algebraically completely integrable in the generalized sense? A second first integral is provided by F =

y14

+

y12 x1

  1 2 3 1 2 3 4y1 x2 − y2 x1 + x1 − y1 x1 − y1 y2 x2 + y2 x1 . 2 2 16

E XERCISE 8.6.– Same question for the following Hamiltonian (Hietarinta 1985): H=

  1 2 1 1 y1 + y22 + y1 x1 x2 + y2 − x21 + x22 . 2 4 2

  2 A second first integral is provided by F = 2y12 + y22 2y2 + x21 + 2x22 . E XERCISE 8.7.– Same question for the following Hamiltonian (Roekaerts 1987): H=

 1 2 y1 + y22 + y1 x1 x2 − y2 x22 . 2

  A second first integral is provided by F = y12 2y2 + x21 . E XERCISE 8.8.– Consider the following Hamiltonian: H1 =

1 2 a 1 (p + p22 ) + (q12 + 4q22 ) + q14 + 4q24 + 3q12 q22 , 2 1 2 4

where a is a constant. The corresponding system, that is, q¨1 = −(a + q12 + 6q22 )q1 ,

q¨2 = −2(2a + 3q12 + 8q22 )q2 ,

[8.16]

is integrable and the second integral is H2 = aq12 q2 + q14 q2 + 2q12 q23 − q2 p21 + q1 p1 p2 . As mentioned in section34.8, this # case can be deduced 4from the paper$(Dorizzi 2 et al. 1983). Let A = be the k=1 z = (q1 , q2 , p1 , p2 ) ∈ C : Hk (z) = bk invariant surface defined (for generic (b1 , b2 ) ∈ C2 ) by the two constants of motion. a) Show that the system [8.16] possesses a three-dimensional family of Laurent solutions (depending on three free parameters)  (q1 , q2 , p1 , p2 ) = t−1/2 , t−1 , t−3/2 , t−2 × a Taylor series, and moreover, these solutions restricted to the surface A are parameterized by two smooth curves of genus 4.

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b) Show that the system of differential equations [8.16] can be written as follows: "

ds1 P6 (s1 )

ds2 −" = 0, P6 (s2 )

s ds s ds " 1 1 − " 2 2 = dt, P6 (s1 ) P6 (s2 )

  where P6 (s) = s −8s5 − 4as3 + 2b1 s + b2 , and the flow can be linearized in terms of genus 2 hyperelliptic functions. c) Let ϕ : A −→ C5 , (q1 , q2 , p1 , p2 ) −→ (z1 , z2 , z3 , z4 , z5 ), be a morphism on the affine variety A, where z1 , . . . , z5 are defined as z1 = q12 , z2 = q2 , z3 = p2 , z4 = q1 p1 , z5 = 2q12 q22 + p21 . Show that this morphism maps the vector field [8.16] into the system [7.57] in five unknowns z1 , z2 , z3 , z4 , z5 (we have seen in exercise 7.9, that the system3 [7.57] possesses three quartic invariants F1 , F2 , F3 [7.58] and the 3 affine variety B = k=1 {z : Fk (z) = ck } ⊂ C5 , defined by putting these invariants 2 more precisely the equal to generic constants, is the affine part of an Abelian surface B, Jacobian of a genus 2 curve and the system of differential equations [7.57] is algebraic 2 complete integrable and the corresponding flows evolve on B). d) Show that the invariant surface A can be completed as a cyclic double cover A of the Abelian surface B2 and the system [8.16] is algebraic complete integrable in the generalized sense. Also analyze the resolution A2 of A as a surface of general type with invariants to be specified (hint: see Lesfari (2020)).

9 The Korteweg–de Vries Equation

The Korteweg–de Vries (KdV) equation, which is a nonlinear partial differential equation of the third order, is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1 + 1 dimensions whose solutions can be exactly specified; it has a soliton-like solution or a solitary wave of sech2 form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and brief overview of this interesting subject. One of the objectives of this chapter is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand–Levitan equations) used to solve it. 9.1. Historical aspects and introduction Korteweg and de Vries established a nonlinear partial differential equation describing the gravitational wave propagating in a shallow channel (Korteweg and de Vries 1895) and possessing remarkable mathematical properties: ∂u ∂u ∂ 3 u = 0, − 6u + ∂t ∂x ∂x3

[9.1]

where u(x, t) is the amplitude of the wave at the point x and the time t. The equation thus bearing their name (abbreviated KdV) presents a solution: the soliton or solitary wave. This model was obtained from Euler’s equations (assuming irrotational flow) by Boussinesq around 1877 (see Boussinesq 1877, p. 360) and rediscovered by Korteweg and de Vries in 1890. The solution to this equation was obtained and interpreted rigorously only in the early 1970s, even though a solitary wave had already been observed in 1834 by the engineer Scott-Russell riding along the Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Edinburgh Glasgow Canal in Scotland; he described his observation of a hydrodynamic phenomenon as follows: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some 30-feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”. Fascinated by this phenomenon, Scott-Russell built a wave pool in his garden and worked to generate and study these waves more carefully. This led to a paper (Scott-Russell 1844) named “The report on waves” published in 1844 by the British Association for the Advancement of Science. A little later, Boussinesq, then Korteweg and de Vries, proposed equation [9.1] to explain this phenomenon. The KdV equation preserves mass, momentum, energy and many other quantities. Many experiments have uncovered the astonishing properties of the solutions of this equation satisfying zero boundary conditions: when |t| −→ ∞, these solutions are decomposed into solitons, that is, in to waves of defined forms progressing at different speeds. These waves propagate over long distances without deformation and one of the remarkable characteristics of solitons is that they are exceptionally stable with respect to ∂3u disturbances; the term u ∂u ∂x leads to shock waves while the term ∂x3 produces a scattering effect. Everyone can contemplate solitons where the tide comes to die on the beaches. In the field of hydrodynamics, for example, tsunamis (tidal waves) are manifestations of solitons. Generally, we group together under the term soliton, solutions of nonlinear wave equations presenting the following characteristic properties: they are localized in space, last indefinitely and retain their amplitude and velocity even at the end of several collisions with other solitons. Solitons have become indispensable for the study of several phenomena, in particular, the study of wave propagation in hydrodynamics and the study of localized waves in astrophysical plasmas. They are involved in the study of signals in optical fibers, charge transport phenomena in conductive polymers, localized modes in magnetic crystals, etc. Industrialized societies have developed, after soliton studies, what may be called solitary lasers. The latter play an important role in the field of telecommunications. Ultra-short light signals sent in certain optical fibers made from a specific material can travel long distances without lengthening or fading. The construction of memories with ultra-fast communication time and low energy consumption is based on the movement of magnetic vortices in the dielectric junction between two superconductors. At the molecular level, the theory of solitons can

The Korteweg–de Vries Equation

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elucidate the contraction mechanism of striated muscles, the dynamics of biological macromolecules such as DNA and proteins. In the peptide and hydrogen chain of proteins, solitons arise from the marriage of dispersion due to intrapeptide vibrations and the nonlinearity due to the interaction of these vibrations with the displacements of peptide groups around their position being balanced. But the theory of solitons has also had an impact on pure mathematics; for example, it provides the answer to the famous Schottky problem, posited a century ago, on the relationships between the periods coming from a Riemann surface. Roughly, it is a question of finding criteria so that a matrix of the periods belonging to the Siegel half-space is the matrix of the periods of a Riemann surface. Geometrically, Schottky’s problem consists of characterizing the Jacobians among all the Abelian mainly polarized varieties. In addition to the KdV equation, examples that may be mentioned among the nonlinear equations having soliton-type solutions are  as follows: 3 the nonlinear 2 ∂ ∂u ∂ u Kadomtsev–Petviashvili equation: ∂∂yu2 − ∂x 4 ∂u = 0, the − 12u − ∂t ∂x ∂x3 2

∂ ψ 2 nonlinear Schrödinger equation: i ∂ψ ∂t + ∂x2 + |ψ| ψ = 0, the Sine Gordon equation: 2 2 ∂ u ∂ u ∂2u ∂2u ∂4u ∂ 2 u2 ∂t2 − ∂x2 + sin u = 0, the Boussinesq equation: ∂t2 − ∂x2 + ∂x4 + ∂x2 = 0, the ∂3u ∂u ∂u ∂ 2 u ∂3u ∂u Camassa–Holm equation: ∂u ∂t − ∂t∂x2 + 3u ∂x = 2 ∂x ∂x2 + u ∂x3 − 2α ∂x , α ∈ R, dxj the Toda lattice (previously studied) described by a system: = yj , dt dyj xj −xj+1 xj−1 −xj = −e + e , consisting of vibrating masses arranged on a circle dt and interconnected by springs whose return force is exponential. Solitons have appeared in many other fields; in particular the nonlinear Klein Gordon equation, the Zabusky–Kruskal equation for the Fermi–Pasta–Ulam model of phonons in anharmonic lattice, and so on.

9.2. Stationary Schrödinger and integral Gelfand–Levitan equations Since the method (used later) of solving the KdV equation is based on the idea of studying it in the form of an equation of a certain operator and using the analogy with quantum mechanics, we will expose certain mathematical notions of this mechanic. The terminology of physicists will be used to describe the properties of the solutions of the stationary Schrödinger equation,   ψ + (λ − u(x))ψ = 0, 2m



≡

d , dx

without stopping on the physical motivations of the introduced notions. We will see that the method of the inverse diffusion is reduced to the solution of a linear integral equation (Gelfand–Levitan equation). In the following, we will simplify the notation by using a system of units in which the Planck constant is  = 1 and the mass of the particle is m = 12 . So consider the equation ψ  + (λ − u(x))ψ = 0,

−∞ < x < ∞

[9.2]

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where ψ (unknown) is the wave function of the particle, the spectral parameter λ is the energy of the particle, the function u(x) is the potential or potential energy of the particle. This potential is assumed to have a compact support, that is, it is different from zero only in some domain. When the particle is free (i.e. u = 0) and has a positive energy (i.e. λ = k 2 ), then equation [9.2] is reduced to ψ  (x) + k 2 ψ = 0,

[9.3]

and admits two linearly independent solutions eikx (describing the particle moving to the right) and e−ikx (describing the particle moving to the left). Let us denote rs cs by E(2) (respectively, E(2) ) the space of the real (respectively, complex) solutions rs cs of equation [9.2] and by E(3) (respectively, E(3) ) the space of the real (respectively, cs complex) solutions of equation [9.3]. The space E(2) (of states of the particle) is the rs complexification of E(2) and all four states of the particle arriving and departing to cs rs cs the right and left belong to the space E(2) . The space E(3) (respectively, E(3) ) has the ikx −ikx following natural basis (cos kx, sin kx) (respectively, (e , e )). Let [α, β] be the bounded support of u. The monodromy operator, denoted by M, of equation [9.2] with a potential of compact support is a linear operator mapping the state space of a free particle with energy λ = k 2 into itself. It is defined in the following way: to a solution of equation [9.3] of a free particle, we assign a solution of the Schrödinger equation coinciding with it to the left of the support, and to this solution, in turn, we assign its value to the right of the support.

Figure 9.1. Particle moving

(We denote by (1) particle moving from left, (2) particle moving to left, (3) particle moving to right and (4) particle moving from right.) This figure does not reflect reality but it is a good approximation. In Figure 9.1, case (i) corresponds to equation [9.3] whose solution is a cos kx + b sin kx where a, b are constants. Case (ii) corresponds to equation [9.2] whose solution is a cos kx + b sin kx if x < α, solution of equation

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[9.2] if α ≤ x ≤ β and c cos kx + d sin kx if x > β where (c, d) = Mu (a, b). So we have, rs rs M : E(3) −→ E(3) ,

 a cos kx + b sin kx −→

a cos kx + b sin kx if x < α c cos kx + d sin kx if x > β

where a, b are constants and (c, d) = Mu (a, b). This means that for each solution of equation [9.3] is associated: (•) the solution of [9.2] which is to the left of α; in this region the solution of [9.3] coincides with that of [9.2], and (••) the solution of [9.2] which is to the right of β. Similarly, the complex monodromy operator of equation [9.2] is defined by M:

cs E(3)

−→

cs E(3) ,

ae

ikx

+ be

−ikx

 −→

aeikx + be−ikx if x < α ceikx + de−ikx if x > β

T HEOREM 9.1.– Let W be the phase plane formed by the representative points x1 rs (i.e. pairs of real numbers) (ψ, ψ  ). Let B(2) : E(2) −→ W, x1  ψ −→ B(2) ψ = (ψ(x1 ), ψ (x1 )), be an operator with ψ a solution of equation [9.2] rs whose initial conditions for x = x1 ∈ R are (ψ(x1 ), ψ  (x1 )). Then, the space E(2) is isomorphic to W and  −1 x1 x2 gxx12 ≡ B(2) B(2) : W −→ W,

(ψ(x1 ), ψ  (x1 )) −→ (ψ(x2 ), ψ  (x2 )),

is a linear isomorphism. x1 is linear. In addition, for any representative point P ROOF.– It is clear that B(2) (ψ, ψ  ) ∈ W , there exists from the existence theorem a solution ψ satisfying the  x1 x1  rs initial condition (ψ(x1 ), ψ (x1 )). Then Im B(2) ≡ B(2) ψ : ψ ∈ E(2) = W .   x1 x1 rs Finally, Ker B(2) ≡ ψ : ψ ∈ E(2) , B(2) ψ = 0 = 0 follows from the uniqueness theorem because the solution satisfying the initial condition at the point x1 is equal to zero. The fact that gxx12 is a linear isomorphism follows from the fact that the inverse of an isomorphism is one. More specifically, if ψ1 and ψ2 are two solutions of equation [9.2], then x1 x1 x1 (ψ(x1 ), ψ  (x1 )) = B(2) ψ = B(2) ψ1 + B(2) ψ2 = (ψ1 (x1 ), ψ1 (x1 ))

+(ψ2 (x1 ), ψ2 (x1 )), and this is equivalent to 

x1 B(2)

−1

((ψ1 (x1 ), ψ1 (x1 )) + (ψ2 (x1 ), ψ2 (x1 )))

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−1  x1 = B(2) (ψ(x1 ), ψ  (x1 )) = ψ = ψ1 + ψ2 , −1 −1   x1 x1 = B(2) (ψ1 (x1 ), ψ1 (x1 )) + B(2) (ψ2 (x1 ), ψ2 (x1 )). This completes the proof.  The isomorphism gxx12 is called the phase transformation from x1 to x2 .

Figure 9.2. Phase transformation x1 rs In the same way, we can define an operator B(3) of E(3) in W that associates with each solution of equation [9.3], its initial condition at the point x1 . In this case, instead of “phase transformation”, there will be “phase point”. A particle propagating from x = −∞ crosses a potential barrier with a transmission coefficient T and a reflection coefficient R if the equation [9.2] where λ = k 2 admits a solution ψ such that: % T eikx , to the right of the barrier ψ= ikx e + Re−ikx , to the left of the barrier

T HEOREM 9.2.– If equation [9.2] where λ = k 2 has a confounded solution with aeikx for x # 0 and with be−ikx for x + 0, then this solution is null. In addition, for all k > 0 the ψ, T and R defined above exist and are unique. cs P ROOF.– Consider in E(2) the Hermitian forms aeikx , aeikx , be−ikx , be−ikx  and ikx −ikx . Let us designate by [., .] the left scalar product1, then ae , ae   i ikx −ikx i  a ia  ikx ikx = |a|2 . ]=  ae , ae  = [ae , ae 2 2 a −ia 

Similarly, we have be−ikx , be−ikx  = −|b|2 and aeikx , ae−ikx  = 0. Therefore, by setting z = z1 eikx + z2 e−ikx where z1 and z2 are the coordinates of the vector 1 [ξ, η] is the oriented area of the parallelogram constructed on the vectors: ξ = ξ1 e1 + ξ2 e2 , η = η1 e1 + η2 e2 of the real  plane where (e1 , e2 ) is a fixed basis in which [e1 , e2 ] = 1. We  ξ1 ξ2  .  show that [ξ, η] =  η1 η2 

The Korteweg–de Vries Equation

247

z in the basis (eikx , e−ikx ), we obtain z, z = |z1 |2 − |z2 |2 , that is, ., . is of type (1, 1). Since the monodromy operator retains this Hermitian form, we deduce that |a|2 = −|b|2 and so a = b = 0. Consider now a particle going to +∞ and let eikx be a solution to the right of the barrier. To the left of the barrier this solution becomes eikx  aeikx + be−ikx .

[9.4]

From what precedes, the coefficient a is non-zero. To get the solution in question, simply divide the two members of [9.4] by a, a1 eikx  eikx + ab e−ikx . Taking T = a1 , R = ab , this shows that T and R are uniquely defined.  We will now demonstrate the Liouville theorem, which will be useful later. T HEOREM 9.3.– Let dx dt = f (x), x = (x1 , ..., xn ), be a system of differential equations whose solutions extend to the whole time axis. Let {g t } be the corresponding group of transformations: g t x = x + f (x)t + o(t2 ), for t small. We denote by D a domain in phase space, D(t) ≡ g t D(0) and by v(t) the volume of

n ∂f D(t). If div f = j=1 ∂xjj = 0, then v(t) = v(0), that is, g t preserves the volume of any domain. P ROOF.– We have v(t) = ∂g t x ∂x

D(t) 2

dx =

 D(0)

∂g t x ∂x dx,

where

∂g t x ∂x

is the Jacobian

matrix, =I+ + o(t ). The determinant of the operator I + ∂f ∂x t is equal to the product of the eigenvalues. These (taking into account their multiplicities) are ∂f ∂f equal to 1 + t ∂xjj where ∂xjj are the eigenvalues of ∂f ∂x . Then det

∂f ∂x t



n  ∂fj ∂g t x + o(t2 ) = 1 + t div f + o(t2 ). =1+t ∂x ∂x j j=1

Therefore,  + dv(t)  v(t) = (1 + t div f + o(t ))dx =⇒ = div f dx. dt t=0 D(0) D(0) +

2



Since t = t0 is not worse than t = 0, we also have

dv(t)  dt 

= t=t0

 D(t0 )

div f dx. 

R EMARK 9.1.– Liouville’s theorem is easily generalized to the case of non-autonomous systems (f = f (x, t)). Indeed, the terms of first degree in the t expression of ∂g∂xx remain the same. But the terms of degree greater than one do not intervene in the demonstration. In other words, Liouville’s theorem is a first-order theorem.

248

Integrable Systems

Let SL(2, R) be the real unimodular group, that is, the set of all real 2 × 2 matrices with determinant one. In other words, SL(2, R) is the group of all linear transformations of R2 that preserve oriented area [., .] (see the notation used in the proof of theorem 9.2). Consider the group SU (1, 1) of (1, 1)-unitary unimodular matrices. This is the set of all complex 2 × 2 matrices with determinant one preserving the Hermitian form |z1 |2 − |z2 |2 (see the notation used  in the proof of a b theorem 9.2). In other words, they are matrices of the form for which c d |a|2 − |b|2 = |c|2 − |d|2 = 1, ac − bd = 0, ad − bc = 1. T HEOREM 9.4.– In the basis (cos kx, sin kx) (respectively, (eikx , e−ikx )), the matrix of the monodromy operator M belongs to the group SL(2, R) (respectively, SU (1, 1)). P ROOF.– We show that the determinant of the monodromy operator of the Schrödinger equation is equal to one. Note that (cos kx, sin kx) is a basis on the sr x x space E(3) . As B(3) cos kx = (cos kx, −k sin kx), B(3) sin kx = (sin kx, k cos kx), so W is provided with a basis in which the matrix of the operator (we use here the same notation for the operator and the matrix) is written as cos kx sin kx x x B(3) = , hence det B(3) = k, independent of x. Let us denote −k sin kx k cos kx by x+ the point x to the left of the support of the potential and by x− the one on the right. We have the following situation: sr sr M : E(3) −→ E(3) ,

a cos kx + b sin kx −→ c cos kx + d sin kx,

(c, d) = Mu (a, b),

−

x sr : E(3) B(3) −→ W,

a cos kx + b sin kx −→ (a cos kx− + b sin kx− , −ak sin kx− + bk cos kx− ), +

x sr : E(3) −→ W, B(3)

c cos kx + d sin kx −→ (a cos kx+ + b sin kx+ , −ak sin kx+ + bk cos kx+ ), +

gxx− : W −→ W, (a cos kx− + b sin kx− , −ak sin kx− + bk cos kx− )

−→ (a cos kx+ + b sin kx+ , −ak sin kx+ + bk cos kx+ ). +

−

+

+

−

x x x x = B(3) oM, and since det B(3) = B(3) , so we We verify directly that: gxx− oB(3) +

have det M = det gxx− . Now g x preserves the areas according to Liouville’s theorem 9.3. Indeed, by putting ψ1 = ψ, ψ2 = ψ  , we rewrite equation [9.2] in the form ψ1 = ψ2 ≡ f1 , ψ2 = (u(x) − λ)ψ1 ≡ f2 . Here, we have f = (f1 , f2 ), t = x

The Korteweg–de Vries Equation

249

+

∂(u(x)−λ)ψ1 2 and div f = ∂ψ = 0. Therefore, det gxx− = 1 and consequently ∂ψ1 + ∂ψ2 det M = 1. For the case of SU (1, 1), we will show that the matrix (also denoted M) of an operator is real and unimodular in the basis (cos kx, sin kx) if and only if it is special (1, 1)-unitary in complex conjugate basis (eikx , e−ikx ). By setting as in the proof of theorem 9.2, z = z1 eikx + z2 e−ikx where z1 and z2 are the coordinates of the vector z in the basis (eikx , e−ikx ), we obtain z, z = |z1 |2 − |z2 |2 , that is, ., . is of type (1, 1). The monodromy operator conserves this Hermitian form. Say that M is real and unimodular in the basis (cos kx, sin kx) which is equivalent to M ∈ GL(2, R) ∩ SL(2, C) or what amounts to the same M ∈ SU (1, 1) or what is equivalent M is (1, 1)-unitary and unimodular in the basis (eikx , e−ikx ). 

R EMARK 9.2.– It is well known that the sum of the transmission and reflection coefficients is equal to one. This property of the Schrödinger equation can be obtained as a corollary of the preceding theorem and thus without the use of the theory of probabilities. Indeed, we use the definition of the monodromy operator, eikx + Re−ikx −→ T eikx , e−ikx + Reikx −→ T e−ikx . Divide the first expression by T = 0 and the second by T = 0, we get  ikx 1 R   ikx e e T T

−→ . −ikx R 1 e−ikx e T T

ikx −ikx So in the basis (e ,e ), the matrix of the inverse of the monodromy operator

is M−1 =

1 R T T R 1 T T

, and therefore, the matrix of the monodromy operator in the 

1 R − ikx −ikx T T . According to theorem 9.4, we have M ∈ ) is M = basis (e , e 1 −R T T SU (1, 1) (det M = 1) and consequently |T |2 + |R|2 = 1. Define the solutions ψ1 (x, λ) and ψ2 (x, λ) of equation [9.2] by the initial conditions: ψ1 (0, λ) = 1, ψ1 (0, λ) = 0, and ψ2 (0, λ) = 0, ψ2 (0, λ) = 1. For the simple case u(x) = 0, we have = √    1 2 4   ψ1 (x, λ) = cos λx = 1 + − 12λ x2+ 24 λ O x6 ,  x + √   [9.5] 1 λ2 x5 + O x7 . ψ2 (x, λ) = √1λ sin λx = x + − 16 λ x3 + 120

√ √ √ θ For λ, we choose, for example, the determination λ = rei 2 where λ = reiθ , r > 0, −π < θ < π. Let α be an arbitrary real number. The function ψ(x, λ) = ψ1 (x, λ) + αψ2 (x, λ) is also solution of equation [9.2] and satisfies the boundary condition ψ  (0, λ) − αψ(0, λ) = 0. For α = 0, we have ψ(x, λ) = ψ1 (x, λ) and for α = ∞, we put ψ(x, λ) = ψ2 (x, λ). We assume that for λ ∈ C and x ≥ 0, we have + x √ √ ψ(x, λ) = cos λx + K(x, t) cos λtdt, [9.6] 0

250

Integrable Systems

where K is to be determined, subject to the condition of having partial derivatives of order one and order two continuous in the set of real pairs (x, t) such that: 0 ≤ t ≤ x. In other √ words, we look for ψ(x, .) as a perturbation of the function x → ψ(x, λ) = cos λx and, precisely, as a transform (I + K)ψ1 (x, .) where K is a Volterra operator in [0, +∞[. We look for the conditions that K(x, t) must satisfy for the function [9.6] to be a solution of the differential equation [9.2]. From equation [9.6], we get √ √ √ √ ∂2ψ dK(x, x) (x, λ) = −λ cos λx + cos λx − λK(x, x) sin λx 2 ∂x dx  + x 2 √ √ ∂K(x, t)  ∂ K(x, t) + cos λx + cos λtdt. [9.7]  2 ∂x ∂x t=x

0

√ Let us calculate the expression λ 0 K(x, t) cos λtdt, by doing two integrations in parts, we get + x √ √ √ λ K(x, t) cos λtdt = λK(x, x) sin λx x

0

  √ ∂K(x, t)  ∂K(x, t)  + cos λx − ∂t t=x ∂t t=0 + x 2 √ ∂ K(x, t) − cos λtdt. ∂t2 0

[9.8]

To calculate expression [9.2], substitute [9.7] and [9.8], 0 = ψ  + (λ − u(x))ψ  √ √ ∂K(x, t) ∂K(x, t) dK(x, x) cos λx = cos λx + + dx ∂t ∂x x=t  √  ∂K(x, t)  − u(x) cos λx − ∂t t=0 + x 2 √ ∂ K(x, t) ∂ 2 K(x, t) + − − u(x)K(x, t) cos λtdt. ∂x2 ∂t2 0 We have ∂ 2 K(x, t) ∂ 2 K(x, t) − u(x)K(x, t) = , ∂x2 ∂t2 with the boundary conditions  ∂K(x, t)  = 0, ∂t t=0 dK(x, x) 1 = u(x). dx 2

[9.9]

[9.10] [9.11]

The Korteweg–de Vries Equation

251

For the initial conditions, we have ψ(0, λ) = 1 and ψ  (0, λ) = K(0, 0). As ψ (0, λ) − αψ(0, λ) = 0, then K(0, 0) = α. Therefore, + 1 x K(x, x) = α + u(t)dt. [9.12] 2 0 

If u(x) has a continuous derivative, then there exists a unique solution of [9.9], satisfying conditions [9.10] and [9.12]. Hence, there exists a satisfying function K(x, t)[9.6]. Let us solve equation [9.6] as an equation of Volterra, we get + x √ cos λx = ψ(x, λ) − K1 (x, t)ψ(t, λ)dt, [9.13] 0

and in the same way as before, we show that K1 (x, t) is solution of the equation ∂ 2 K1 (x, t) ∂ 2 K1 (x, t) = − u(t)K1 (x, t), 2 ∂x ∂t2  1  with the conditions ∂K ∂t − αK1 t=0 = 0, K1 (x, x) = α +

1 2

x 0

u(t)dt.

For the case α = √ ∞, we look for ψ(x, λ) as a perturbation of the function x → ψ(x, λ) = √1λ sin λx (see expression [9.5]) or what is equivalent as a transform (I + K)ψ1 (x, .) where K is a Volterra operator in [0, +∞[. In other words, we set λ ∈ C and x ≥ 0, √ √ + x sin λx sin λx ψ(x, λ) = √ L(x, t) √ + dt, [9.14] λ λt 0 where L is a function to be determined, subject to the condition of having partial derivatives of order one and order two continuous in the set of real pairs (x, t) such that: 0 ≤ t ≤ x. By reasoning as before, we obtain the relation ∂ 2 L(x, t) ∂ 2 L(x, t) − u(x)L(x, t) = , 2 ∂x ∂t2 x with the conditions: L(x, x) = 0, L(x, x) = 12 0 u(t)dt. By solving equation [9.14], we obtain √ + x sin λx √ L1 (x, t)ψ(t, λ)dt. [9.15] = ψ(x, λ) + λ 0 The functions L(x, t) and L1 (x, t) have the same properties as the functions K(x, t) and K1 (x, t) previously obtained. Recall that for every function ∞ ∞ f (x) ∈ L2 (R), we have the Parseval identity, 0 f 2 (x)dx = −∞ F 2 (λ).dρ(λ), ∞ where F (λ) = 0 f (x)ψ(x, λ)dx, is the Fourier transform of f (x) and ρ(λ) a

252

Integrable Systems

monotone function, bounded on any finite interval. The sequence of functions n Fn (λ) = 0 f (x)ψ(x, λ)dx converges in quadratic mean (with respect to the ∞ spectral measure ρ(λ)) to F (λ), that is, limn→∞ −∞ (F (λ) − Fn (λ))2 dρ(λ) = 0. √ We choose ρ(λ) in the form ρ(λ) = π2 λ + σ(λ) if λ > 0, and ρ(λ) = σ(λ) if  xthe condition: λ ∞< 0, where σ(λ) is a measure with compact support satisfying |λ|.|dσ(λ)| < +∞. For 0 < b < y < a < x, the functions ψ(t, λ)dt and a √ −∞ y In other words, we have the cos λtdt are orthogonal with respect to ρ(λ). b √   y  ∞  x orthogonality relation: I ≡ −∞ a ψ(t, λ)dt cos λtdt dρ(λ) = 0. Indeed, b by integrating equation [9.13] from b to y, we obtain +

y

cos

√

+

+

y

b

+

y

ψ(t, λ)dt −

λtdt = +

b

+

y b

+

+

b

y

ψ(s, λ)ds +

y

−

K1 (t, s)ψ(s, λ)ds, 0

b

ψ(t, λ)dt −

=

t

dt

K1 (t, s)dt

0

b y

ψ(s, λ)dt

K1 (t, s)dt.

b

s

By definition, this function is expressed using the transform (in ψ(t, λ)) of a null function outside the interval ]b, y[. Since ]b, y[∩]a, x[= ∅, we deduce from Parseval’s equality that I = 0. To obtain the Gelfand–Levitan integral equation, we proceed as follows: according to equation [9.6], we have +

+

x

x

ψ(t, λ)dt =

cos

a

+

√

+

cos

=

√

t

dt

a x

+

x

λtdt + +

K(t, s) cos 0

a a

λtdt +

cos

√

+

+

x

cos

+

√

+

K(t, s)dt a

x

λsds

a

λsds,

x

λsds

0

a

√

K(t, s)dt, s

by virtue of Lebesgue–Fubini’s theorem. Therefore, +

+

∞

x

I =

cos −∞

+

+ λtdt

a

∞

+

a

+

cos −∞

+

y

× b

= 0.

√

0

√

+

y

cos

√

λtdt dρ(λ)

b

+

x

λsds

x

K(t, s)dt + a

√ cos λtdt dρ(λ),

cos a

√

+

x

λsds s

K(t, s)dt

The Korteweg–de Vries Equation

253

This expression can be written using the definition of ρ(λ) in the form +

+

∞

x

cos

I = −∞

+

√

a

+

∞

a

+

cos −∞

+

0

y

×

cos b

2 π

+

+ λtdt

+

−∞

+

λtdt dσ(λ)

b

+

+

x

λsds

x

√

cos

y

cos

√

+



x

λsds

K(t, s)dt s

λtdt dσ(λ)

b

+

a

√

a

+ λtdt

a

x

K(t, s)dt +

√ 2 + cos λsds π −∞ 0 + y √ cos λtdt dσ(λ), × ∞



λtdt dσ(λ) cos

+

cos

√

a

√

+

∞

√

y

+

x

x

K(t, s)dt + a

cos

√

+

x

λsds

a

K(t, s)dt

s

b

= 0. Since b < y < a < x, then given the Parseval identity, the third term is equal to zero while the fourth is equal to +

y

+

a

cos

√

+

0

b

+

+

x

K(t, s)dt + a

y

cos

√

+

x

λsds

a

K(t, s)dt ds

s

x

ds

=

+

x

λsds

K(t, s)dt.

b

a

Therefore, √ √ √ √ (sin λx − sin λa)(sin λy − sin λb) I = dσ(s) λ −∞ + x + x + + ∞ + a √ √ cos λsds K(t, s)dt + cos λsds + +

∞

−∞

+ ×

cos +

+

b y

+

a

√

λsds dσ(λ)

x

ds b

= 0.

0

y

K(t, s)dt, a

a

x s

K(t, s)dt

254

Integrable Systems

By setting + F (x, y) ≡

∞

sin

√

−∞

λx sin λ

√

λy

dσ(λ),

⎧x ⎨ a K(t, s)dt, 0 ≤ s ≤ a x G(x, s) ≡ K(t, s)dt, a ≤ s ≤ x ⎩ s 0, s > x

the equation above becomes F (x, y) − F (x, b) − F (a, y) + F (a, b) + + ∞ + x √ + G(x, s) cos λsds + +

−∞

+

y

cos

√

λsds dσ(λ)

b

x

ds b

0

y

K(t, s)dt = 0. a

This last equation can still be written, doing an integration by parts and noticing that G(x, x) = 0, F (x, y) − F (x, b) − F (a, y) + F (a, b)  √  √ √ + ∞ + x ∂G(x, s) sin λs sin λy − sin λb √ √ dσ(λ) + ds ∂s λ λ −∞ 0 + y + x + ds K(t, s)dt = 0. [9.16] b

a

But +

∞ −∞

+

x

= 0

+

 √  √ √ ∂G(x, s) sin λs sin λy − sin λb √ √ dσ(λ), ds ∂s λ λ 0

+ 

√ √ √ √  ∞ sin λs sin λy − sin λs sin λb ∂G(x, s) dσ(λ) ds, ∂s λ −∞

+

x

x

∂G(x, s) (F (s, y) − F (s, b)) ds, ∂s 0  + x ∂F (s, y) ∂F (s, b) ds, G(x, s) =− − ∂s ∂s 0 + x + a ∂F (s, y) ∂F (s, b) ds =− K(t, s)dt − ∂s ∂s 0 a

=

The Korteweg–de Vries Equation

255

+ x ∂F (s, y) ∂F (s, b) ds K(t, s)dt , − ∂s ∂s a s + x + t ∂F (s, y) ∂F (s, b) ds, = dt − ∂s ∂s a 0 +

x



−

so equation [9.16] becomes + +

0

∂F (s, y) ∂F (s, b) − ∂s ∂s

ds

x

ds b

dt a

+

y

+

+ t

x

F (x, y) − F (x, b) − F (a, y) + F (a, b) + K(t, s)dt = 0. a

Deriving this expression with respect to y and then with respect to x (the support of the measure σ is compact), we obtain ∂2F + ∂x∂y

+

x

K(x, s) 0

∂ 2 F (s, y) + K(x, y) = 0. ∂s∂y 2

∂ F , we finally obtain the Gelfand–Levitan integral By setting f (x, y) ≡ ∂x∂y equation for the function x −→ K(x, y) valid for 0 < y < x, + x f (x, y) + K(x, y) + K(x, s)f (s, y)ds = 0, y ≤ x. [9.17] 0

For the case α = ∞, that is, ψ(x, λ) = ψ2 (x, λ), just integrate the two members of equation [9.15] from 0 to x and use a similar reasoning. Under the continuity assumption of K, equation [9.17] must be checked for x = 0 and x = y. Note also that if we set x in the previous equation, then we will obtain the so-called Fredholm linear integral equation. We can prove that, conversely, equation [9.17] gives a single continuous solution in the set of pairs of real numbers such that: 0 ≤ t ≤ x. We will not look for the solution at this level, it will be done later (in the next section) when we treat the KdV equation. 9.3. The inverse scattering method Let us first examine some particular solutions of the KdV equation [9.1], of the kind of progressive waves u(x, t) = s(x − ct), where c is the phase velocity. By ∂s ∂s ∂3s replacing this expression in [9.1], we obtain −c ∂x − 6s ∂x + ∂x 3 = 0. By integrating this equation with respect to x and imposing the boundary condition that s and its ∂2s derivatives decrease for |x| −→ ∞, we get −cs − 3s2 + ∂x 2 = 0, hence −cs −   ∂s 2 3 2s + ∂x = 0, and the exact expression of the solution s requires the use of elliptic

256

Integrable Systems

∂s functions. Suppose that ∂x (0) = 0, in which case the solution of this last equation √ 2 c c is s(x − ct) = − 2 sech 2 (x − ct), where sech denotes the hyperbolic secant, that 1 is, cosh . Therefore, u(x, 0) = u0 sech2 xl , u0 ≡ − 2c , l2 ≡ 4c . This expression shows that u is removed for an infinitely long time in the position u 0, then it reaches the value u0 , is reflected on this point and returns again in the position of u 0. This solution is called a soliton. To obtain this solution, we can use the so-called Bäcklund transformations for the KdV equation. When solitons collide, dimensions and speeds of solutions do not change after collision. This remarkable phenomenon has suggested the idea of conservation laws. And indeed, Kruskal, Zabusky, Lax, Gardner, Green and Miura (Gardner et al. 1967; Lax 1968) have been able to find a whole  series offirst integrals for the KdV equation. These integrals are of the form Pn u, ..., u(n) dx, where Pn is a polynomial. Indeed, the conservation equations that can be deduced ∂Qn n from the KdV equation take the following general form: ∂P ∂t + ∂x = 0, where Pn and Qn form a series of functions of which we discuss the first three as follows: (i) ∂ ∂2u 2 The KdV equation can be written in the form ∂u = 0. Hence, ∂t + ∂x −3u + ∂x2

P1 = u, Q1 = −3u2 + u

∂2u ∂x2 .

(ii) Multiply the KdV equation by u, this gives

∂u ∂u ∂3u − 6u2 + u 3 = 0, ∂t ∂x ∂x

∂ ∂t



u2 2



Hence, P2 =

∂ + ∂x u2 2 ,

∂2u 1 −2u3 + u 2 − ∂x 2 2

Q2 = −2u3 + u ∂∂xu2 −



1 2

∂u ∂x

2 

 ∂u 2 ∂x

= 0.

. (iii) We have

  ∂u ∂2u ∂u ∂ 3 u 2 3u − = 0, − 6u + ∂x2 ∂t ∂x ∂x3  ∂u ∂ 2 u 2 ∂u + 3u + ∂t ∂x ∂x∂t  3 ∂u ∂ 2 u ∂ 2 u ∂ 3 u ∂ 2 u ∂u ∂u ∂ 2 u 3 ∂u 2∂ u −18u = 0. + 6u − − + 3u − ∂x ∂t3 ∂x ∂x2 ∂x2 ∂x3 ∂x2 ∂t ∂x ∂x∂t Therefore, ∂ ∂t

1 u + 2 3



∂u ∂x

2 

∂ + ∂t

9 ∂2u 1 − u4 + 3u2 2 − 2 ∂x 2



∂2u ∂x2

2

∂u ∂u − ∂x ∂t

 = 0.

 2 9 4 1 ∂2u ∂u 2 ∂2u , Q = − u + 3u − − ∂u 2 2 3 ∂x 2 ∂x 2 ∂x ∂x ∂t .   ∂ If u vanishes for x → ∞, we get ∂t Pn dx = 0, then Pn dx are first integrals of Consequently, P3 = u3 +

1 2

 ∂u 2

The Korteweg–de Vries Equation ∂y ∂y ∂ 3 y that ∂y , ∂x , ∂t3 decay ∂x (x, t) and suppose ∂t  2 ∂y ∂y ∂3y written as ∂t − 3 ∂x + ∂t3 = 0. Hence,

the KdV equation. Let u(x, t) = |x| → ∞. The KdV equation is ∂ ∂t

+



∞

y(x, t)dx = 3 −∞

Since u = ∂ ∂t

+

∞

+

−∞ ∂y ∂x ,

2

+

∞

(x, t)dx = 3

when

u2 (x, t)dx = constant.

−∞

we also have

∞

y(x, t)dx = −∞

∂y ∂x

257

∂ ∂t

+

∞

+

x

u(z, t)dzdx, −∞

−∞

∞ + ∞  ∂ u(z, t)dz  − u(z, t)dx, ∂t −∞ −∞ −∞ + ∞ ∂ xu(x, t)dx, = − ∂t −∞

∂ = x ∂t

+

x

2

because by hypothesis u2 and ∂∂xu2 tend to 0 when |x| → ∞. Comparing the two ∞ ∂ expressions obtained, we obtain a new first integral ∂t xu(x, t)dx = constant. −∞ Lax (1968) showed that the equation of KdV is equivalent to the following ∂2 equation: dA = [B, A] = BA − AB, where A = − ∂x 2 + u(x, t), dt ∂3 ∂ ∂ (Sturm–Liouville operator), B = −4 ∂x3 + 6u ∂x + 3 ∂x . We deduce that the spectrum of A is conserved: if A is a symmetric operator (A = A) and T an orthogonal transformation (T = T −1 ), then the spectrum of T −1 AT coincides with that of A. The appearance of an infinite series of first integrals is easily explained by the Lax equation. The Sturm–Liouville equation Aψ = λψ, where λ is a real parameter, can be written in the form ∂2ψ + (λ − u(x, t))ψ = 0. ∂x2

[9.18]

This equation reminds us of the unidimensional and stationary Schrödinger equation. In the following, we will see that the complete solution of the KdV equation is closely related to the solution of this equation. We will look at solutions for which u decreases fast enough for x −→ ±∞. It should be noted that there are other interesting conditions to know: the case where u(x, t) tends to different constants for |x| −→ ∞ and the one where u(x, t) is periodic in x. So consider equation [9.18] where u(x, t) is the solution of the KdV equation [9.1]. It is assumed that after a certain time, equation [9.18] has N bound states with energy λn = −kn2 , n = 1, 2, ..., N and continuous states with for energy λ = k 2 . We draw u from

258

Integrable Systems

equation [9.18] and replace it in equation [9.1]. After a long calculation, after multiplying by ψ 2 , we get the expression  ∂Υ ∂ψ ∂λ 2 ∂ ψ .ψ + − Υ = 0, [9.19] ∂t ∂x ∂x ∂x 3

∂ ψ ∂ψ where Υ ≡ ∂ψ ∂t + ∂x3 − 3(u + λ) ∂x . For the study of the discrete part of the spectrum λn (t) = −kn2 (t), we show the following result: n T HEOREM 9.5.– If ψn (measurable and square integrable function) and ∂ψ ∂x tend to zeros when |x| goes to infinity, then λn (t) = constant and the solution of equation 2 [9.18] is given by ψn (t) = cn (0)ekn (x−4kn t) , where cn (0) is determined by the initial condition u(x, 0) = u0 (x) of the KdV equation. ∞ 2 ∂ψn ∂Υ n P ROOF.– Just integrate equation [9.19], this gives ∂λ ∂t . −∞ ψn dx+ψn ∂x − ∂x Υ = n 0. By hypothesis, ψn ∈ L2 and ψn , ∂ψ ∂x tend to zero when |x| goes to infinity, so ∂ψn ∂Υ ψn ∂x − ∂x Υ tends to 0 for |x| → ∞ and we deduce  that λn (t) = constant.

∂ψ ψ ∂Υ ∂x − ∂x Υ = 0. Let us   ∂ψ (ψ ∂Υ ∂x − ∂x Υ) integrate this expression twice, = ψA2 , that is, Υ = ψA2 , hence, Υ = ψ2 ψ  A(t) ψ ψ2 dx + B(t)ψ, where A(t) and B(t) are integration constants. So we have

Now, since

∂λ ∂t

= 0, then equation [9.19] becomes

∂ψn ∂ψn ∂ 3 ψn − 3(u + λn ) + = ψn ∂t ∂x3 ∂x

+

∂ ∂x

An dx + Bn ψn . ψn2

[9.20]

Note that An (t) = 0 because ψn satisfies [9.20] and decreases to zero for t → −∞. Let us consider u ∼ = 0 for x → −∞ because otherwise ψn would not have the decay assumption. Multiply [9.20] by ψn and integrate +

∞ −∞

ψn

∂ψn dx + ∂t

+

 + ∞ ∂ 3 ψn ∂ψn dx = B ψn − 3λ ψ ψn2 dx. n n n ∂x3 ∂x −∞ −∞ ∞

This expression can be written by adding and subtracting

∂ψn ∂ 2 ψn ∂x ∂x2 ,

 2  + ∞ ∂ψ 3 1 1 ∂ψn2 ∂ ∂ 2 ψn n dx + ψn dx − λn ψn2 − ∂x2 2 2 ∂x −∞ ∂x −∞ 2 ∂t + ∞ ψn dx. = Bn +

∞

−∞

We have Bn (t) = 0 because ψn ∈ L2 and decreases to zero when x → −∞. Since u ∼ = 0 for x → −∞, then from equation [9.18], it comes ψn (x, t) = cn (t)ekn x ,

The Korteweg–de Vries Equation

259

 n  k x 3 n x → −∞. By replacing the latter in equation [9.20], we obtain ∂c = ∂t + 4cn kn e 3 2 −4kn t kn (x−4kn t) 0, hence cn (t) = cn (0)e . Consequently, ψn (x, t) = cn (0)e . For the study of the continuous part of the spectrum λ(t) = k 2 (t), we proceed as follows: we assume that a stationary plane wave propagates from x = −∞ and meets a potential u(x, t) with a transmission coefficient T and a reflection coefficient R. In this case, equation [9.18] admits a solution ψ such that: ⎧ ⎨ ψ=

⎩

T (k, t)eikx , x → +∞ (i.e. to the right of the potential barrier) eikx + R(k, t)e−ikx , x → + − ∞ (i.e. to the left of the potential barrier)

where |R|2 + |T |2 = 1. T HEOREM 9.6.– If u 0 for |x| → ∞, then we have T (k, t) = T (k, 0) and R(k, t) = 3 R(k, 0)e−8ik t , where R(k, 0) and T (k, 0) are determined by the initial condition u(x, 0) = u0 (x) of the KdV equation. P ROOF.– Choose λ = constant since the spectrum for λ > 0 is continuous. So equation [9.20] remains valid, + ∂ψ ∂ 3 ψ ∂ψ A − 3(u + λ) dx + Bψ. [9.21] + =ψ ∂t ∂x3 ∂x ψ2 For u ∼ = 0, when x → +∞, we replace ψ = T (k, t)eikx , λ = k 2 in equation A 3 [9.21] and we get ∂T e−2ikx dx + BT . For this equation to preserve ∂t − 4ik T = T meaning when x → +∞, we must have A = 0, hence ∂T − (4ik 3 + B)T = 0. ∂t

[9.22]

Similarly, for u ∼ = 0, when x → −∞, we replace ψ = eikx + R(k, t)e−ikx , 2 λ = k in equation [9.21] and we get 

∂R + 4ik 3 R − BR e−ikx − (4ik 3 + B)eikx ∂t + dx = A(eikx + Re−ikx ) . 2ikx 2 e + R e−2ikx + 2R

For x → +∞, the equation above preserves a sense if A = 0 and is written as 

∂R 3 + 4ik R − BR e−ikx − (4ik 3 + B)eikx = 0. ∂t

260

Integrable Systems

For 4ik 3 + B = 0, that is, B = −4ik 3 , equation [9.22] implies that T (k, t) = 3 −8ik3 t T (k, 0) while the condition ∂R . ∂t +4ik R−BR = 0 gives us R(k, t) = R(k, 0)e  The knowledge of cn (t), kn (t), n = 1, 2, ..., N and R(k, t) allows us to express u(x, t) for any time; it is the problem of the inverse diffusion. The latter is reduced to the solution K(x, y; t) (to simplify the notations, the reader can obviously use K(x, y) instead of K(x, y; t)) of the Gelfand–Levitan linear integral equation: + x K(x, y; t) + I(x + y, t) + I(y + z, t)K(x, z; t)dz = 0, y ≤ x [9.23] −∞

∞

N 1 R(k, t)e−ik(x+y) dk + n=1 c2n (t)ekn (t)(x+y) . The where I(x + y, t) = 2π −∞ solution u(x, t) of the KdV equation is then given (see [9.11]) by u(x, t) = 2

d K(x, x; t). dx

[9.24]

The nonlinear KdV equation is transformed into the linear Gelfand–Levitan equation. The initial problem is thus completely solved. This method presents two major simplifications. First, in the analytical approach of the solution of the KdV equation, it suffices at each stage to solve only linear equations. Then t only appears parametrically and more than for all t the Gelfand–Levitan equation seems superficially to be an integral equation of two variables, actually x intervenes as a parameter and so we have to do to a family of integral equations for the functions K(x, y) of a single variable y. Before dealing with the general case, that is, the case of distinct N solitons, let us return first √ to the case of a soliton and therefore consider the solution u(x, t) = − 2c sech2 2c (x − ct), of the KdV equation obtained previously with the following initial condition: u(x, 0) = −2 sech2 x, where by convention we put c = 4. The Schrödinger equation [9.18] is written as ∂2ψ + (2 sech2 x + λ)ψ = 0. ∂x2

[9.25]

To study equation [9.25], one poses ψ = A sechα x.w(x),

[9.26]

where A is an arbitrary amplitude, α2 = −λ and w satisfies the following equation: 2 ∂2w ∂w 2 ∂x2 − 2α tanh x ∂x + (2 + α − α ) sech x.w = 0. By doing the substitution u = 1 2 (1−tanh x), the last equation comes down to a hypergeometric differential equation 2 or Gaussian equation: u(1 − u) ∂∂uw2 + (c − (a + b + 1)u) ∂w ∂u − abw = 0, where a, b, c denote constants and are equal to a = 2 + α, b = −1 + α, c = 1 + α. This equation

The Korteweg–de Vries Equation

261

presents three regular singular points: u = 0, u = 1, u = ∞. The solution of this equation for u = 0 is w ≡ F (a, b, c, u) = 1 + +

ab u a(a + 1)b(b + 1) u2 . + . c 1! c(c + 1) 2!

[9.27]

a(a + 1)...(a + n − 1)b(b + 1)...(b + n − 1) un . + ··· c(c + 1)...(c + n − 1) n!

For x → ∞ (i.e. when u → 0), we have w → 1. According to [9.26], we have ψ = A2α (ex + e−x )−α .w(x), and this one tends to Ae2α e−αx , x → ∞. To represent a plane wave Aeikx going to +∞, we will put α = −ik. The asymptotic form of the wave function for x → −∞ (u → 1) is obtained by transforming the hypergeometric function using the well-known functional relation: F (a, b, c, u) =

Γ(c)Γ(c − a − b) F (a, b, a + b − c + 1, 1 − u) Γ(c − a)Γ(c − b)

+(1 − u)c−a−b

Γ(c)Γ(a + b − c) F (c − a, c − b, c − a − b + 1, 1 − u), Γ(a)Γ(b)

∞ where Γ(z) = 0 e−t ez−1 dt, Re z > 0, is the Euler Gamma function. Taking into account [9.27] and the expression above, relation [9.26] becomes D

 1+

ab (1 − u) + · · · a+b−c+1  E (c − a)(c − b) Γ(c)Γ(a + b − c) 1+ (1 − u) + · · · . + (1 − u)c−a−b Γ(a)Γ(b) c−a−b+1

Γ(c)Γ(c − a − b) ψ = A sech x Γ(c − a)Γ(c − b) α

When u → 1 (x → −∞), we have (1 − u)c−a−b → e−2αx and since α = −ik, then  Γ(c)Γ(a + b − c) ikx Γ(c − a − b)Γ(a)Γ(b) e + . ψ −→ Aeα Γ(a)Γ(b) Γ(c − a)Γ(c − b)Γ(a + b − c) This last expression combined with the fact (already seen) that ψ tends to Γ(a)Γ(b) Ae2α e−α when x → ∞ gives us the transmission coefficient T = Γ(c)Γ(a+b−c) and Γ(c−a−b)Γ(a)Γ(b) the reflection coefficient R = Γ(c−a)Γ(c−b)Γ(a+b−c) . Here, we have k1 = 1, √ c(0) = 2, R(k, 0) = 0. For an individual soliton, equation [9.1] has a precise solution. It turns out that the soliton of amplitude u0 has only one discrete level with eigenvalue λ = u20 , while the next level corresponds to the point λ = 0 (with the respective eigenfunction ψ = tanh x) and already belongs to the continuous

262

Integrable Systems

spectrum. The Gelfand–Levitan equation [9.23] where I(μ, t) = c21 (t)ek1 μ = c21 (0)e−8k1 t ek1 t = 2e−8t+μ is written as K(x, y; t) + 2e

−8t+x+y

+ 2e

−8t+y

+

x

ez K(x, z; t)dz = 0. −∞

By putting K(x, y, t) = f (x)ey , into this equation, we obtain f (x) + 2e−8t+x + −x e f (x) = 0, hence f (x) = −2 1+ee8t−2x . Therefore, solution [9.24] of the KdV equation in the case of a solitary wave is given as −8t+2x

u(x, t) = 2

d 2 = −2 sech2 (x − 4t). K(x, x, t) = − dx cosh2 (x − 4t)

This illustrates the method and the correspondence between eigenvalue and solution. We will now look at the case of N -solitons through the procedure suggested by Gardner et al. (1967) and use the results of Wadati and Toda (1972). In order to solve the Gelfand–Levitan equation [9.23], where R(k, t) = 0, one poses K(x, y) =

N 

wn (x, t)ekn y ,

[9.28]

n=1

where wn are functions to be determined. By replacing this expression in the Gelfand– Levitan equation, we obtain the following linear system: ⎧

N (k1 +km )x 2 k1 x ⎪ + m=1 c21 (t) e k1 +km wm (x, t) = 0, ⎪ ⎨ w1 (x, t) + c1 (t)e .. . ⎪ ⎪

N (kN +km )x ⎩ wN (x, t) + c2N (t)ekN x + m=1 c2N (t) e kN +km wm (x, t) = 0. Define the following notations: 



A = c2n (t)e(kn +km )x ,

⎛

⎞ w1 ⎜ ⎟ W = ⎝ ... ⎠ , wN

⎛

⎞ c21 (t)ek1 x ⎜ ⎟ .. G=⎝ ⎠, . 2 kN x cN (t)e

 e(kn +km )x 2 = I + A, P ≡ (Pnm ) = δnm + cn (t) kn + km

[9.29]

The Korteweg–de Vries Equation

263

where I is the unit matrix. The system above is written, P W = −G, and it is easy to show that it has a unique solution. From equation [9.28], we draw ⎞ ek1 x ⎟ ⎜ h ≡ ⎝ ... ⎠ , ekN x ⎛

K(x, x) = h w = −h P −1 G,

P −1 =

αnm , det P

where αnm is the cofactor of P . Or d Pnm = c2m ekn x .ekm x , dx

det P =

N  

δnm + c2n (t)

n=1

e(kn +km )x kn + km

αnm ,

so K(x, x) = −

 αnm d 1 d d Pnm = − (det P ) = − ln det P, det P dx det P dx dx n,m 2

d d K(x, x) = −2 dx and according to [9.24], u = 2 dx 2 ln det P . Therefore, we have the following theorem: 2

d T HEOREM 9.7.– The solution of the KdV equation is u = −2 dx 2 ln det P , where P 3 −4kn t , with kn > 0 distinct. is defined by [9.29] and whose cn (t) = cn (0)e

The function obtained in theorem 9.7 is negative for all x, continuous and behaves like the exponential when |x| → ∞. To get an idea of the behavior of solitons and in particular their asymptotic behavior, suppose that k1 < k2 < ... < kN −1 < kN . But before this, we need the following lemma (Muir 1960) and the remark below: L EMMA 9.1.–      Δ≡   

1 a1 −b1 1 a2 −b1

.. .

1 a1 −b2 1 a2 −b2

1 1 an −b1 an −b2

... ... .. . ...

1 a1 −bn 1 a2 −bn 1 an −bn

   / /  n(n−1)  j ε(m+1)n > 0 if n > m, and εnm < εn(m−1) < ... < εn(m+1) < 0 if n < m. Replace these expressions in the determinant above and approximate the elements of the diagonal (for j < n) as c2 c2 follows: 1+ 2kjj e2kj (ξn −εjn t) ∼ = 2kjj e2kj (ξn −εjn t) , j < n, t → ∞ (since for j < n, we have εjn < 0 and 1 + ex ∼ = ex for x → ∞). We put in factor the common expressions: e2k1 (ξn −ε1n t) , e2k2 (ξn −ε2n t) ,...,e2kn−1 (ξn −ε(n−1)n t) . By turning t to infinity, we have (since εjn > 0 for n ≤ j ≤ N ): det P =            C         

c21 2k1 c2 c1 k2 +k1

c1 c2 k1 +k2 c22 2k2

.. .

cn−1 c1 cn−1 c2 kn−1 +k1 kn−1 +k2 cn c1 kn ξn cn c2 kn ξn kn +k1 e kn +k2 e

where C ≡

... .. .

0 .. .

0 .. .

0

0

...

j=1

       det P = C      

c1 cn kn ξn k1 +kn e c2 cn kn ξn k2 +kn e

.. .

c2n−1 cn cn−1 kn ξn kn +kn−1 e

cn−1 cn kn ξn kn−1 +kn e c2 1 + 2knn e2kn ξn

0 .. .

0 .. .

0

0

... ... ... .. .

/n−1

c1 cn−1 k1 +kn−1 c2 cn−1 k2 +kn−1

...

2kn−1

 0 . . . 0   0 . . . 0  .. . . . . . ..   0 . . . 0  ,  0 . . . 0  1 . . . 0  .. . . ..  . . . 0 ... 1

e2kj (ξn −εjn ) . Obviously, we have c21 2k1 c2 c1 k2 +k1

.. .

c1 c2 k1 +k2 c22 2k2

cn−1 c1 cn−1 c2 kn−1 +k1 kn−1 +k2 cn c1 kn ξn cn c2 kn ξn kn +k1 e kn +k2 e

... ... .. . ... ...

c1 cn−1 k1 +kn−1 c2 cn−1 k2 +kn−1 c2n−1 2kn−1 cn cn−1 kn ξn kn +kn−1 e

c1 cn kn ξn k1 +kn e c2 cn kn ξn k2 +kn e

.. .

cn−1 cn kn ξn kn−1 +kn e c2 1 + 2knn e2kn ξn

       .      

The Korteweg–de Vries Equation

This determinant is still written in the form   1 1  . . . k1 +k1 n−1   2k1 k1 +k2 1 1 n−1 . . . k2 +k1 n−1  9  k2 +k 2k2 1 2  det P = C cl  .. .. ..  .   . . l=1    k 1+k k 1+k . . . 2k 1  n−1 1 n−1 2 n−1  1 1  . . . k1 +k1 n−1 k1 +k2  2k1 1 1  . . . k2 +k1 n−1  k2 +k1 2k2 n 9  .. .. +C c2l  . .  1 1 1 l=1  kn−1 +k1 kn−1 +k2 . . . 2kn−1  1 1 1  k +k kn +k2 . . . kn +kn−1 n 1 for n ≥ 2. For n = 1, it equals 1 +

c21 2k1 ξ1 . 2k1 e

1 k1 +kn 1 k2 +kn

.. .

1 kn−1 +kn 1 2kn

267

      ,     

Using the previous lemma, we get,

det P = n−1 9

⎛ e2ki (ξn −εin t) ⎝

i=1

n−1 9 j=1

( 2

cj

⎞ / n−1 2 2 9 (k − k )) ( (k − k )) i j i j i 2 is a positive integer (the number of particles) and x1 , ..., xN are fixed on the closure of the#locus, that is, the geometrical position of the points given by the equations, $

 Δ = (x1 , ..., xN ) ∈ CN : i=j ℘ (xi − xj ) = 0, xi = xj , j = 1, ..., N . Show that Δ is non-empty (for triangular positive integers N , i.e. for numbers of the form N = g(g+1) where g is the number of gaps in the spectrum or the genus of the 2 corresponding algebraic curve) and study its geometry (see Airault et al. 1977). b) Show

that if xi = xi (t), j = 1, 2, 3 evolve according to the law x˙ i = −12 i=j ℘ (xi − xj ), then the function [9.32] is an elliptic solution of the 3

∂u ∂ u integrable KdV equation: ∂u ∂t − 6u ∂x + ∂x3 = 0, and is connected with the

N 1 Calogero–Moser system described by the Hamiltonian: H = 2 i=1 yi2 −

g(g+1) , with yi , xi , i = 1, ..., N , being canonical 2 i=j ℘ (xi − xj ), N = 2 variables.

c) Show that the Jacobi inversion with two-gap of the  s problem  s2associated  s1 zdzsolutions  s2 zdz dz KdV equation is determined by ∞1 dz + = −8it + c , + = 1 ∞ w w ∞ w ∞ w / 5 2 2ix + c2 , where c1 , c2 are constants, w = j=1 (z − zj ), sj = sj (x, t), j = 1, 2 and zj , j = 1, ..., 5, are expressed in terms of integrals of motion. d) Show that after an appropriate change of variables and corresponding choices of zj , the Jacobi inversion problem above coincides with that associated with the Hénon– Heiles equations for the case (i), section 7.6 (hint: see Eilbeck and Enolskii 1994). e) Discuss that equation [9.31] allows the coalescence of three particles xi and

n m show that the potential takes the form: U(x) = 6 i=1 ℘ (x − xi )+ i=1 ℘ (x − xi ), 3n + m = N . f) Consider the two-gap potential for the above potential normalized by its expansion near x = 0 as U(x) = x62 + α1 x2 + α2 x4 + α3 x6 + α4 x8 + O(x10 ), where α1 , α2 , α3 , α4 are functions of the moduli g2 , g3 of the underlying elliptic curve. Show that this potential satisfies the Novikov equation Novikov (1974): δS1 δS2 0 operator a−1 δSδu−1 + a0 δS δu + a1 δu + a2 δu = 0, where δ is the variational   of2 the calculus of variations, a1 , a2 , are constants and S = udx, S = u dx, −1 0   2   1  ∂u 2  2 2 1 ∂ u + u3 dx, S2 = − 52 u2 ∂∂xu2 + 52 u4 dx, are the first S1 = 2 ∂x 2 ∂x2

274

Integrable Systems

integrals of the KdV equation. Show that the algebraic curve associated with this potential has the form (Belokolos and Enol’skii 1989): w2 = z 5 −

35 63 1 α1 z 3 − α2 z 2 + 2 2 4



567 2 1377 1287 α1 + 297α3 z + α1 α2 − α4 . 2 4 2 [9.33]

g) Consider the trace formulas (Novikov 1974) written for

the elliptic potential

3 5 in the form: s1 + s2 = − j=1 ℘ (x − xj ) + 12 j=1 zj − C2 and 

2

3

5 5 + s1 s2 = 3 i,j=1 ℘ (x − xi ) ℘ (x − xj ) − 38 g2 + 12 i,j=1 zi zj − 38 j=1 zj i r}, the C-algebra, that is, the set of infinite matrices with support in a band around the diagonal (see Jacobi matrices, section 5.1). The product of two matrices belonging, respectively, to N and M, is defined in the usual way. Note that N is a Lie algebra and M is a N -module. 2 and M > are defined by Their extensions N 2 −→ N −→ 0, 0 −→ Cc −→ N

> −→ M −→ 0, 0 −→ Cc −→ M

2 = N ⊕ Cc, M > = M ⊕ Cc, where c is a central element, that is, [c, A] = with N 2 > We note ei,j = (δki .δlj )kl the elementary matrices, [c, B] = 0, ∀A ∈ N , ∀B ∈ M. that is, the matrices whose coefficients are all zero except the one of the line i and the column j, which is equal to 1. Since a Jacobi matrix has no trace, we consider

280

Integrable Systems

the matrix A[J, B] where A ∈ N , B ∈ M and J is the matrix defined by J =

i∈Z ε(i)ei,i , where ε(i) = +1 if i < 0 and −1 if i ≥ 0. The elements of the matrix A[J, B] are null except for a finite number, so it is indeed a finite matrix and we define the cocycle of A ∈ N and B ∈ M using the formula ρ(A, B) = 12 T r(A[J, B]) =

1 2 i,j (ε(i) − ε(j))aij bji . Therefore, the bracket [, ] of A ∈ N and B ∈ M is 2  2 is a non-trivial defined by [A, B] = [A + αc, B + βc] = [A, B] + ρ(A, B)c and N >f = Mf ⊕ Cc is a trivial central extension of Mf = central extension of N while M

{(aij ) ∈ M : (i, j) −→ (aij ) with finished support}. Let us put Ei = n∈Z en,n+i , whereFei,i = (δki .δij )kl are the elementary matrices defined above. The subspace E = i∈Z CEi is a commutative subalgebra of N . The subalgebra of N defined by 2 = E ⊕ Cc is called Heisenberg subalgebra. We have setting E  [E i , Ej ] = iδi,−j c.

[10.10]

Consider the previous example and replace q(x) with the Fourier series [10.9]. Let H be a functional of q. Its Fréchet derivative in terms of the coordinates ϕk is written as ∞ ∞   δH ∂ϕk δH ikx δH . e . [10.11] = = α−1 δq δϕk ∂q δϕk k=−∞

k=−∞

We substitute [10.10] and [10.11] in [10.8] and we specify the Fourier coefficients; −1 ∂H n n ∂ϕn . Since the symplectic structure is given by we get the relation α ∂ϕ ∂t = −iα

∞ ∂H n the matrix of the Poisson brackets, we have ∂ϕ m=−∞ {ϕn , ϕm }1 ∂ϕm . So, ∂t = {ϕn , ϕm }1 = −iα−2 nδm+n,0 . By putting −iα−2 = 1, we obtain the Heisenberg algebra (where {, } plays the role here of the bracket [,2] [10.10]).  T HEOREM 10.4.– In the case N = 2 (theorem 10.2), one obtains the Virasoro algebra, c its structure is given by {ϕm , ϕn }2 = (m − n)ϕm+n + 12 (m3 − m)δm+n,0 . of the unit circle: S 1 = {z ∈ P ROOF.– Let Diff(S 1 ) #be the group of diffeomorphisms  4$ d C : |z| = 1} and F = f (z) dz : f (z) ∈ C z, z1 , the set of vector fields (Laurent’s polynomials). F can be seen as the tangent space Diff(S 1 ) at its unit point, so F is a d Lie algebra with respect to the bracket [, ]. By setting ϕm = −z m+1 dz , we obtain   d d [ϕm , ϕn ] = (n + 1)z m+n+1 − (m + 1)m+n+1 = −(m + n)z m+n+1 , dz dz = (m − n)ϕm+n . ∼ C and ρ(ϕm , ϕn ) = 1 (m3 − m)δm,−n . The vector We show that H 2 (F, C) = 12 space F ⊕ Cc is called the Virasoro algebra, it is a central extension of the algebra of complex vector fields on the circle. The bracket is given by the formula c [10.12] [ϕm , ϕn ] = (m − n)ϕm+n + (m3 − m)δm,−n . 12

KP–KdV Hierarchy and Pseudo-differential Operators

281

Let us now consider the example of the KdV equation. We have N = 2 and   δH dq dL = = (L∇H)+ − L(∇HL)+ = ∂ 3 + 2(∂q + q∂) . dt dt δq   3  ∂ + 2(∂q + q∂) δF In this case, the bracket is written {H, F }2 = δH δq δq , and  3  we have {q(x), q(y)}2 = ∂ + 2(∂q + q∂) δ(x − y). By reasoning as previously,

δH i δH 3 m we obtain α ∂ϕ n (n − m)ϕm+n δϕn + 2α (m − 4βm) δϕ−m , where (ϕk )k∈Z ∂t = i are the Fourier coefficients of q. By setting 4β = 1, α = 6i c and taking into account the Fourier series [10.9], we obtain ⎛ ⎛ ⎞ ⎞ .. .. ⎞ ⎛ . ⎟ −nth column nth column ⎜ . ⎟ ∂ ⎜ ⎜ ϕm ⎟ = ⎝ ⎠ ⎜ δH ⎟ . ↓ ↓ ⎝ δϕm ⎠ ∂t ⎝ . ⎠ c .. (m3 − m) ... (m − n)ϕm+n mth line −→ 12 .. . c (m3 − m)δm+n,0 , that is, the Consequently {ϕm , ϕn }2 = (m − n)ϕm+n + 12 Virasoro structure (Gervais 1985) (where {, }2 plays the role here of the bracket [, ] [10.12]). 

10.3. KP hierarchy and vertex operators Consider the pseudo-differential operator of infinite order ∂ [10.13] ∂x where u1 , u2 , ... are functions of class C ∞ depending on an infinity of independent variables x ≡ t1 , t2 , ... The compound operator Ln is calculated according to the rules [10.1] and [10.2]. We obtain L = ∂ + u1 ∂ −1 + u2 ∂ −2 + · · · ,

∂≡

Ln = ∂ n + pn,2 ∂ n−2 + · · · + pn,n + pn,n+1 ∂ −1 + · · · = ∂n +

n 

pn,j ∂ n−j +

∞ 

pn,n+j ∂ −j ,

j=1

j=2

derivatives in relation to x. The differential where pn,j are polynomials in uj and their

n part Ln+ of Ln being equal to Ln+ = ∂ n + j=2 pn,j ∂ n−j , we deduce that L1+ = ∂,

L2+ = ∂ 2 + 2u2 ,

L3+ = ∂ 3 + 3u2 ∂ + 3(u3 + ∂u2 ), ...

[10.14]

The dependency between the functions u1 , u2 , ... and the variables x = t1 , t2 , ... is provided by the following system of partial differential equations: ∂L = [Ln+ , L], ∂tn

n ∈ N∗

[10.15]

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Integrable Systems

The set of these equations is called the KP hierarchy. It is a hierarchy of isospectral deformations of the pseudo-differential operator [10.13]. We prove (see Date et al. 1983) the following result: T HEOREM 10.5.– There is an equivalence between [10.15] and the equations ∂ m ∂ n L − L = [Ln+ , Lm + ], ∂tn + ∂tm +

[10.16]

as well as their dual forms ∂ n ∂ m L− − L = −[Ln− , Lm − ], ∂tn ∂tm −

[10.17]

where Ln− = Ln − Ln+ . Equations [10.15] determine an infinite number of commutative vector fields on algebra A = A+ ⊕ A− . ∂L P ROOF.– Note that since Ln = Ln+ + Ln− , then ∂t = [Ln+ , L] = −[Ln− , L] ∈ n A− . Equations [10.15] define an infinite number of vector fields on A. Since ∂t∂n and [Ln+ , .] are derivations, then

∂Lm n m n m n m = [Ln+ , Lm + ] + [L+ , L− ] = −[L− , L+ ] − [L− , L− ], ∂tn  1  1 n m n m = [L+ , L+ ] − [Ln− , Lm −[Ln− , Lm +] + − ] + [L+ , L− ] , 2 2  1 m n  1 n m n = [L+ , L+ ] − [Ln− , Lm [L+ , L− ] − [Lm −] + − , L+ ] . 2 2 Similarly, we have (just swap n and m)  1 n m  1 m n ∂Ln n = [L+ , L+ ] − [Lm [L+ , L− ] − [Ln− , Lm − , L− ] + +] . ∂tm 2 2 Hence,

∂Lm ∂tn

−

∂Ln ∂tm

n m = [Ln+ , Lm + ] − [L− , L− ]. Or

∂Ln ∂ m ∂ m ∂ n ∂ n ∂Lm − = L + L − L − L , ∂tn ∂tm ∂tn + ∂tn − ∂tm + ∂tm − =

∂ n ∂ m ∂ n ∂ m L+ − L+ + L− − L , ∂tn ∂tm ∂tn ∂tm −

∂ ∂ ∂ n m n n n m m then ∂t∂n Lm + − ∂tm L+ − [L+ , L+ ] = − ∂tn L− + ∂tm L− − [L− , L− ]. Since the expression on the left belongs to A+ and the one on the right belongs to A− , then the result comes from the decomposition A = A+ ⊕ A− since obviously A+ ∩ A− = ∅.

KP–KdV Hierarchy and Pseudo-differential Operators

283

To show that the vector fields defined by these equations commute, we put X(L) = n [Lm + , L] and Y (L) = [L+ , L]. Hence,     [X, Y ](L) = (XY − Y X)(L) = X [Ln+ , L] − Y [Lm + , L] ,  4  n 4  m 4 = X(Ln+ ) − Y (Lm + ), L + L+ , X(L) − L+ , Y (L) ,  4  n 4  m n 4 m = X(Ln+ ) − Y (Lm + ), L + L+ , [L+ , L] − L+ , [L+ , L] ,  4 m n = X(Ln+ ) − Y (Lm + ) − [L+ , L+ ], L , according to Jacobi’s identity and taking into account [10.16], we deduce that the vector fields in question commute.  By specifying the quantifiers of ∂ k in [10.16], one obtains an infinity of nonlinear partial differential equations (Cherednick 1978) forming the KP hierarchy. These equations connect infinitely many functions uj to infinitely many variables tj . For example, for m = 2, n = 3, relations [10.16] and [10.14] determine two expressions based on u2 and u3 . After eliminating u3 , we immediately obtain the KP equation:  ∂u2 ∂u2 ∂ ∂ 3 u2 ∂ 2 u2 4 = 0. [10.18] − 6u2 − 3 2 − ∂t2 ∂t1 ∂t3 ∂t1 ∂t31 3

∂u2 ∂ u2 ∂u2 2 By solving the system: ∂u ∂t2 = 0, 4 ∂t3 − 6u2 ∂t1 − ∂t31 = 0, we obtain particular solutions of [10.18]. Note that the last equation is precisely the KdV equation. The KP equation is therefore a generalization of the KdV equation, to which it is reduced 2 when ∂u ∂t2 = 0. Equations [10.15] and [10.16] imply the existence of the following pseudo-differential operator of degree 0 (wave operator) W ∈ I + A− :

W = 1 + w1 (t)∂ −1 + w2 (t)∂ −2 + · · ·

[10.19]

with t = (t1 , t2 , ...) ∈ C∞ . The inverse W −1 of W is also a pseudo-differential operator of the form W −1 = 1 + v1 (t)∂ −1 + v2 (t)∂ −2 + · · · and can be calculated term by

term. Indeed, by definition, we have W W −1 = 1. Using the fact that ∞ m! m k u m−k ∂ u = u, ∂ m ∂ n = ∂ m+n , as well as the formulas k=0 k!(m−k)! (∂ u∂ )∂ described in example 10.1, we specify the quantifiers of ∂ −1 , ∂ −2 , ... in the equation W W −1 = 1 and we determine relations between wm and vm . We finally obtain for W −1 the following expression: W −1 = 1 − w1 ∂ −1 + (−w2 + w12 )∂ −2 + (w3 + 2w1 w2 − w1 ∂w1 − w13 )∂ −3 + · · · In terms of W , the operator L [10.13] can be written in the form L = W.∂.W −1 .

[10.20]

According to [10.13] and [10.19], we deduce the relations: u2 = ∂w2 , u3 = −∂w2 − w1 ∂w1 , u4 = −∂w3 + w1 ∂w2 + (∂w1 )w2 − w12 ∂w1 − (∂w1 )2 . We have the following theorem:

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Integrable Systems

T HEOREM 10.6.– Equations [10.15] or what amounts to the same (according to theorem 10.5, equations [10.16]) are equivalent to the existence of the wave operator W [10.19] such that the system of differential equations LW = W ∂,

[10.21]

∂W = −Ln− W, ∂tn

[10.22]

has a solution (which can be inductively obtained).

∞ j T HEOREM 10.7.– Let ξ(t, z) = j=1 tj z , z ∈ C, be the phase function with m m m ξ(t,z) m ξ(t,z) =z e . There is an equivalence between [10.18] ∂ ξ(t, z) = z and ∂ e and [10.22] and the following problem: there is a wave function ψ (Baker–Akhiezer function)   Ψ(t, z) = 1 + w1 (t)z −1 + w2 (t)z −2 + · · · eξ(t,z) = W eξ(t,z) , [10.23] where z ∈ C, W is identified as [10.19] and such that: LΨ = zΨ,

∂Ψ = Ln+ Ψ. ∂tn

[10.24]

P ROOF.– Indeed, we have, from [10.23], ∂Ψ ∂W ξ(t,z) = e + W z n eξ(t,z) , ∂tn ∂tn = −Ln− W eξ(t,z) + z n W eξ(t,z) , −Ln− Ψ

n

according to [10.20]

+ z Ψ,

according to [10.21]

= −Ln− Ψ + Ln Ψ,

according to [10.22]

=

=

Ln+ Ψ.

In other words, Ψ satisfies [10.23] and [10.24] is equivalent to the fact that W satisfies [10.19] and [10.22].  Introduce the conjugation ∂ ∗ = −∂ and let L∗ = 1+(−∂)−1 u1 +(−∂)−2 u2 +· · · , and W ∗ = 1 + (−∂)−1 w1 + (−∂)−2 w2 + · · · , be the adjoints of L and W such that: L∗ = −(W ∗ )−1 .∂.W ∗ . T HEOREM 10.8.– The adjoint wave function Ψ∗ (t, z) = (W ∗ (t, ∂)) ∗ n ∗ ∗ satisfies the following relations: L∗ Ψ∗ = zΨ∗ , ∂Ψ ∂tn = −(L+ ) Ψ .

−1 −ξ(t,z)

e

KP–KdV Hierarchy and Pseudo-differential Operators

285

P ROOF.– The reason is as in the proof of the previous theorem.  Therefore, the knowledge of Ψ implies the knowledge of W and also of W ∗ and L.

k Define the following residues: Res ak z = a−1 , Res ak ∂ k = a−1 , and consider z

∂

the following lemma (Dickey 1997): L EMMA 10.1.– We have Res((P exz ).(Qe−xz )) = Res P Q∗ , where P and Q are two z

∂

pseudo-differential operators and Q∗ is the adjoint of Q. 



P ROOF.– Indeed, we have Res((P exz ).(Qe−xz )) = Res pk z k ql (−z)l = z z



l ∗ k l l k+l=−1 (−1) pk ql , and Res P Q = Res kl pk ∂ (−∂) ql = k+l=−1 (−1) pk ql . ∂



∂

Moreover, we have (Date et al. 1983; Dickey 1997):  Res (∂ k Ψ).Ψ∗ = Res ∂ k W eξ(t,z) (W ∗ )−1 e−ξ(t,z) , z z   = Res ∂ k W exz (W ∗ )−1 e−xz , x ≡ t − 1, z

= Res ∂ k W.W −1 = Res ∂ k = 0. ∂

∂

This bilinear identity can be written in the following symbolic form: Res (Ψ(t, z).Ψ∗ (t , z)) = 0 ∀t, t . Therefore, we have

z=∞

T HEOREM 10.9.– Ψ(t, z) is a wave function for the KP hierarchy if and only if the residue identity is satisfied: Res (Ψ(t, z).Ψ∗ (t , z)) = 0 ∀t, t

[10.25]

z=∞

or what amounts to the same if and only if + 1 √ Ψ(t, z).Ψ∗ (t , z)dz = 0, 2π −1 γ with γ a closed path around z = ∞ (such that:

[10.26] 

dz √ γ 2π −1

= 1).

D EFINITION 10.2.– A τ (t) function is defined by the Fay differential identity: {τ (t − [y1 ]), τ (t − [y2 ])} + (y1−1 − y2−1 )(τ (t − [y1 ])τ (t − [y2 ]) −τ (t)τ (t − [y1 ] − [y2 ])) = 0, where y1 , y2 ∈ C∗ and {u, v} is the Wronskian u v − uv  .

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Integrable Systems

 T HEOREM 10.10.– Let us put [s] = following identities:

2 3 s, s2 , s3 , ... . The τ function satisfies the

i) Fay identity: F(t, y0 , y1 , y2 , y3 ) ≡ (y0 − y1 )(y2 − y3 )τ (t + [y0 ] + [y1 ])τ (t + [y2 ] + [y3 ]) +(y0 − y2 )(y3 − y1 )τ (t + [y0 ] + [y2 ])τ (t + [y2 ] + [y1 ]) +(y0 − y3 )(y1 − y2 )τ (t + [y0 ] + [y3 ])τ (t + [y1 ] + [y2 ]), = 0. ii) Fay differential identity: {τ (t − [y1 ]), τ (t − [y2 ])} + (y1−1 − y2−1 )(τ (t − [y1 ])τ (t − [y2 ]) −τ (t)τ (t − [y1 ] − [y2 ])) = 0, where y1 , y2 ∈ C∗ and {u, v} is the Wronskian u v − uv  . This identity can still be ∞ −1 −1 j j 1 τ (t−[λ ]+[μ ]) written in the form as ∂ −1 ψ(t, λ)ψ ∗ (t, μ) = μ−λ e j=1 tj (μ −λ ) . τ (t) The equation τ˙ = X(t, λ, μ)τ determines a vector field on the infinite dimension manifold of the τ functions where X(t, λ, μ) is the vertex operator (of Date–Jimbo– Kashiwara–Miwa) for the KP equation. P ROOF.– According to Sato theory (Sato 1989; Sato and Sato 1982), the functions Ψ and Ψ∗ can be expressed in terms of a tau function as follows: −1 ]) ξ(t,z) Ψ(t, z) = W eξ(t,z) = τ (t−[z e , Ψ∗ (t, z) = (W ∗ )−1 e−ξ(t,z) = τ (t) τ (t+[z −1 ]) −ξ(t,z) e . τ (t)

By replacing these expressions in the residue formula [10.25] or

[10.26], we obtain a bilinear relation for the τ functions. Indeed, equation [10.26] is  written as: γ eξ(t−t ,z) τ (t − [z −1 ])τ (t − [z −1 ])dz = 0. Using the following change:  t ← t + s and t ← t + s, we obtain γ eξ(−2s,z) τ (t − s − [z −1 ]) τ (t + s + [z −1 ])dz = 0. Again using the transformation, s ← t + 12 ([y0 ]+ [y1 ] + [y2 ] + [y3 ]), t ← 12 ([y0 ] − [y1 ] − [y2 ] − [y3 ]), and taking into account that ∞ −1 j −1 e 1 (ab ) .j = 1 − ab−1 , we obtain, via the residue theorem + 1 − zy0 0 = τ (t − s − [z −1 ])τ (t + s + [z −1 ])dz, /3 (1 − zy ) γ j j=1

 √  1 − zy0 −1 −1 Res = 2π −1 τ (t − s − [z ])τ (t + s + [z ]) , /3 y1−1 ,y2−1 ,y3−1 j=1 (1 − zyj ) √ 2π −1 F(t, y0 , y1 , y2 , y3 ), = (y1 − y2 )(y2 − y3 )(y3 − y1 )

KP–KdV Hierarchy and Pseudo-differential Operators

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where F(t, y0 , y1 , y2 , y3 ) ≡ (y0 − y1 )(y2 − y3 )τ (t + [y0 ] + [y1 ])τ (t + [y2 ] + [y3 ]) +(y0 − y2 )(y3 − y1 )τ (t + [y0 ] + [y2 ])τ (t + [y2 ] + [y1 ]) +(y0 − y3 )(y1 − y2 )τ (t + [y0 ] + [y3 ])τ (t + [y1 ] + [y2 ]). The relation F(t, y0 , y1 , y2 , y3 ) = 0 is the Fay identity. In addition, by making the ∂F transformation in the expression (y1 y2 )−1 ∂y |y0 =y3 =0 and replacing t by t − [y1 ] − 0 [y2 ], we obtain the Fay differential identity which allows to define the τ functions: {τ (t − [y1 ]), τ (t − [y2 ])} + (y1−1 − y2−1 ) (τ (t − [y1 ])τ (t − [y2 ]) − τ (t)τ (t − [y1 ] − [y2 ])) = 0, where y1 , y2 ∈ C∗ and {u, v} is the Wronskian u v−uv  . Consider the Fay differential identity above and replace t with t + [y1 ]. We obtain {τ (t), τ (t + [y1 ] − [y2 ])} + (y1−1 − y2−1 ) (τ (t)τ (t + [y1 ] − [y2 ]) − τ (t)τ (t − [y2 ])) = 0. By putting λ = y −1 , μ = y2−1 , we obtain, after having multiplied the expression ∞1 j j 1 obtained by τ (t) e 1 tj (μ −λ ) , the following formula: τ (t + [λ−1 ]) −  tj λj τ (t − [μ−1 ])  tj μj e e τ (t) τ (t)   −1 j j τ (t + [λ ] − [μ−1 ]) 1 ∂ e tj (μ −λ ) . = μ − λ ∂x τ (t) ∞

j

j

∞

j −1 (λ−j −μ−j )

∂

1 ∂tj Let X(t, λ, μ) = μ−λ e 1 tj (μ −λ ) e 1 , λ = μ, be the vertex operator (of Date–Jimbo–Kashiwara–Miwa) for the KP equation, then X(t, λ, μ)τ and τ + X(t, λ, μ)τ are also τ functions. Therefore, τ˙ = X(t, λ, μ)τ determines a vector field on the infinite dimension manifold of functions τ . We deduce, according 1 to Dickey (1997), that ∂ −1 (Ψ∗ (t, λ)Ψ(t, μ)) = τ (t) X(t, λ, μ)τ (t). 

/ R EMARK 10.2.– Let Δ(s1 , ..., sn ) = 1≤j 0} be the Siegel half-space, Λ = Zg ⊕ ZZg a lattice in Cg and T = Cg /Λ a principally polarized Abelian variety. The following three conditions are equivalent (see Shiota 1986): 1 p(∂t )f (t).g(t) ≡ p



∂ , ∂ , ... ∂s1 ∂s2



  f (t + s)g(t − s)

and g(t) are two differentiable functions.

s=0

where p is any polynomial, f (t)

290

Integrable Systems

(i) There

are vector fields v1 , v2 , v3 on Cg and a quadratic form 3 q(t) = ∈ Cg , the function k,l=1 qkl (t)tk tl such that for all z 

3 τ (t) = eq(t) θ k=1 tk vk + z satisfies the KP equation. The theta divisor does not contain an Abelian subvariety of T for which the vector v1 is tangent. (ii) T is isomorphic to the Jacobian variety of a reduced non-singular complete curve of genus g. (iii) There is a matrix V =

(v1 , v2 , ...) of order g × ∞, vk ∈ Cg , of rank g ∞ g and a quadratic form Q(t) = k,l=1 qkl (t)tk tl such that for all z ∈ C , Q(t) θ (Vt + z), is a τ function for KP hierarchy. τ2(t) = e 10.4. Exercises E XERCISE 10.1.– Use [10.1] to compare u(x).∂ and ∂.u(x) (for some function u(x)) and show that these expressions are not equal to each other. What can be concluded about the multiplication of differential operators?



E XERCISE 10.2.– Let P (x, ∂x ) = j aj (x)∂xj , Q(x, ∂x ) = j bj (x)(−∂)jx , be two pseudo-differential operators with coefficients depending on x. Show that 



P (x, ∂x )Q (x, ∂x )

+

 −

δ(x − y) =

P (x, ∂x )exz Q(y, ∂y )e−yz

dz H(x − y), 2πi

where the integral is taken over a small circle around z = ∞, δ is the customary δ-function and H the Heaviside function H(x) = ∂x−1 δ(x). E XERCISE 10.3.– Given the pseudo-differential operator L, we can change the wave operator W (this operator is not unique) as follows, W (t) −→ W 0 , where W0 is

(t)W ∞ a constant coefficients pseudo-differential operator W0 = 1 + j=1 aj ∂ −j , aj ∈ C. Show that this modification has the following effect on the wave function Ψ: Ψ(t, z) = W (t)e

∞

j=1 tj z

j

2 = Ψ(t, z)e−

−→ Ψ

∞

bj j=1 j

z −j

for some appropriate bj , and the τ -function τ (t) −→ τ2 = τ (t)e

,

∞

j=1

b j tj

.

E XERCISE 10.4.– Show that ∂ −1 .(∂.u(x)) = u(x), (∂.∂ −1 )u(x) = ∂.(∂ −1 u(x)), ∂ 2 .(∂ −2 .(∂ −1 + x2 ∂ −3 )) = ∂ −1 + x2 ∂ −3 . E XERCISE 10.5.– Given an ordinary operator L, show (Dickey 1991) that there are 1 two operators L−1 and L n such that: L.L−1 = L−1 .L = 1,  1 pseudo-differential n Ln = L.

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291

E XERCISE 10.6.– Let A be the Lie algebra operator and Res :  of a pseudo-differential  A −→ C ∞ (S 1 ), the residue given by ak ∂ −k −→ a−1 (x) (i.e. coefficient of ∂ −1 ). Show that for all L1 , L2 ∈ A, we have Res[L1 , L2 ] = 0. E XERCISE 10.7.– We use the notation  from section 10.3. Consider the set of equations [10.15] (KP hierarchy). Show that Res Lk dx are first integrals of equation [10.15] and vector fields defined by this equation commute.

References

Abenda, S. and Fedorov, Y. (2000). On the weak Kowalewski-Painlevé property for hyperelliptically separable systems. Acta Appl. Math., 60, 137–178. Abraham, R. and Marsden, J. (1978). Foundations of Mechanics, 2nd edition. Benjamin Cummings Publishing Company, Reading, MA. Adler, M. (1979). On a trace functional for formal pseudo differential operators and the symplectic structure of the KdV type equations. Invent. Math., 50(3), 219–248. Adler, M. and van Moerbeke, P. (1980a). Completely integrable systems, Euclidean Lie algebras, and curves. Adv. in Math., 38, 267–379. Adler, M. and van Moerbeke, P. (1980b). Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. in Math., 38, 318–379. Adler, M. and van Moerbeke, P. (1982a). The algebraic complete integrability of geodesic flow on SO(4). With an appendix by D. Mumford. Invent. Math., 67, 297–331. Adler, M. and van Moerbeke, P. (1982b). Kowalewski’s asymptotic method, Kac-Moody lie algebras and regularization. Comm. Math. Phys., 83(1), 83–106. Adler, M. and van Moerbeke, P. (1984). Geodesic flow on SO(4) and the intersection of quadrics. Proc. Nat. Acad. Sci. U.S.A., 81, 4613–4616. Adler, M. and van Moerbeke, P. (1985). Algebraic completely integrable systems: A systematic approach, I, II, III. Séminaire de Mathématique, Report no. 110, SC/MAPA, UCL, London. Adler, M., and van Moerbeke, P. (1987). The intersection of four quadrics in P6 . Abelian surfaces and their moduli. Math. Ann., 279, 25–85. Adler, M. and van Moerbeke, P. (1988). The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows on SO(4) – A two-dimensional family of Lax pairs. Comm. Math. Phys., 113(4), 659–700.

Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

294

Integrable Systems

Adler, M. and van Moerbeke, P. (1989). The complex geometry of the KowalewskiPainlevé analysis. Invent. Math., 97, 3–51. Adler, M. and van Moerbeke, P. (1991). The Toda lattice, Dynkin diagrams, singularities and abelian varieties. Invent. Math., 103(2), 223–278. Adler, M., van Moerbeke, P., Vanhaecke, P. (2004). Algebraic Integrability, Painlevé Geometry and Lie Algebras. Springer-Verlag, Berlin, Heidelberg. Airault. H., McKean, H.P., Moser, J. (1977). Rational and elliptic solutions of the KdV equation and a related many-body problem. Comm. Pure Appl. Math., 30, 94–148. Ankiewicz, A. and Pask, C. (1983). The complete Whittaker theorem for two-dimensional integrable systems and its application. J. Phys. A: Math. Gen., 16, 4203–4208. Appel’rot, G. (1894). The problem of motion of a rigid body about a fixed point. Uchenye Zap. Mosk. Univ. Otdel. Fiz. Mat. Nauk, 11(3), 1–112. Arbarello, E., Cornalba, M., Griffiths, P., Harris, J. (1985). Geometry of Algebraic Curves, Volume 1. Springer-Verlag, Berlin. Arnold, V.I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York. Arnold, V.I. (1989). Mathematical Methods in Classical Mechanics. Springer-Verlag, New York. Arnold, V.I. and Givental, A.B. (1990). Symplectic geometry. In Dynamical Systems IV, Arnold, V.I. and Novikov, S.P. (eds). Springer, Berlin, Heidelberg. Arutyunov, G. (2019). Elements of Classical and Quantum Integrable Systems. Springer, Cham. Babelon, O., Bernard, D., Talon, M. (2003). Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge. Baker, S., Enolskii, V.Z., Fordy, A.P. (1995). Integrable quartic potentials and coupled KdV equations. Phys. Lett., 201A, 167–174. Bechlivanidis, C. and van Moerbeke, P. (1987). The Goryachev-Chaplygin top and the Toda lattice. Comm. Math. Phys., 110(2), 317–324. Belokolos, A.I. and Enol’skii, V.Z. (1989). Isospectral deformations of elliptic potentials. Russ. Math. Surveys, 44, 155–156. Belokolos, A.I., Bobenko, V.Z., Enol’skii, V.Z., Its, A.R., Matveev, V.B. (1994). Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer-Verlag, Berlin, Heidelberg. Berry, M.L. (1978). Regular and irregular motions. In Topics in Nonlinear Dynamics, Jorna, S. (ed.). American Institute of Physics, New York. Bialy, M. (2010). Integrable geodesic flows on surfaces. Geom. Funct. Anal., 20, 357–367.

References

295

Bobylev, D. (1896). On a certain particular solution of the differential equations of rotation of a heavy rigid body about a fixed point. Trudy Otdel. Fiz. Nauk Obsc. Estestvozn., 8, 21–25. Bolsinov, A.V., Morales-Ruiz, J., Zung, N.T. (2016). Geometry and Dynamics of Integrable Systems. Springer, Basel. Bolsinov, A.V., Izosimov, A.M., Tsonev, D.M. (2017). Finite-dimensional integrable systems: A collection of research problems. J. Geom. Phys., 115, 2–15. Boussinesq, J. (1877). Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants. Acad. des Sci. Inst. Nat. France, XXIII, 1–680. Chaplygin, S.A. (ed.) (1948). A new case of rotation of a rigid body, supported at one point. Collected Works I. Gostekhizdat, USSR. [in Russian]. Cherednick, I.V. (1978). Differential equations for the Baker-Akhiezer functions of algebraic curves. Funct. Anal. Appl., 12, 45–54. Christiansen, P.L., Eilbeck, J.C., Enolskii, V.Z., Kostov, N.A. (1995). Quasi-periodic of the coupled nonlinear Schrödinger equations. Proc. R. Soc. Lond. A, 451, 685–700. Clebsch, A. (1871). Der Bewegung eines starren Körpers in einen Flüssigkeit. Math. Ann., 3, 238–268. Conte, R., Musette, M., Verhoeven, C. (2005). Completeness of the cubic and quartic Hénon-Heiles Hamiltonians. Theor. Math. Phys., 144, 888–898. Date, E., Jimbo, M., Kashiwara, M., Miwa, T. (1983). Transformation groups for soliton equations. Proc. RIMS Symp. Nonlinear Integrable Systems, Classical and Quantum Theory (Kyoto, 1981), Jimbo, M., Miwa, T. (eds). World Scientific, Singapore. Deift, P., Lund, F., Trubowitz, E. (1980). Nonlinear wave equations and constrained harmonic motion. Comm. Math. Phys., 74, 141–188. Dickey, L. (1991). Soliton Equations and Integrable Systems. World Scientific, Singapore. Dickey, L. (1993). Additional symmetries of KP, Grassmannian and the string equation. Mod. Phys. Lett. A, 8, 1259–1272. Dickey, L. (1997). Lectures on classical W-algebras (Cortona Lectures). Acta Appl. Math., 47, 243–321. Dorizzi, B., Grammaticos, B., Ramani, A. (1983). A new class of integrable systems. J. Math. Phys., 24, 2282. Dubrovin, B.A. (1981). Theta functions and non-linear equations. Russian Math. Surv., 36(2), 11–92. Dubrovin, B.A. and Novikov, S.P. (1974). Periodic and conditionally periodic analogues of multi-soliton solutions of the Korteweg-de Vries equation. Soviet Physics JETP, 40, 1058–1063.

296

Integrable Systems

Dubrovin, B.A., Novikov, S.P., Fomenko, A.T. (1984). Modern Geometry–Methods and Applications. Part I. Springer-Verlag, New York. Dubrovin, B.A., Novikov, S.P., Fomenko, A.T. (1985). Modern Geometry–Methods and Applications. Part II. Springer-Verlag, New York. Dubrovin, B.A., Novikov, S.P., Fomenko, A.T. (1990). Modern Geometry–Methods and Applications. Part III. Springer-Verlag, New York. Eilbeck, J.C. and Enolskii, V.Z. (1994). Elliptic solutions and blow-up in an integrable Hénon-Heiles system. Proc. Roy. Soc. Edinburgh, Sect. A, 124, 1151–1164. Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B., Leykin, D.V. (1993). Linear r-matrix algebra for systems separable in parabolic coordinates. Phys. Lett., 180A, 208–214. Eilbeck, J.C., Enolskii, V.Z., Kuznetsov, V.B., Tsiganov, A.V. (1994). Linear r-matrix algebra for classical separable systems. J. Phys. A.: Math. Gen., 27, 567–578. Euler, L. (1765). Theoria motus corporum solidorum seu rigidorum. A.F. Rose, Rostock, Greifswald. Faddev, L.D. and Takhtajan, L.A. (2007). Hamiltonian Methods in the Theory of Solitons. Springer, Cham. Fay, J. (1973). Theta Functions on Riemann Surfaces. Springer-Verlag, Cham. Ferguson, W.E. (1980). The construction of Jacobi and periodic Jacobi matrices with prescribed spectra. Math. Comp., 35, 1220–1230. Flaschka, H. (1974). The Toda lattice I. Existence of integrals. Phys. Rev. B, 3(9), 1924–1925. Fordy, A.P. (1991). The Hénon-Heiles system revisited. Physica D, 52, 204–210. Garay, M. and van Straten, D. (2010). Classical and quantum integrability. Mosc. Math. J., 10, 519–545. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M. (1967). Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett., 19, 1095–1097. Garnier, R. (1919). Sur une classe de systèmes différentiels abéliens déduits de la théorie des équations linéaires. Ren. Circ. Math. Palermo, 43(4), 155–191. Gavrilov, L. and Angel Zhivkov, A. (1998). The complex geometry of the Lagrange top. Enseign. Math., 44, 133–170. Gavrilov, L., Ouazzani-Jamil, M., Caboz, R. (1992). Bifurcations des tores de Liouville du potentiel de Kolosoff U = ρ + ρ1 − k cos ϕ. CRAS, 315(I), 289–294. Gelfand, I.M. and Dickey, L. (1968). Family of Hamiltonian structures connected with integrable nonlinear differential equations. Funct. Anal. Appl., 2, 92–93. Gelfand, I.M. and Levitan, B.M. (1955). On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl., 2(1), 253–304. Gerd, R. and Matthias, S. (2013). Differential Geometry and Mathematical Physics Part I: Manifolds, Lie Groups and Hamiltonian Systems. Springer, Dordrecht.

References

297

Gervais, J.-L. (1985). Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson bracket. Phys. Lett. B, 160(4–5), 277–278. Goryachev, D. (1900). On the motion of a rigid material body about a fixed point in the case A=B=4C. Mat. Sb., 21(3). Grammaticos, B., Dorozzi, B., Ramani, A. (1983). Integrability of Hamiltonians with third and fourth-degree polynomial potentials. J. Math. Phys., 24, 2289–2295. Grammaticos, B., Dorozzi, B., Ramani, A. (1984). Hamiltonians with higher-order integrals and the “weak-Painlevé” concept. J. Math. Phys., 25, 3470. Griffiths, P.A. (1985). Linearizing flows and a cohomological interpretation of Lax equations. Amer. J. Math., 107, 1445–1483. Griffiths, P.A. and Harris, J. (1978). Principles of Algebraic Geometry. Wiley Interscience, Hoboken. Guillemin, V. and Sternberg, S. (1984). Symplectic Techniques in Physics. Cambridge University Press, Cambridge. Haine, L. (1983). Geodesic flow on SO(4) and Abelian surfaces. Math. Ann., 263, 435–472. Haine, L. (1984). The algebraic complete integrability of geodesic flow on SO(N ) and Abelian surfaces. Comm. Math. Phys., 94(2), 271–287. Halphen, G.-H. (1888). Traité des fonctions elliptiques et de leurs applications. Gauthier-Villars, Paris. Hénon, M. and Heiles, C. (1964). The applicability of the third integral of motion; some numerical experiments. Astron. J., 69, 73–79. Hess, W. (1890). Uber die Euler hen Bewegungsgleichungen und tlber eine neue par dare L(isung des Problems der Bewegung eines starren Korpers un einen festen punkt. Math. Ann., 37(2), 153–181. Hietarinta, J. (1984). Classical versus quantum integrability. J. Math. Phys., 25, 1833–1840. Hietarinta, J. (1985). How to construct integrable Fokker-Planck and electromagnetic Hamiltonians from ordinary integrable Hamiltonians. J. Math. Phys., 26, 1970–1975. Hietarinta, J. (1987). Direct methods for the search of the second invariant. Phys. Rep., 147, 87–154. Hitchin, N.J. (1983). On the construction of monopoles. Comm. Math. Phys., 89(2), 145–190. Hitchin, N.J. (1994). Stable bundles and integrable systems. Duke Mathematical Journal, 54(1), 91–114. Hitchin, N.J. (2013). Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces. Oxford University Press, Oxford.

298

Integrable Systems

Holmes, P.J. and Marsden, J.E. (1983). Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J., 32, 273–310. Husson, E. (1906). Recherche des intégrales algébriques dans le mouvement d’un solide pesant autour d’un point fixe. Ann. Fac. Sc. Univ. Toulouse, Series 2, 8(1906), 73–152. Jacobi, C. (1850). Sur la rotation d’un corps. J. Reine Angew. Math., 39, 293–350. Jacobi, C. (1969). Vorlesungen über Dynamik, Königsberg Lectures of 1842–1843. Chelsea Publishing Co., New York. Kac, M. and van Moerbeke, P. (1975). On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv. in Math., 16, 160–169. Kadomtsev, B.B. and Petviashvili, V.I. (1970). On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl., 15(6), 539–541. Kalla, C. (2011). Fay’s identity in the theory of integrable systems. Thesis, Institut de Mathématiques de Bourgogne, Dijon. Kirchoff, G. (1876). Vorlesungen über Mathematische Physik, Volume 1, Mechanik. Teubner, Leipzig. Knörrer, H. (1980). Geodesics on the ellipsoïd. Invent. Math., 59, 119–143. Knörrer, H. (1982). Geodesics on quadrics and a mechanical problem of C. Neumann. J. Reine Angew. Math., 334, 69–78. Koizumi, S. (1976). Theta relations and projective normality of abelian varieties. Am. J. Math., 98, 865–889. Kolossoff, G. (1903). Zur Rotation eines Körpers im Kowalewski’schen Falle. Mathematische Annalen, 56, 265–272. Korteweg, D.J. and de Vries, G. (1895). On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag., 39, 422–443. Kostant, B. (1979). The solution to a generalized Toda lattice and representation theory. Adv. Math., 34(3), 195–338. Kötter, F. (1892). Uber die Bewegung eines festen Körpers in einer Flüssigkeit I, II. J. Reine Angew. Math., 109(51–81), 89–111. Kötter, F. (1900). Die von Steklow und Lyapunov entdeckten intgralen Fälle der Bewegung eines Körpers in einen Flüssigkeit Sitzungsber. König. Preuss. Akad. Wiss., 6, 79–87. Krichever, I.M. (1976). Algebraic-geometric construction of Zakhorov-Shabat equations and their periodic solutions. Sov. Math., Dokl., 17, 394–397. Kowalewski, S. (1889). Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math., 12, 177–232. Lagrange, J.L. (1888). Mécanique analytique, Volume 11. Oeuvres de Lagrange, Gauthier-Villars, Paris.

References

299

Landau, L.D. and Lifshitz, E.M. (1935). On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Zeitsch. der Sow., 8, 153–169. Laurent-Gengoux, C., Miranda, E., Vanhaecke, P. (2011). Action-angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Not., 8, 1839–1869. Lax, P. (1968). Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21, 467–490. Lesfari, A. (1986). Une approche systématique à la résolution du corps solide de Kowalewski. C. R. Acad. Sc., Paris, 302(I), 347–350. Lesfari, A. (1988). Abelian surfaces and Kowalewski’s top. Ann. Scient. École Norm. Sup., Paris, 21(4), 193–223. Lesfari, A. (1999). Completely integrable systems: Jacobi’s heritage. J. Geom. Phys., 31, 265–286. Lesfari, A. (2003). Le système différentiel de Hénon-Heiles et les variétés Prym. Pacific J. Math., 212(1), 125–132. Lesfari, A. (2007). Abelian varieties, surfaces of general type and integrable systems. Beiträge zur Algebra und Geometrie, 48(1), 95–114. Lesfari, A. (2008a). Cyclic coverings of abelian varieties and the generalized Yang Mills system for a field with gauge groupe SU(2). Int. J. Geom. Methods Mod. Phys., 5(6), 947–961. Lesfari, A. (2008b). Fonctions et intégrales elliptiques. Surv. Math. Appl., 3, 27–65. Lesfari, A. (2009). Integrable systems and complex geometry. Lobachevskii J. Math., 30(4), 292–326. Lesfari, A. (2010). Théorie spectrale et problèmes non-linéaires. Surv. Math. Appl., 5, 151–190. Lesfari, A. (2011). Algebraic integrability: The Adler-van Moerbeke approach. Regul. Chaotic Dyn., 16(3–4), 187–209. Lesfari, A. (2013). Etude des équations stationnaire de Schrödinger, intégrale de Gelfand-Levitan et de Korteweg-de-Vries. Solitons et méthode de la diffusion inverse. Aequat. Math., 85, 243–272. Lesfari, A. (2014). Rotation d’un corps solide autour d’un point fixe. Rend. Sem. Mat. Univ. Pol. Torino, 72(1–2), 255–284. Lesfari, A. (2015a). Géométrie et intégrabilité algébrique. Rend. Mat. Appl., (7), 36(1–2), 27–76. Lesfari, A. (2015b). Introduction à la géométrie algébrique complexe. Hermann, Paris. Lesfari, A. (2019a). Réalisation du flot géodésique sur le groupe SO(n) comme flot sur des orbites de Kotant-Kirillov. Surv. Math. Appl., 14, 231–259.

300

Integrable Systems

Lesfari, A. (2019b). Spectral theory, Jacobi matrices, continued fractions and difference operators. Fundamental Journal of Mathematics and Applications, 2(1), 63–90. Lesfari, A. (2020). Generalized algebraic completely integrable systems. Surv. Math. Appl., 15, 169–216. Lesfari, A. (2021). Géométrie symplectique, calcul des variations et dynamique hamiltonienne. ISTE Editions Ltd, London. Li, C.Z., He, J.S., Wu, K., Cheng, Y. (2010). Tau function and Hirota bilinear equations for the extended bigraded Toda Hierarchy. J. Math. Phys., 51, 043514. Liouville, R. (1896). Sur le mouvement d’un corps solide pesant suspendu par l’un de ses points. Acta Math., XX, 239–284. Lyapunov, A. (1894). On a property of the differential equations of the problem of motion of a rigid body having a fixed point. Comm. Soc. Math. Kharkow, 4(3), 123–140. Manakov, S.V. (1976). Remarks on the integrals of the Euler equations of the n-dimensional heavy top. Fund. Anal. Appl., 10(4), 93–94. van Moerbeke, P. (1976). The spectrum of Jacobi matrices. Invent. Math., 37, 45–81. van Moerbeke, P. (1989). Introduction to algebraic integrable systems and their Painlevé analysis. Theta Functions Bowdoin, Part 1, 107–131, Proc. Sympos. Pure Math., Vol. 49, Amer. Math. Soc., Providence. van Moerbeke, P. (1994). Integrable foundations of string theory. In Lectures on Integrable Systems, Babelon, O., Cartier, P., Kosmann-Schwarzbach, Y., Verdier, J.-L. (eds). World Scientific, River Edge. van Moerbeke, P. (2011). Random matrix theory and integrable systems. In The Oxford Handbook of Random Matrix Theory, Akemann, G., Baik, J., Di Francesco, P. (eds). Oxford University Press, Oxford. van Moerbeke, P. and Mumford, D. (1979). The spectrum of difference operators and algebraic curves. Acta Math., 143, 93–154. McKean, H.P. and van Moerbeke, P. (1975). The spectrum of Hill’s equation. Invent. Math., 30, 217–274. Mironov, A.E. (2010). Polynomial integrals of a mechanical system on a twodimensional torus. Izv. Math., 74, 805–817. Mishchenko, A.S. and Fomenko, A.T. (1978). Euler equations on finite-dimensional Lie groups. Math. USSR-Izv., 12(2), 371–389. Moishezon, B.G. (1967). On n-dimensional compact varieties with n algebraically independent meromorphic functions. Amer. Math. Soc. Transl., 63, 51–177. Moser, J.K. (1965). On the volume elements on a manifold. Trans. Amer. Math. Soc., 120, 286–294.

References

301

Moser, J.K. (1980). Geometry of quadrics and spectral theory. In The Chern Symposium 1979, Hsiang, W.-Y., KobayashiI, S., Singer, M., Wolf, J., Wu, H.-H., Weinstein, A. (eds). Springer, Cham. Muir, T. (1960). A Treatise on the Theory of Determinants. Dover Publications, New York. Mumford, D. (1967a). On the equations defining Abelian varieties I. Invent. Math., 1, 287–354. Mumford, D. (1967b). On the equations defining Abelian varieties II. Invent. Math., 3, 75–135. Mumford, D. (1967c). On the equations defining Abelian varieties III. Invent. Math., 3, 215–244. Mumford, D. (1983a). Tata Lectures on Theta I. Birkhaüser, Basel. Mumford, D. (1983b). Tata Lectures on Theta II. Birkhaüser, Basel. Nahm, W. (1981). All self-dual multi-monopoles for all gauge groups. CERNTH3172. Presentation, Int. Summer Inst. on Theoretical Physics, Freiburg, West Germany, August 31–September 11. Neumann, C. (1859). De problemate quodam mechanics, quod ad primam integralium ultraellipticorum classem revocatur. J. Reine Angew. Math. (Crelles journal), 56, 46–63. Novikov, S.P. (1974). The periodic problem for KdV equation. F. Anal. Pril., 8, 53–66. Painlevé, P. (1975). Oeuvres, tomes 1, 2, 3. Édition du CNRS, Paris. Pelayo, A. (2017). Hamiltonian and symplectic symmetries: An introduction. Bull. Amer. Math. Soc., 54, 383–436. Perelomov, A.M. (1990). Integrable Systems of Classical Mechanics and Lie Algebras. Birkhäuser Verlag, Basel. Piovan, L. (1992). Cyclic coverings of Abelian varieties and the Goryachev-Chaplygin top. Math. Ann., 294, 755–764. Poincaré, H. (1905). Leçons de mécanique céleste. Gauthier-Villars, Paris. Poinsot, L. (1851). Théorie nouvelle de la rotation des corps. Journal de mathématiques pures et appliquées, 1re série, Volume 16, 9–129. Ramani, A., Dorozzi, B., Grammaticos, B. (1982). Painlevé conjecture revisited. Phys. Rev. Lett., 49, 1539–1541. Ratiu, T. (1982). Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body. Amer. J. Math., 104, 409–448. Ratiu, T. and van Moerbeke, P. (1982). The Lagrange rigid body motion. Ann. Inst. Fourier, 32(1), 211–234. Ravoson, V., Gavrilov, L., Caboz, R. (1993). Separability and Lax pairs for Hénon-Heiles system. J. Math. Phys., 34, 2385–2393.

302

Integrable Systems

Roekaerts, D. and Schwarz, F. (1987). Painléve analysis, Yoshida’s theorems and direct methods in the search for integrable Hamiltonians. J. Phys. A, Math. Gen., 20, L127–L133. Rosemann, S. and Schöbel, K. (2015). Open problems in the theory of finite-dimensional integrable systems and related fields. J. Geom. Phys., 87, 396–414. Rudolph, G. and Schmidt, M. (2013). Differential Geometry and Mathematical Physics, Part I. Manifolds, Lie Groups and Hamiltonian Systems. Springer, Dordrecht. Sato, M. (1989). The KP hierarchy and infinite-dimensional Grassmann manifolds. Proc. of Sympos. Pure Math., 49, 51–66. Sato, M. and Sato, Y. (1982). Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. Lect. Notes in Num. Appl. Anal., 5, 259–271. Scott-Russell, J. (1844). Report on waves. Report, 14th meeting of the British Association for Advancement of Science, 311-3-90. Shiota, T. (1986). Characterization of Jacobian varieties in terms of soliton equations. Invent. Math., 83, 333–382. Siegel, C.L. (1955). Meromorphic funktionen anf kompakten mannigfaltigkeiten. Nachrichten der Akademie der Wissenschaften in Göttingen, Math.-Phys. Klasse, 4, 71–77. Skrypnyk, T. (2019). Symmetric separation of variables for the Clebsch system. J. Geom. Physics, 135, 204–218. Sokolov, V. (2017). Algebraic structures related to integrable differential equations. Sociedade Brasileira de Mathematica, Ensaios Mathematicos, 31, 1–108. Steklov, V.A. (1893). Über die Bewegung eines festen Körper in einer Flüssigkeit. Math. Ann., 42, 273–374. Steklov, V.A. (1896). A certain case of motion of a heavy rigid body having a fixed point. Trudy Otdel. Fiz. Nauk. Obsh. Lyubit. Estestvozn., 8(2), 19–21. Sternberg, S. (1983). Lectures on Differential Geometry, 2nd edition. Chelsea, New York. Symes, W. (1980). Systems of Toda type, inverse spectral problems and representation theory. Invent. Math., 59, 13–53. Teschil, G. (2000). Jacobi Operators and Completely Integrable Nonlinear Lattices. American Mathematical Society, Providence. Toda, M. (1967). Wave propagation in anharmonic lattices. J. Phys. Soc. of Japan, 23, 501–506. Tsiganov, A.V. (1999). The Lax pairs for the Holt system. J. Phys. A, Math. Gen., 32(45), 7983–7987.

References

303

Valent, G. (2010). On a class of integrable systems with a cubic first integral. Comm. Math. Phys., 299, 631–649. Vanhaecke, P. (2001). Integrable Systems in the Realm of Algebraic Geometry. Springer-Verlag, Berlin. Wadati, M. and Toda, M. (1972). The exact N-soliton solution of the Kortewegde Vries equation. J. Phys. Soc. Japan, 32, 1403–1411. Weil, A. (1983). Euler and the Jacobians of Elliptic Curves. Birkhauser Boston, Inc., Boston. Weinstein, A. (1971). Symplectic manifolds and their Lagrangian submanifolds. Adv. in Math., 6, 329–346. Weinstein, A. (1977). Lectures on symplectic manifolds. CBMS Regional Conference Series in Mathematics, no. 29, AMS. Whittaker, E.T. (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge. Wojciechowski, S. (1985). Integrability of one particle in a perturbed central quartic potential. Physica Scripta, 31, 433–438. Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevskii, L.P. (1980). Soliton Theory, Inverse Scattering Method. Nauka, Moscow. Ziglin, S.L. (1983a). Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics I. Funktsional. Anal. i Prilozhen., 16(3), 30–41, 96. Ziglin, S.L. (1983b). Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics II. Funktsional. Anal. i Prilozhen., 17(1), 8–23.

Index

A

Bäcklund transformation, 269 auto-, 270 Baker–Akhiezer function, 273, 284 Bobylev–Steklov top, 90 Boussinesq equation, 243, 279 brachistochrone, 65 Burgers equation, 269, 270

Camassa–Holm equation, 243, 270 canonical basis, 2 transformation, 55, 66 Carleman’s condition, 138 Cartan homotopy formula, 18 Casimir functions, 74 Cauchy problem, 159 central element, 279 force, 65 Clebsch’s case, 91 coadjoint orbit, 36 representation, 36 cocycle, 280 coisotropic space, 2 Cole–Hopf transformation, 269 commutator, 15 commute, 11, 27 complete, 73 completely integrable system, 67, 68, 74 constant of motion, 27 cyclic coordinate, 59

C

D

Calogero –Moser system, 124, 273 system, 123

Darboux coordinates, 22 theorem, 23

action, 49 -variables, 74 adjoint orbit, 36 wave function, 284 Adler–Kostant–Symes theorem, 104 algebra of differential operators, 275 W, 278 algebraic integrable system, 165 amplitude, 63 angle-variables, 74 angular momentum, 97 Arnold–Liouville theorem, 68, 73 B

Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc.

306

Integrable Systems

Davey–Stewartson equations, 270, 271 Dym equation, 271 type equation, 272 dynamical system, 8 E elliptic potential, 273 energy functional, 97 Euler –Arnold equation, 99, 107, 109 –Lagrange equation, 51 –Poinsot motion, 78 differential equation, 85 equation, 51 top, 32, 78, 108, 173 extremal, 51 F Fay differential identity, 286 identity, 286 first integral, 27 fixed singularity, 160 Flaschka variables, 150 flow, 9 Fréchet derivative, 280 Frobenius theorem, 48 functions in involution, 27 G Gardner equation, 269 transformation, 269 Gelfand Dickey reduction, 277 generalized algebraic completely integrable systems, 221, 222 periodic Toda systems, 213 generating function, 56 geodesic, 51 flow, 96 flow on an ellipsoid, 114 flow on SO(4), 33 flow on SO(n), 107, 210

Goryachev–Chaplygin top, 90, 121, 231 Griffiths theorem, 149 Gross–Neveu system, 214 H Hénon–Heiles system, 31, 156, 191, 229 Hamilton –Jacobi equation, 57 canonical equations, 31, 54 Hamiltonian (see also Jaynes–cummings–Gaudin Hamiltonian), 25, 31, 54, 67 system, 25, 67 vector field, 30 harmonic oscillator, 35, 60, 75, 100 heat equation, 269, 270 Heaviside function, 290 Heisenberg algebra, 31, 279, 280 subalgebra, 280 Hesse–Appel’rot top, 90 Hirota bilinear equations, 289 symbol, 289 Holt system, 125 I, J integral action, 49 curve, 9 interior product, 16 isospectral deformation, 104 isotropic space, 2 Jacobi elliptic function, 63 geodesic flow on an ellipsoid, 114 identity, 26, 31 matrix, 129 theorem, 58 Jaynes–cummings–Gaudin Hamiltonian, 126 K Kac–van Moerbeke lattice, 212 Kadomtsev–Petviashvili equation, 243

Index

Kaup–Kuperschdmit system, 123 KdV equation, 241 Kepler problem, 61, 100 kinetic energy, 97 Kirchhoff equations, 34, 91 Kolossof potential, 215 Korteweg–de Vries system, 123 modified, 269 Kostant–Kirillov orbit, 36 Kowalewski matrix, 161 top, 33, 83, 119, 175 L Lagrange (see also system) equation, 51 top, 82, 115, 153, 236 Lagrangian, 50, 53 manifold, 5 section, 6 space, 2 Lax equation, 103 pair, 103 left-invariant, 96 left angular velocity, 97 Legendre transformation, 52 Leibniz rule, 26 Levi–Civita symbol, 154 Lie bracket, 18 derivative, 15, 16 light–cone coordinates, 126 Liouville (see also Sturm–Liouville equation) –Arnold–Adler–van Moerbeke theorem, 171 form, 4 integrable system, 67, 74 theorem, 64, 247 Lotka–Volterra system, 157 Lyapunov–Steklov’s case, 93 M, N Manakov geodesic flow, 109, 200 Mittag–Leffler problem, 148

307

Miura transformation, 269 mobile singularity, 160 moment of inertia operator, 98 Moser’s lemma, 22 Nahm’s equations, 154 Neumann problem, 114 Noether theorem, 28 normal order, 275 Novikov equation, 273 O, P one-parameter group of diffeomorphism, 9 orientable, 5 Parseval identity, 251 phase space, 31, 55 Poisson bracket, 15, 26, 28 manifold, 27 structure, 26, 27 theorem, 27 pseudo-differential operator, 275 pull-back, 16 R, S RDG integrable potential, 223, 225 rectification theorem, 48 regular value, 48 Riccati equation, 269 right angular velocity, 97 Sawada-Kotera system, 123 Schrödinger equations, 118, 125, 243, 269, 271, 272 Schur polynomials, 288 Siegel half–space, 289 simple pendulum, 62 sine–Gordon equation, 125, 270, 271 special orthogonal group, 37 spectral curve, 104 parameter, 103 spherical pendulum, 100 Sturm–Liouville equation, 257 Sutherland system, 124 symplectic basis, 2

308

Integrable Systems

diffeomorphism, 6 form, 2, 3 manifold, 3 matrix, 66 morphism, 6 space, 2 structure, 2, 3 transformation, 66 symplectomorphism, 6 system integrable by quadrature, 73 of Lagrange equations, 28 T, V tau function, 286 tensor, 98 Toda lattice, 150 trajectory, 9 trivial invariants, 74

variation, 50 variational problem, 49 vector field, 7 vertex operator, 287 Virasoro algebra, 280 structure, 281 Volterra group, 276 volume form, 5 W, Y wave function, 284 operator, 283 weight-homogeneous, 161, 225 Yang–Mills equations, 119 field, 93

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