Nonlinear reaction-diffusion systems : conditional symmetry, exact solutions and their applications in biology 978-3-319-65467-6, 3319654675, 978-3-319-65465-2

Table of contents :
Front Matter ....Pages i-xiii
Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications (Roman Cherniha, Vasyl’ Davydovych)....Pages 1-44
Q-Conditional Symmetries of Reaction-Diffusion Systems (Roman Cherniha, Vasyl’ Davydovych)....Pages 45-76
Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems (Roman Cherniha, Vasyl’ Davydovych)....Pages 77-118
Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems (Roman Cherniha, Vasyl’ Davydovych)....Pages 119-154
Back Matter ....Pages 155-160

Citation preview

Lecture Notes in Mathematics  2196

Roman Cherniha Vasyl’ Davydovych

Nonlinear ReactionDiffusion Systems Conditional Symmetry, Exact Solutions and their Applications in Biology

Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg

2196

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

Roman Cherniha • Vasyl’ Davydovych

Nonlinear Reaction-Diffusion Systems Conditional Symmetry, Exact Solutions and their Applications in Biology

123

Vasyl’ Davydovych Institute of Mathematics National Academy of Science Kyiv, Ukraine

Roman Cherniha Institute of Mathematics National Academy of Science Kyiv, Ukraine

ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-65465-2 DOI 10.1007/978-3-319-65467-6

ISSN 1617-9692 (electronic) ISBN 978-3-319-65467-6 (eBook)

Library of Congress Control Number: 2017950000 Mathematics Subject Classification (2010): Primary: 35K57, 35K58, 35K61, 35Q79, 35Q92, 92D25, 92D99; Secondary: 17B80, 17B81, 34A05, 35N05, 35N10 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

‘. . . And this devotion [to biology] is achieved only by deep understanding of beauty, infinity, symmetry, and harmony in nature.’ Taras Shevchenko, the great Ukrainian poet

Preface

Nowadays it is widely known that nonlinear reaction-diffusion systems (RDSs) are governing equations for many very important nonlinear models used to describe various processes in physics, biology, ecology, chemistry, etc. Since the 1970s they have been studied extensively by means of different mathematical methods. Construction of particular exact solutions for nonlinear partial differential equations (PDEs), especially for nonlinear RDSs, is an important problem. Finding exact solutions that have a physical, chemical or biological interpretation is of fundamental importance. The well-known principle of linear superposition cannot be applied to generate new exact solutions of nonlinear PDEs and systems of PDEs. Thus, the classical methods are not applicable for solving these equations. Of course, a change of variables can sometimes be found that transforms a given nonlinear PDE into a linear equation, but finding exact solutions of most nonlinear PDEs generally requires new methods. The most powerful methods for construction of exact solutions of nonlinear PDEs are symmetry-based methods. These methods originate from the Lie method, which was created by the prominent Norwegian mathematician Sophus Lie at the end of the nineteenth century. The method was essentially developed using modern mathematical language by L.V. Ovsiannikov, G. Bluman, N. Ibragimov, W.F. Ames and some other researchers in the 1960s and 1970s. Although the technique of the Lie method is well known, the method still attracts the attention of researchers and new results are published on a regular basis. However, it is well known that some nonlinear PDEs and systems of PDEs arising in applications have poor Lie symmetry. For example, the Fisher and Fitzhugh– Nagumo equations, which are widely used in mathematical biology, are invariant only under time and space translations. The Lie method is not efficient for such equations since it enables only those exact solutions to be constructed, which can be easily obtained without using this method. Taking this fact into account, other symmetry-based methods were developed. The best known among them is the method of nonclassical symmetries proposed by G. Bluman and J. Cole in 1969. Even though this approach was suggested almost 50 years ago, its successful applications for solving nonlinear equations were accomplished only in the 1990s vii

viii

Preface

owing to D.J. Arrigo, P. Broadbridge, P. Clarkson, J.M. Hill, E.L. Mansfield, M.C. Nucci, P. Olver, E. Pucci, G. Saccomandi, E.M. Vorob’ev, P. Winternitz and others. A prominent role in applications and further development of the nonclassical symmetry method belongs to the Ukrainian school of symmetry analysis, which was created in the early 1980s and led by W.I. Fushchych (V.I. Fyshchich) until 1997 when he passed away. In particular, a concept of conditional symmetry was worked out and its applications to a wide range of nonlinear PDEs were realized by M. Serov, I. Tsyfra, R. Zhdanov, R. Popovych, R. Cherniha and others. Notably, following Fushchych’s proposal dating back to 1988, we use the terminology ‘Q-conditional symmetry’ instead of ‘nonclassical symmetry’. It turns out that the problem of finding Q-conditional symmetry gets to be much more complicated in the case of the two- and multi-component nonlinear systems of PDEs. To the best of our knowledge, the pioneering papers devoted to the search for Q-conditional symmetries of systems of reaction-diffusion equations appeared only in the early 2000s, i.e., about 30 years later than the seminal Bluman and Cole work was published. Moreover, the majority of such papers were published during the last decade. This book is devoted to searching for conditional symmetries of nonlinear RDSs and their application for constructing exact solutions. Properties of exact solutions obtained for several nonlinear systems (especially the diffusive Lotka–Volterra system (DLVS)) arising in real-world applications are studied in order to provide their biological, ecological and/or physical interpretation. The book is mostly based on authors’ papers published during the last 10 years. Notably, several misprints and inexactnesses arising in those papers were corrected during the book preparation. To the best of our knowledge, this is the first monograph devoted to search and application of conditional symmetries in the case of systems of nonlinear PDEs (not scalar PDEs!). The reader does not need to study symmetry-based methods in detail in order to use the exact solutions obtained in an explicit form and to construct new solutions using conditional symmetry operators derived in the book. Each chapter contains both ideas for further theoretical generalizations and examples of realworld applications. In Chap. 1, all the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and others were derived in the 2000s, this is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation. In Chap. 2, the recently developed theoretical background for searching Q-conditional symmetries of systems of evolution PDEs is presented. We generalize the standard notation of Q-conditional symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of symmetry

Preface

ix

of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some special properties, which distinguish it from the standard conditional symmetry. The general class of two-component nonlinear RDSs is examined in order to find the Q-conditional symmetry operators. The relevant systems of so-called determining equations are solved under additional restrictions. As a result, several RDSs possessing conditional symmetry are constructed. In particular, it is shown that DLVS, the Belousov–Zhabotinskii system (with the correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry. In Chap. 3, two- and three-component DLVSs are examined in order to find Q-conditional symmetries, to construct exact solutions and to provide their biological interpretation. A complete description of Q-conditional symmetries of the first type (a special subset of nonclassical symmetries) of these nonlinear systems is derived. An essential part of this chapter is devoted to the construction of exact solutions of the systems in question using the symmetries obtained. Starting from examples of travelling fronts (finding such solutions is important from the applicability point of view), we concentrate mostly on finding exact solutions with a more complicated structure. As a result, a wide range of such exact solutions are constructed for the two-component DLVS and some examples are presented for the three-component DLVS. Moreover, a realistic interpretation for two and three competing species is provided for some exact solutions. In Chap. 4, two classes of two-component nonlinear RDSs are studied in order to find Q-conditional symmetries of the first type, to construct exact solutions and to show their applicability. The first class involves systems with the constant coefficient of diffusivity, while the second one contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the RDSs belonging to the above classes and admitting Q-conditional symmetries of the first type. The systems obtained allow us to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravitydriven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented. The book is a monograph. Its academic level is suitable for graduate students and higher. Some parts of the book may be used in ‘Mathematical Biology’ and ‘Nonlinear Partial Differential Equations’ courses for master students and in the final year of undergraduate studies. Nowadays such courses are common in all leading universities all over the world. The book was typeset in LaTeX using the Springer templates; the figures were drawn using the computer algebra package MAPLE. The authors thank their Ukrainian colleagues for fruitful discussions, valuable critique and helpful suggestions, which helped us to write this modest work. We are especially grateful to Sergii Kovalenko, Anatoly Nikitin, Oleksii Pliukhin and Mykola Serov.

x

Preface

R. Ch. is indebted to John R. King for valuable discussions in 2013–2015 when he was Marie Curie Fellow at the University of Nottingham. R. Ch. is also grateful to George Bluman, Phil Broadbridge, Malte Henkel, Changzheng Qu, and Jacek Waniewski for fruitful discussions and valuable comments, which inspired us to do further research and generated new ideas. Last but not least, we are grateful to Eva Hiripi, who invited us to write this book, and for her continuous support during the preparation of the monograph. Finally, we express our thanks to our mothers, Stefaniya and Valentyna, for their love and support! R. Ch. reaffirms his love and thankfulness to his wife, Nataliya, for her love, patience and understanding throughout the process of writing the book. Kyiv, Ukraine Kyiv, Ukraine May 2017

Roman Cherniha Vasyl’ Davydovych

Contents

1

Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Nonlinear Reaction-Diffusion Equations in Mathematical Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Nonclassical Symmetry: Historical Review, Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Examples of Q-Conditional Symmetries of Some Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Determining Equations for the General Reaction-Diffusion-Convection Equation.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations with Constant Diffusivity . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Exact Solutions of Some Equations Arising in Biological Models .. . 1.7 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations with Variable Diffusivity .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8 Exact Solutions of Some Equations with Power-Law Diffusivity and Their Interpretation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2 Q-Conditional Symmetries of Reaction-Diffusion Systems. . . . . . . . . . . . . . 2.1 Reaction-Diffusion Systems and Their Applications .. . . . . . . . . . . . . . . . . 2.2 Q-Conditional Symmetry for Systems of Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Systems of Determining Equations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Conditional Symmetries of Reaction-Diffusion Systems with Constant Diffusivities . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law Diffusivities . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 4 9 13 17 21 33 36 40 41 45 45 48 52 56 61 72 73 xi

xii

Contents

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Lotka–Volterra System and Its Application .. .. . . . . . . . . . . . . . . . . . . . 3.2 The Two-Component Diffusive Lotka–Volterra System . . . . . . . . . . . . . . 3.2.1 Determining Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Q-Conditional Symmetry.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Reductions to Systems of Ordinary Differential Equations and Exact Solutions .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 The Three-Component Diffusive Lotka–Volterra System.. . . . . . . . . . . . 3.3.1 Lie Symmetry .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Determining Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.3 Q-Conditional Symmetry of the First Type.. . . . . . . . . . . . . . . . . . . 3.3.4 Exact Solutions and Their Interpretation . .. . . . . . . . . . . . . . . . . . . . 3.4 A Hunter-Gatherer–Farmer Population Model . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

89 98 99 101 102 107 112 115 117

4 Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems . .. . . . . . . . . . . . . . . . . . . . 4.1 Determining Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Reaction-Diffusion Systems with Constant Diffusivities . . . . . . . . . . . . . 4.2.1 Q-Conditional Symmetry of the First Type.. . . . . . . . . . . . . . . . . . . 4.2.2 Reductions, Exact Solutions and Their Interpretation . . . . . . . . 4.3 Reaction-Diffusion Systems with Variable Diffusivities .. . . . . . . . . . . . . 4.3.1 Q-Conditional Symmetry of the First Type.. . . . . . . . . . . . . . . . . . . 4.3.2 Reductions and Exact Solutions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Application to a Physically Motivated Problem .. . . . . . . . . . . . . . 4.4 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

119 119 122 123 132 137 137 143 145 147 151

77 77 79 79 82

A List of Reaction-Diffusion Systems and Exact Solutions . . . . . . . . . . . . . . . . 155 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 157

Acronyms

BVP CSCP DE DLVS FNE iff KPPE ODE PDE RDCE RDE RDS w.r.t.

Boundary value problem Conditional symmetry classification problem Determining equation Diffusive Lotka–Volterra system Fitzhugh–Nagumo equation if and only if Kolmogorov–Petrovskii–Piskunov equation Ordinary differential equation Partial differential equation Reaction-diffusion-convection equation Reaction-diffusion equation Reaction-diffusion system With respect to

xiii

Chapter 1

Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications

Abstract All the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and the others were derived in the 2000s, it is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation.

1.1 Nonlinear Reaction-Diffusion Equations in Mathematical Biology Since the seventeenth century when Leibnitz and Newton discovered differential and integral calculus, differential equations have been the most powerful tools for mathematical modelling of various processes in physics, chemistry biology, medicine, ecology, etc. Of course, pioneering models were created in order to express some classical laws in physics and astronomy. Probably the first nontrivial biological model was created in 1838 by Verhulst in order to describe the time evolution of species population (in particular, the total world population of people). His model is usually called the logistic model and has the form (in dimensionless variables) dU D U.1  U/; dt

U.0/ D N0 > 0

and is the classical example in any textbook on Mathematical Biology. Its exact solution is well known UD

N0 et 1 C N0 .et  1/

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6_1

(1.1)

1

2

1 Scalar Reaction-Diffusion Equations. . .

Fig. 1.1 Solution (1.1) of the logistic equation for N0 D 1:25I 0:75I 0:05 (green, blue and red curve, respectively)

and depending on the value N0 suggests three different scenarios for the population evolution, which are presented in Fig. 1.1. There are several other mathematical models based on ordinary differential equations (ODEs) and used for describing biological processes, which can be exactly solved. For example, the simplified Fitzhugh–Nagumo model dU D rU.1  U/.U  /; dt

U.0/ D N0 ; 0 <  < 1:

The situation is changed drastically if one constructs mathematical models in order to take into account both time and space so that an unknown function (or vector-function) is the function u of the variables x D .x1 : : : xn / and t. The most known biological model in this direction was created by Fisher in 1937 and is based on the nonlinear partial differential equation (PDE) ut D u C u.1  u/; where  is the Laplace operator and u.t; x/ means the concentration of cells (population, drugs, molecules, etc.). The model was proposed in order to describe the spread in space of a favoured gene in a population. In contrast to the logistic model, the Fisher equation cannot be exactly solved taking into account any reasonable initial and/or boundary conditions even in one-dimensional space. The

1.1 Nonlinear Reaction-Diffusion Equations in Mathematical Biology

3

only known exact solution is in the form of a travelling front (see Sect. 1.6 for details). We remind the reader that a plane wave solution, which is nonnegative, bounded and satisfies the zero Neumann conditions at infinity is usually called travelling front. In the seminal work [43] (see formulae (13) and (16) therein) the equation ut D u C u.1  u/2 was introduced in order to generalize the Fisher equation. Notably, the Fisher equation is not presented in [43], nevertheless many authors wrongly cite this paper in this regard and even propose to call the Fisher equation the Fisher–Kolmogorov– Petrovskii–Piskunov equation. In our opinion, the above equation with the cubic nonlinearity can be called the Kolmogorov–Petrovskii–Piskunov equation (KPPE). Since 1937 when the Fisher equation was derived, a wide range of reactiondiffusion equations (RDEs) have been introduced in order to describe processes in biology, ecology and medicine (see the well-known books [13, 52, 53, 58] and the recent books [45, 73]). One may unite them in the form of the general RDE ut D r  .D.u/ru/ C F.u/

(1.2)

  (hereafter the standard notation r D @x@1 ; : : : ; @x@n is used). In (1.2) the variable diffusivity D.u/ > 0 (typically it is a constant) arises in more and more modelling situations of biomedical importance from diffusion of genetically engineered organisms in heterogeneous environments to the effect of white and grey matter in the growth and spread of brain tumours [52]. The function F.u/ > 0 represents a source (sink) of cells and typically is called the reaction term. In particular, the most typical forms of the nonconstant diffusivity and the reaction term are D.u/ D um and F.u/ D 1 up  2 uq (see Sects. 11.3 and 13.4 in [52]). For example, the powerlaw diffusivity occurs as an extension of the classical diffusion model, when there is an increase in diffusion due to population pressure (see Sect. 11.3 in [52] and the references therein). There are processes, in which the convective transport plays an essential role (e.g., the evolution of fish population in a river cannot be described properly, if one neglects the stream velocity). As a result one needs to introduce the term V  ru into RDE (1.2). In particular, the velocity vector V can be a function of the concentration u (if V D constant then the equation obtained is equivalent to RDE (1.2)). Thus, the RDE with a nonlinear convective term ut D r  .D.u/ru/ C V.u/  ru C F.u/; which describes three transport mechanisms (diffusion, convection and reaction/kinetics) is obtained. The problem, which occurs with finding exact solutions for the models based on the Fisher equation and its various generalizations, is very natural because

4

1 Scalar Reaction-Diffusion Equations. . .

they are nonlinear PDEs. Typically, the mathematical models used for description of biomedical processes are based on nonlinear PDEs and usually they are not linearizable (in contrast to many problems occurring in physics!). The well-known principle of linear superposition cannot be applied to generate new exact solutions to nonlinear PDEs. This means that classical methods (the Fourier method, the methods of the Laplace transformations and the Green function, etc.) are not applicable for solving such PDEs. Thus, the construction of particular exact solutions for these equations is a nontrivial problem. Finding exact solutions that have a physical, chemical or biological interpretation is of fundamental importance.

1.2 Nonclassical Symmetry: Historical Review, Definitions and Properties In 1969, Bluman and Cole [9] introduced an essential generalization of Lie symmetry (other terminology used for this symmetry is ‘Lie point symmetry’) using the (1C1)-dimensional linear heat equation ut D uxx : The crucial idea used for introducing the notion of a new kind of symmetry, later called nonclassical symmetry, is to change the classical criterion of Lie symmetry by inserting into the criterion an additional equation, which is produced by the symmetry in question. As a result, a so-called system of determining equations (DEs) in finding the symmetry in question becomes nonlinear (in contrast to that for finding Lie symmetry). Integration of the system of DEs is the most difficult task for finding nonclassical symmetry of the PDE in question. At the end of the 1980s and in the 1990s, the notion of nonclassical symmetry was further developed and applied to various PDEs in [5–7, 17, 23, 24, 33, 35, 38, 46, 55– 57, 60, 61, 66–68, 77, 78] and many other papers (see an extensive overview in [71]). A new generalization of Lie symmetry, conditional symmetry, was suggested by Fushchych and his collaborators [34], [36, Sect. 5.7]. Note that the notion of nonclassical symmetry can be derived as a particular case from conditional symmetry but not vice versa (see, e.g., a highly nontrivial example in [20]). In the mid-1990s, the notion of generalized conditional symmetry was introduced [31, 48, 69, 76], which again can be considered as a special case of conditional symmetry. Taking this into account, to avoid any misunderstanding we use the terminology Q-conditional symmetry [36] instead of nonclassical symmetry. In fact, there are several types of non-Lie symmetries at the present time and each of them can be called a nonclassical symmetry.

1.2 Nonclassical Symmetry: Historical Review, Definitions and Properties

5

Now we present a definition of Q-conditional symmetry for an arbitrary PDE with smooth coefficients. Let us consider the k-order PDE   (1.3) L t; x; u; u; : : : ; u D 0; k  1; 1

k

where u D u.t; x/ is an unknown function, u means a totality of s-order derivatives s of u.t; x/ (s D 1; 2; : : : ; k) and L is a given smooth function. Similarly to Lie symmetries, Q-conditional symmetries are constructed in the form of the first-order differential operators Q D  0 .t; x; u/@t C  1 .t; x; u/@x C .t; x; u/@u ; . 0 /2 C . 1 /2 ¤ 0;

(1.4)

where the operator coefficients  0 .t; x; u/;  1 .t; x; u/ and .t; x; u/ should be found using the well-known criterion. Throughout this book we use the notation Q.u/ D 0 for the first-order PDE  0 ut C  1 ux   D 0; which is generated by the Q-conditional symmetry (1.4). Rigorously speaking, one should write the invariance surface condition (see, e.g., [8] for details) ˇ ˇ Q.u  u0 .t; x// ˇ

uDu0 .t;x/

D 0;

which leads exactly to the above first-order PDE for arbitrary solution u D u0 .t; x/. Definition 1.1 Operator (1.4) is called the Q-conditional symmetry of PDE (1.3) if the following invariance criterion is satisfied: ˇ Q .L/ ˇˇ k

M

D 0;

(1.5)

where the differential operator Q is the k-order prolongation of operator (1.4) and k the manifold M is defined by the system of equations L D 0;

Q.u/ D 0;

@pCq Q.u/ D 0; @ t p xq

1 pCq k1

in the prolonged space of the variables t; x; u; u; : : : ; u : 1

k

Remark 1.1 The operator Q is the kth-order prolongation of the operator Q. Its k

coefficients are expressed via the functions  0 ;  1 and  by the well-known formulae (see, e.g., [59, 64]).

6

1 Scalar Reaction-Diffusion Equations. . .

The definition generalizes the classical definition of Lie symmetry. We remind the reader that the latter is defined by the same criterion, however the simpler manifold ML D fL D 0g is used instead of M . In particular, each Lie symmetry is automatically a Q-conditional symmetry. The above definition allows us to formulate the main properties of Q-conditional symmetries, which can be easily proved. Property 1.1 If operator (1.4) is the Q-conditional symmetry of PDE (1.3), then this symmetry can be multiplied by an arbitrary smooth function M.t; x; u/ and the operator obtained is again the Q-conditional symmetry of the same PDE. Property 1.2 In contrast to Lie symmetries, all possible Q-conditional symmetries of PDE (1.3) form a set, which is not any Lie algebra in the general case. Property 1.3 Similarly to Lie symmetries, each Q-conditional symmetry (1.4) of PDE (1.3) guarantees reduction of this equation to an ODE. Proof of Property 1.1 To avoid cumbersome calculations, we restrict ourselves to the case of second-order PDEs, i.e., Eq. (1.3) with k D 2: L.t; x; u; ut ; ux ; utt ; utx ; uxx / D 0:

(1.6)

The proof for the arbitrary order k can be realized in the same way. Let us assume that operator (1.4) is a Q-conditional symmetry of Eq. (1.6). Thus, according to Definition 1.1, the condition ˇ Q .L/ ˇˇ 2

M

D 0; M D fL D 0; Q.u/ D 0; Dt .Q.u// D 0; Dx .Q.u// D 0g

is fulfilled. Hereafter we use the notations Dt D @t C ut @u C utt @ut C utx @ux ; Dx D @x C ux @u C utx @ut C uxx @ux : Let us show that the operator Y D M.t; x; u/Q; where M is an arbitrary smooth function, is also a Q-conditional symmetry of Eq. (1.6). Thus, we need to show that the criterion ˇ ˇ Y .L/ ˇ 2

M

is fulfilled.

D 0; M  D fL D 0; Y.u/ D 0; Dt . Y.u// D 0; Dx . Y.u// D 0g

1.2 Nonclassical Symmetry: Historical Review, Definitions and Properties

7

Obviously the manifold M  coincides with the manifold M in the prolonged space of variables, because Dt . Y.u// D MDt .Q.u// C Q.u/Dt .M/; Dx . Y.u// D MDx .Q.u// C Q.u/Dx .M/: The second prolongations of the operators Q and Y have the forms Q D Q C t @ut C x @ux C tt @utt C tx @utx C xx @uxx 2

(1.7)

and  Y D Y C t @ut C x @ux C tt @utt C tx @utx C xx @uxx : 2

Hereafter  and  with indices are the coefficients of the second prolongation of the operators Q and Y and are calculated by the well-known formulae (see, e.g., [59, 64]) t D Dt ./  ut Dt . 0 /  ux Dt . 1 /; x D Dx ./  ut Dx . 0 /  ux Dx . 1 /; tt D Dt .t /  utt Dt . 0 /  utx Dt . 1 /;

(1.8)

tx D Dx .t /  utt Dx . 0 /  utx Dx . 1 /; xx D Dx .x /  utx Dx . 0 /  uxx Dx . 1 / and t D Dt .M/  ut Dt .M 0 /  ux Dt .M 1 /; x D Dx .M/  ut Dx .M 0 /  ux Dx .M 1 /; tt D Dt .t /  utt Dt .M 0 /  utx Dt .M 1 /; tx D Dx .t /  utt Dx .M 0 /  utx Dx .M 1 /;  D Dx .x /  utx Dx .M 0 /  uxx Dx .M 1 /: xx

Let us show that the coefficients t ; x ; tt ; tx and xx are related to the corresponding coefficients with stars on the manifold M . In fact,     t D M Dt ./  ut Dt . 0 /  ux Dt . 1 / C Dt .M/    0 ut   1 ux D Mt  Dt .M/Q.u/ D Mt ;

8

1 Scalar Reaction-Diffusion Equations. . .

    x D M Dx ./  ut Dx . 0 /  ux Dx . 1 / C Dx .M/    0 ut   1 ux D Mx  Dx .M/Q.u/ D Mx ; because Q.u/ D 0 on M . Similarly     tt D M Dt .t /  utt Dt . 0 /  utx Dt . 1 / C Dt .M/ t   0 utt   1 utx D Mtt  Dt .M/Dt .Q.u// D Mtt ;     tx D M Dx .t /  utt Dx . 0 /  utx Dx . 1 / C Dx .M/ t   0 utt   1 utx  xx

D Mtx  Dx .M/Dt .Q.u// D Mtx ;     D M Dx .x /  utx Dx . 0 /  uxx Dx . 1 / C Dx .M/ x   0 utx   1 uxx D Mxx  Dx .M/Dx .Q.u// D Mxx ;

because Dt .Q.u// D 0 and ˇ Dx .Q.u// D 0.ˇ ˇ ˇ Y Thus, we obtain .L/ ˇ  D M Q .L/ ˇ D 0. 2 2 M M The proof is now complete. Let us consider a class of the k-order evolution equations

t u

  ut D F t; x; u; ux ; : : : ; u.k/ ; k  1; x .s/

(1.9)

where F is an arbitrary smooth function and u D u.t; x/; ux D @@xus ; s D 1; 2; : : : ; k. In order to find the Q-conditional symmetry (1.4) of this equation, one needs to examine two essentially different cases: a)  0 ¤ 0;

s

b)  0 D 0;  1 ¤ 0:

In fact, taking into account Property 1.1, the symmetry in question has essentially different forms in these cases, namely: Q D @t C .t; x; u/@x C .t; x; u/@u

(1.10)

Q D @x C .t; x; u/@u

(1.11)

in case a) and

in case b). It turns out that the systems of DEs for finding Q-conditional symmetries (1.10) and (1.11) are essentially different and we will discuss this issue at the end of Sect. 1.4. Hereafter we concentrate mostly on case a). In this case, Definition 1.1 can be simplified. In fact, one notes that differential consequences of equation Q.u/ D 0

1.3 Examples of Q-Conditional Symmetries of Some Reaction-Diffusion. . .

9

with respect to (w.r.t.) the variables t and x lead to the second- and higherorder PDEs involving the time derivatives utt ; uttt ; : : : and the mixed derivatives utx ; utxx ; : : :, which do not occur in any evolution equation. This means that the PDEs obtained do not play any role in criterion (1.5), hence the definition can be reformulated as follows. Definition 1.2 Operator (1.10) is called the Q-conditional symmetry for an evolution equation of the form (1.9) if the following invariance criterion is satisfied: ˇ Q .ut  F/ˇˇ k

M

D 0;

(1.12)

where the manifold M is formed by two equations fut D F; Q.u/ D 0g. Notably, Definition 1.1 should still be applied to evolution equations provided one searches for Q-conditional symmetries of the form (1.11).

1.3 Examples of Q-Conditional Symmetries of Some Reaction-Diffusion Equations Let us consider a nontrivial example of the application of Definition 1.2 to the fast diffusion equation  1  Ut D U  2 Ux ; x

(1.13)

which has been extensively studied by symmetry-based methods [5, 37, 62]. 1 First of all we simplify Eq. (1.13) by applying the substitution u D U 2 . As a result, we obtain the equation uut D uxx :

(1.14)

Let us apply Definition 1.2 to construct Q-conditional symmetry operators of the form (1.10). The invariance criterion (1.12) takes the form ˇ Q .uut  uxx /ˇˇ D 0; 2

M

where the manifold M is fuut D uxx ; Q.u/ D 0g, while Q is defined by 2 formula (1.7). Now we consider Eq. (1.14) as a manifold in the prolonged space of independent variables t; x; u; ut ; ux ; utx ; utt ; uxx :

10

1 Scalar Reaction-Diffusion Equations. . .

Acting on (1.14) by the operator Q, we arrive at the second-order PDE 2

xx D ut C ut :

(1.15)

The explicit forms t and xx are defined by formulae (1.8) with  0 D 1 and  1 D ; hence t D t C u ut  ux .t C u ut / ; xx D xx C 2xu ux C uu u2x C u uxx    ux xx C 2xu ux C uu u2x  2x uxx  3u ux uxx : To obtain the system of DEs, one needs to insert into (1.15) the above expressions for t and xx and exclude derivatives ut and uxx using (1.14) and the condition Q.u/  ut C ux   D 0 generated by operator (1.10). As a result, one arrives at the equation   xx C 2xu ux C uu u2x  ux xx C 2xu ux C uu u2x C u.  ux / .u  2x  3u ux /

(1.16)

D .  ux / C ut C uu .  ux /  uux .t C u .  ux // : Since the unknown functions  and  do not depend on the derivative ux , we can split Eq. (1.16) w.r.t. ux ; u2x and u3x . Thus, we obtain the system of DEs uu D 0;

(1.17)

uu D 2xu  2uu ;

(1.18)

 C .t C 2x  2u / u C 2xu  xx D 0;

(1.19)

2 C .2x  C t /u  xx D 0:

(1.20)

In contrast to the case when Lie symmetries are studied, system (1.17)–(1.20) is nonlinear. To the best of our knowledge, the general solution of the system of DEs (1.17)–(1.20) is unknown. One may easily check that the partial solution  D 0;  D

6 x2

of system (1.17)–(1.20) leads to a Q-conditional symmetry operator Q D @t C

6 @u ; x2

(1.21)

p which generates the known symmetry Q D @t C 12x2 U@U of Eq. (1.13) [5]. Because each symmetry of Eq. (1.13) is spanned by the basic operators of the four-

1.3 Examples of Q-Conditional Symmetries of Some Reaction-Diffusion. . .

11

dimensional Lie algebra [63] h@t ; @x ; 2t@t C x@x ; x@x C 2U@U i ; we conclude that operator (1.21) is indeed a non-Lie operator. Now we use the symmetry derived above in order to show that Property 1.2 holds. Let us take the operator @t C ˛@x C

6 @u ; ˛ ¤ 0; x2

(1.22)

which is a linear combination of two Q-conditional symmetries of (1.14). It is easily seen that operator (1.22) is not Q-conditional symmetry operator of Eq. (1.14) because the functions  D ˛ and  D x62 do not satisfy (1.19). Thus, Property 1.2 is indeed true. Let us consider now the second-order equations of reaction-diffusion type ut D uxx C f .u/;

(1.23)

which form the most important subclass of evolution equations (1.4) because they are widely used in mathematical modelling (as demonstrated in the previous section). In the early 1990s, several papers were published in order to find Qconditional symmetries of RDEs (the pioneering papers [7, 26, 36, 55, 72] are the most important among others). Here we briefly present the main results derived in those works. Theorem 1.1 Equation (1.23) is Q-conditionally invariant under operator Q D @t C .t; x; u/@x C .t; x; u/@u ;

(1.24)

if and only if (iff) it is locally equivalent to the equation ut D uxx C 3 u3 C 1 u C 0 ;

(1.25)

where 0 ; 1 and 3 are arbitrary constants. It should be noted that the RDE with the most general cubic term ut D uxx C 3 u3 C 2 u2 C 1 u C 0

(1.26)

2 provided 3 ¤ 0. In the case is reduced to (1.25) by the substitution u ! u  3 3 3 D 0, one should also set 2 D 0, otherwise the relevant equation does not admit any operator of the form (1.24). In [7, 26], RDE (1.26) was examined in order to show how the form of Q-conditional symmetry depends on the roots of the cubic term arising in the given equation.

12

1 Scalar Reaction-Diffusion Equations. . .

Theorem 1.2 RDE (1.25) is Q-conditional invariant under operator (1.24) iff: Case 1. u ¤ 0; 3 ¤ 0; Q D @t C

 3p 3 3 u3 C 1 u C 0 @u I 23 u@x C 2 2

Case 2. u D 0; 0 D 0; 3 ¤ 0; Q D @t C b@x  bx u@u ; where the function b.t; x/ is an arbitrary solution of the nonlinear system bt  3bxx C 2bbx D 0; bxxx  bbxx C 1 bx D 0I

(1.27)

Case 3. u D 0; 0 D 2 D 3 D 0; Q D @t C q@x C .a C bu/@u ; where the triplet of the functions .a; b; q/ is the general solution of the system at D axx  2aqx C 1 a; bt D bxx  2bqx C 21 ax ;

(1.28)

qt D qxx  2qqx  2bx : In Case 1, Theorem 1.1 was firstly proved by Serov in [72]; later this was independently shown in [7, 26] without any restriction on the function . We note that the general solution of the nonlinear system (1.27) was constructed in an explicit form (for details see [7]). Therefore, the Q-conditional operators have the form (one may set 1 D 0I ˙1 without loss of generality): 3 3 Q D @t  @x  2 u@u ; 1 D 0; x x p p ! p ! 2 2 3 2 3 Q D @t C tan x @x  cos2 x u@u ; 1 D 1; 2 2 2 2 p p ! p ! 3 2 3 2 2 2 tanh x @x C cosh x u@u ; 1 D 1; Q D @t  2 2 2 2 p ! p p ! 2 2 3 2 3 2 coth x @x  sinh x u@u ; 1 D 1: Q D @t  2 2 2 2

1.4 Determining Equations for the General Reaction-Diffusion-Convection. . .

13

xCc1 There is also the time-dependent solution b D 2tCc of system (1.27) with 1 D 0, 0 however this leads to an operator Q, which is equivalent to the known Lie symmetry operators of the equation in question. It should be noted that the DEs (1.28) (with 1 D 0) for searching symmetries of the linear heat equation, which is a subcase of Case 3, were derived in [9] but not solved therein. Many authors tried to build the general solution of those equations [26, 66, 74]. The most general results were obtained in [4, 35, 49]. In the papers [35] and [49], it was proved that the general solution is expressed in terms of three solutions of the linear heat equation, while the authors of [4] showed how the general solution is also obtainable via the matrix Cole–Hopf transformation.

1.4 Determining Equations for the General Reaction-Diffusion-Convection Equation Here we consider the general nonlinear RDE with a convection term Ut D .A.U/Ux /x C B.U/Ux C C.U/;

(1.29)

where U D U.t; x/ is an unknown function and A.U/; B.U/; C.U/ are smooth functions, which are assumed to be known and A.U/  0 (strictly speaking, the function A.U/ should also have a finite number of roots or no roots). Our aim is to extend the results for the standard RDE on equations of the form (1.29). It is easily shown that the Kirchhoff substitution Z uD

A.U/du  A0 .U/

transforms Eq. (1.29) to the equivalent form uxx D F 0 .u/ut C F 1 .u/ux C F 2 .u/;

(1.30)

where 1 ˇˇ B.U/ ˇˇ 1 ; F .u/ D  ; ˇ ˇ A.U/ UD.A0 /1 .u/ A.U/ UD.A0 /1 .u/ ˇ ˇ F 2 .u/ D C.U/ˇ ; 0 1

F 0 .u/ D

UD.A /

.u/

 1 and A0 is the inverse function of A0 .U/. Thus, Eqs. (1.29) and (1.30) are locally equivalent provided A.U/ is an arbitrary smooth function.

14

1 Scalar Reaction-Diffusion Equations. . .

Theorem 1.3 ( [24]) Equation (1.30) is Q-conditionally invariant under the operator (up to equivalent representations generated by multiplying on the arbitrary smooth function M.t; x; u/): Q D  0 .t; x; u/@t C  1 .t; x; u/@x C .t; x; u/@u ;

(1.31)

iff the functions  0 ;  1 ;  satisfy the following equations: Case 1.  0 D 1;   1 1 uu D 0; uu D 2u1 F 1   1 F 0 C 2xu ;   1    F   1 F 0 u  t1 C 2 1 x1  3u1  F 0

(1.32)

1 D 0; Cx1 F 1 C 3u1 F 2  2xu C xx       0  F C F 2 u C 2x1  u F 0 C F 2 C t F 0 C x F 1  xx D 0:

Case 2.  0 D 0;

 1 D 1;

   x C u  F 1  F 2 Fu0 (1.33)   2  D xx C 2xu C 2 uu  2 Fu1  x F 1  Fu2 C u F 2 F 0 C t F 0 : Proof First of all, we note that two essentially different cases should be examined, i.e., Q-conditional symmetries of the form (1.10) and (1.11) should be found separately. Let us consider the case  0 D 1. In order to construct the relevant system of DEs, we apply Definition 1.2 to Eq. (1.30) with an arbitrary given triplet . F 0 ; F 1 ; F 2 /. The manifold M consists of two equations, hence two derivatives, say ut and uxx , can be expressed via other variables (we remind the reader that the invariance criterion acts in the prolonged space of variables up to the second-order derivatives). So, using the equation Q.u/ D 0 we obtain ut D  1 ux C ;

(1.34)

while the second-order derivative uxx D F 0 . 1 ux C / C F 1 ux C F 2 can be derived from Eq. (1.30), taking into account Eq. (1.34).

(1.35)

1.4 Determining Equations for the General Reaction-Diffusion-Convection. . .

15

Now we calculate the second prolongation of Q, i.e., the operator Q (1.7). Its 2 coefficients are calculated via formulae (1.8) and take the forms   t D t C ut u  ux t1 C ut u1 ;  1  x D x C ux u  ux x C ux u1 ;   1 C utt u1 tt D tt C 2ut tu C u2t uu C utt u  ux tt1 C 2ut tu1 C u2t uu 2utx t1 C ut u1 ;  1 tx D tx C ux tu C ut ux C ut ux uu C utx u  ux tx1 C ux tu1 C ut ux 1 1 1 1 1 1 Cut ux uu C utx u  utx x C ux u  uxx t C ut u ;  1 1 1 C 2ux xu C u2x uu C uxx u1 xx D xx C 2ux xu C u2x uu C uxx u  ux xx   2uxx x1 C ux u1 : (1.36) Thus, having operator (1.7) with coefficients (1.36) and formulae (1.34)–(1.35) for excluding the derivatives ut and uxx , we apply Definition 1.2 to Eq. (1.30) and derive the expression     1 C u2x 2u1 F 1   1 F 0 C 2x1 u  uu u3x uu   1 Cux  F 1 1 F 0 u t1C2 1 x13u1  F 0C 1 F 1C3u1 F 22xuCxx   C.F 0 C F 2 /u C 2x1  u .F 0 C F 2 / C t F 0 C x F 1  xx D 0: (1.37) Expression (1.37) can be split w.r.t. u3x ; u2x ; u1x and u0x because all the functions arising therein do not depend on the variable ux . As a result, one arrives exactly at the system of DEs (1.32). DEs for finding Q-conditional symmetries of the form (1.11) (i.e.,  0 D 0) must be derived using Definition 1.1. The manifold M consists of three equations in this case because the differential consequence of the equation Q.u/ D 0 w.r.t. the variable x belongs to M . As a result, three derivatives ut , ux and uxx can be expressed via other variables. Indeed, the equation Q.u/ D 0 takes the form ux D ;

(1.38)

differentiating (1.38) w.r.t. x, we obtain the second equation uxx D x C u ux  x C u ;

(1.39)

while the third equation is the given reaction-diffusion-convection equation (RDCE) from class (1.30). Obviously, having (1.38) and (1.39), we arrive at ut D

 1  x C u  F 1  F 2 : F0

(1.40)

16

1 Scalar Reaction-Diffusion Equations. . .

The second prolongation of (1.11) has the form (1.7) with the coefficients t D t C ut u ; x D x C ux u ; tt D tt C 2ut tu C u2t uu C utt u ; tx D tx C ux tu C ut ux C ut ux uu C utx u ; xx D xx C 2ux xu C u2x uu C uxx u :

(1.41)

Applying operator (1.7) with coefficients (1.41) to Eq. (1.30) and excluding the derivatives ut ; ux and uxx using formulae (1.38), (1.39) and (1.40), we arrive exactly at Eq. (1.33). The proof is now complete. t u The system of the nonlinear equations (1.32) and the nonlinear equation (1.33) are very complicated and it is impossible to construct their general solutions without some restrictions on the functions F 0 .u/; F 1 .u/ and F 2 .u/. Here we present a partial solution of (1.32) under the restrictions F 1 D .u C 4 / F 0 C 33 u C 2 ;

  F 2 D P3 .u/ F 0 C 3 ;

where P3 .u/ D 0 C 1 u C 2 u2 C 3 u3 . 2 R;  D 0; : : : ; 4/ and F 0 .u/ is an arbitrary smooth function. In this case, the solution of system (1.32) has the form  1 D u C 4 ;

 D P3 .u/:

Thus, the equation uxx D F 0 .u/ Œut C .u C 4 /ux  P3 .u/ C .33 u C 2 /ux  3 P3 .u/ is Q-conditional invariant under the operator Q D @t C .u C 4 /@x C P3 .u/@u : It should be noted that this operator does not coincide with any Lie symmetry of Eq. (1.33) because any RDCE can admit Lie symmetry operators of the form (1.31) only under the restriction u1 D 0 [24]. Let us consider an example. The equation ut D uxx C u2 .1  u/

(1.42)

is often called the Zeldovich equation [27] (the terminology ‘the Huxley equation’ is also used [7]). One may easily check that (1.42) admits only a trivial Lie symmetry generated by the operators of time and space translations (see, e.g., [24]). On the other hand, this equation admits Q-conditional symmetry. In fact, inserting F 0 D 1; F 1 D 0; F 2 D u2 .u  1/

1.5 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations. . .

17

into (1.32), we arrive at the system of DEs 1 1 D 0; uu D 2 1 u1 C 2xu ; uu   1 D 0; u1 C t1 C 2 1 x1  3u1   3u1 .u3  u2 / C 2xu  xx       3 2 1 3 2   C u  u u C 2x  u  C u  u C t  xx D 0:

(1.43)

Integrating the first two equations of system (1.43), we obtain  1 D a.t; x/u C b.t; x/;  D 

a2 3 u C .ax  ab/u2 C d.t; x/u C e.t; x/; 3

(1.44)

where a; b; d and e are unknown smooth functions. In order to find these functions, one needs to substitute (1.44) into the last two equations of system (1.43). Splitting the expressions obtained w.r.t. the different exponents of u, and making corresponding calculations, one derives two operators    1 3 3 2 Q D @t ˙ p u  @x C u  u3 @u 3 2 2 of Eq. (1.42). The above operators of Q-conditional symmetry were independently found in the 1990s by several authors [7, 26, 36, 57]. Hereafter we do not consider the problem of constructing Q-conditional symmetries in case 2 of Theorem 1.3 because it is equivalent (up to a set of transformations) to solving of RDCE (1.29) [77]. Of course, some particular solutions can be found using, for example, the algorithm proposed in [56]. In Sects. 1.5 and 1.7, two subcases of the nonlinear RDCE (1.29), which are most relevant for modelling a wide range of processes in biology, ecology and medicine, are examined in order to find all possible Q-conditional symmetry operators. Note that we always search for purely conditional symmetry operators, i.e., those that are inequivalent to Lie symmetry operators.

1.5 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations with Constant Diffusivity Let us consider a particular case of Eq. (1.30): ut D uxx C uux C C.u/;

 2 R:

(1.45)

It follows from Theorem 1.3 that all Q-conditional symmetries generated by the operator Q D @t C  1 .t; x; u/@x C .t; x; u/@u

(1.46)

18

1 Scalar Reaction-Diffusion Equations. . .

can be found by solving the system of DEs (1.32). Note that we do not consider the case  D 0 since this case has been studied in [26, 55, 72] and the result is presented in Sect. 1.2. Now let us formulate a theorem, which gives complete information on Q-conditional symmetries of Eq. (1.45). Theorem 1.4 ( [18]) Equation (1.45) is Q-conditional invariant under the operator (1.46) up to equivalent representations generated by transformations of the form t ! t; x ! c 1 x C c2 t C c3 t 2 ;

(1.47)

u ! c4 C c5 t C c6 u; iff it has one of the following forms. Case 1. The Burgers equation ut D uxx C uux is Q-conditional invariant under the operator  Q D @t C

    q 2 2 3 u C q @x C a C bu  u  u @u ; 2 2 4

(1.48)

where the triplet of functions .a; b; q/ is the general solution of the system at D axx  2aqx; bt D bxx  2bqx C ax ; qt D qxx  2qqx  2bx : Case 2. The equation ut D uxx C uux C 0 C 2 u2 is Q-conditional invariant under the operator   2 Q D @t C u C @x C .0 C 2 u2 /@u ;  where 0 ; 2 6D 0 are arbitrary constants. Case 3. The equation ut D uxx C uux C 0 C 1 u C 3 u3

(1.49)

1.5 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations. . .

19

is Q-conditional invariant under the operators Qi D @t C pi u@x C

3pi .0 C 1 u C 3 u3 /@u ; i D 1; 2; 2pi  

where pi are the roots of the quadratic equation 2p2 C p C 93  2 D 0;

(1.50)

and 0 ; 1 ; 3 6D 0 are arbitrary constants, and the operators Q D @t C b@x C . bxx  bx u/@u ;

(1.51)

where the function b.t; x/ is an arbitrary solution of the overdetermined system .  3/bxx C 2bbx C bt D 0; bxxx  bbxx C 1 bx D 0; 2

 bxxx C where D

 33 ,

b2x

(1.52)

C 3 1 bx C 30 b D c0 ;

0 1 3 6D 0 and c0 2 R.

Proof The proof of the theorem is based on solving the DEs (1.32), which can be essentially simplified if one considers RDCE (1.45). In fact, we obtain the following system for the function C.u/ and the coefficients  1 and  of operator (1.46): 1 D 0; uu

 C

t1

C

2 1 x1



1 uu D 2u1 .u C  1 / C 2xu ;

2u1 

C

ux1

C 2xu 

1 xx

C

3u1 C.u/

(1.53) D 0;

(1.54)

 .  C.u//u C .2x1  u / .  C.u// C t  ux  xx D 0:

(1.55)

Subsystem (1.53) is easily integrated and its general solutions has the form a  1 D a.t; x/uCb.t; x/;  D  .aC/u3 C.ax ab/u2 Cd.t; x/uCe.t; x/; 3

(1.56)

where a; b; d and e are arbitrary smooth functions. Obviously, one needs to consider two different cases, namely: (I) a 6D 0 and (II) a D 0. Consider case (I). Substituting (1.56) into (1.54), we immediately establish that the function C.u/ can be at maximum a cubic polynomial w.r.t. the variable u: C.u/ D 0 C 1 u C 2 u2 C 3 u3 ;

(1.57)

20

1 Scalar Reaction-Diffusion Equations. . .

where the constants i .i D 0; : : : ; 3/ are determined by the functions a; b; d and e. The relevant formulae have the form 2a2 C a C 93  2 D 0; b.  2a/ D 32 ; d.  2a/ D 3a1 ; e.  2a/ D 3a0 ;

(1.58)

provided 3 6D 0. If 3 D 0, then the first equation (of course, with 3 D 0) of (1.58) is again obtained and then a D  or a D 2 . The value a D  is examined below. The value a D 2 leads to the requirement 2 D 0 therefore the function C.u/ (see (1.57)) is linear. According to Theorem 1 [18], the RDCE (1.45) with the linear C.u/ can be reduced to the form ut D uxx C uux

(1.59)

ut D uxx C uux C 1 u:

(1.60)

or

Equation (1.59) is, of course, the well-known Burgers equation [14, 15] and its Q-conditional symmetry is easily obtained by substitution of C.u/ D 0, a D 2 and (1.56) into the DEs (1.54)–(1.55). After the relevant calculations, we obtain their general solution leading to operator (1.48) with coefficients satisfying (1.49). The analogous procedure was realized for Eq. (1.60), however, only Lie symmetry operators were found. Now we consider the case 3 6D 0 and the subcase 3 D 0 and a D . Solving the system of algebraic equations (1.58) w.r.t. a; b; d; e and substituting the expressions obtained into DE (1.55), we arrive at the general solution (1.57) and  1 D u C 2 ;  D 2 u 2 C 1 u C 0 ;

(1.61)

32  1 D pu  2p ;  3p   D 2p 3 u3 C 2 u2 C 1 u C 0 ;

(1.62)

if 3 D 0 and a D , and

if 3 6D 0. Here the constant p is the solution of the quadratic equation (1.50). So, operator (1.46) with coefficients (1.61) forms the Q-conditional symmetry of RDCE ut D uxx C uux C 0 C 1 u C 2 u2 ;

(1.63)

1.6 Exact Solutions of Some Equations Arising in Biological Models

21

while this operator with coefficients (1.62) forms two Q-conditional symmetries of RDCE ut D uxx C uux C 0 C 1 u C 2 u2 C 3 u3 :

(1.64)

It should be noted that there is only a single Q-conditional symmetry, if 83 C 2 D 0. Finally, we note that Eqs. (1.63) and (1.64) are reduced to the same those with 1 D 0 and 2 D 0, respectively, using local substitutions of the form (1.47) with c3 D c5 D 0. Thus, the examination of the case (I) is now complete and cases 1, 2 and 3 (except operator (1.51)) of the theorem are obtained. The examination of case (II) is rather similar and leads to Q-conditional symmetry operator (1.51) only, in which the function b.t; x/ is the general solution of the nonlinear system (1.52). It should be noted that this system is compatible (for example, b D constant is a solution if c0 D 30 b). However, its general solution is unknown in the case 0 1 3 6D 0. The theoretical background of this difficulty follows from paper [40]. Indeed, one easily checks that both third-order ODEs arising in (1.52) cannot be linearized by point and contact transformations (see Theorems 2.1 and 5.1 in [40]). In the case 0 1 3 D 0, the general solution was found but the operators obtained coincide with the Lie symmetry operators listed in Theorem 1 [18]. Thus, the proof is now complete. t u Remark 1.2 Particular cases of Theorem 1.4 were derived a long time ago in papers [6] (case 1) and [17] (subcases of cases 2 and 3). Remark 1.3 System (1.52) with  D 0 is equivalent to the integrable system (1.27).

1.6 Exact Solutions of Some Equations Arising in Biological Models In this section, we apply the Q-conditional symmetry operators listed in Theorem 1.4 for finding exact solutions of some RDCEs arising in applications. First we note that nonlinear RDCEs, except for a very few examples (like the Burgers equation), are nonintegrable equations. A common task is to construct particular exact solutions in the form of travelling fronts. Because exact solutions of this kind can be successfully found using several different techniques, we do not go into details here but note that a wide range of travelling wave solutions for RDCEs (especially with power-law coefficients) are presented in [39]. Hereafter we concentrate on exact solutions with more complicated structures and typically they contain the travelling waves as particular cases. We start from the nonlinear equation, which was introduced by Fisher in 1937 [29]. In the (1C1)-dimensional case, the Fisher equation reads as ut D uxx C u.1  u/:

(1.65)

22

1 Scalar Reaction-Diffusion Equations. . .

Fig. 1.2 Exact solution (1.66) of the Fisher equation (1.65)

This equation describes the spread in space of a favoured gene in a population. The Fisher equation (1.65) is not integrable and there have been many attempts to construct its exact solutions taking into account any reasonable initial and boundary conditions. In particular, the appropriate exact solution in the form of the travelling front    2 1 1 5 uD ; 1  tanh p x  p t 4 2 6 6

(1.66)

satisfying the natural conditions at infinity x D ˙1 W u D 1; u D 0; was found in [1]. The above conditions are realistic because they coincide with the steady-state solutions of (1.65). Thus, the travelling front (1.66) connects the stable steady-state point u D 1 with the unstable one u D 0 and is presented in Fig. 1.2. If one generalizes (1.65) to the form ut D uxx C u.1  2 u/; then solution (1.66) can be rewritten as 1 u.t; x/ D 2

r 1 C c exp

1 .x  ˛t/ 6

!!2 ;

(1.67)

1.6 Exact Solutions of Some Equations Arising in Biological Models

23

p

where ˛ D 5p6 1 , while 1 , 2 and c are arbitrary constants. If c > 0 and 1 > 0 then (1.67) can be reduced to the form (1.66), if c < 0 then this solution possesses singularities and the solution is complex for each negative 1 . The Fisher equation is invariant w.r.t. the discrete transformations x ! x, hence formula (1.67) produces another travelling front solution by replacing x with x and the front obtained is moving in the opposite direction. However, the Fisher equation possesses only a trivial Lie symmetry (the operators of time and space translations, i.e., Pt and Px ) and has no Q-conditional symmetry (see Theorem 1.1). It turns out that its natural generalization ut D uxx C uux C u.1  2 u/

(1.68)

(here u is a velocity that is proportional to the concentration u) has essentially different symmetry properties. Equation (1.68) was introduced by Murray in [50] as a generalization of the Fisher equation by adding the simplest nonlinear convection term, therefore it can be named the Murray equation. Equation (1.68) is invariant w.r.t. the trivial Lie algebra spanned by the operators Pt D @t and Px D @x [24], i.e., the Lie method leads only to the plane wave ansatz (generated by the linear combination Pt C ˛Px ) u D .!/; ! D x  ˛t; ˛ 2 R;

(1.69)

which reduces (1.68) to the second-order ODE ! ! C .˛ C /! C .1  2 / D 0:

(1.70)

The nonlinear ODE (1.70) (with the arbitrary coefficients ; 1 ; 2 and ˛) cannot be integrated, however, a set of particular solutions (for some values of ˛) were found, hence the exact solutions uD

1 C c exp. x C 2 t/; 2

u D c exp. x C . 2 C 1 /t/;     1   2 2 2 u D C c exp  x  C 1 t ; D 1 

(1.71)

of the Murray equation (1.68) were obtained [18]. In the case 1 D 2 D 1 and c > 0, solution (1.71) is a typical travelling front, possessing similar properties to the exact solution (1.66) of the Fisher equation. Moreover, one may note that this solution with an appropriate  < 0 coincides with the numerical solution pictured in Murray’s book [51, Fig. 11.5b]. It turns out that a new solution can be constructed using the Q-conditional invariance of the Murray equation (1.68). In fact, according to Theorem 1.4 and

24

1 Scalar Reaction-Diffusion Equations. . .

Remark 1.2 this equation admits the Q-conditional symmetry operator   2 Q D @t  u C @x C .1 u  2 u2 /@u :  Using this operator one can construct the exact solution (for details see [24])   1 C c1 exp x C 2 t 2 ; D : (1.72) uD 2 C c0 exp.1 t/  This solution at c0 6D 0 is not of the plane wave form (1.68), hence one cannot be found using the Lie method (see ansatz (1.69)). Note that (1.72) with c0 < 0 is the blow-up solution because it increases infinitely for the finite time t0 D 1 1 ln j2 =c0 j. Let us consider the Fitzhugh–Nagumo equation (FNE) ut D uxx C u.u  ı/.1  u/; 0 < ı < 1;

(1.73)

which is a simplification of the classical model describing nerve impulse propagation [30, 53, 54]. This nonlinear RDE was extensively studied in [7, 26, 57], where the Q-conditional symmetry operator Q D 2@t C

p 2 .3u  ı  1/ @x C 3u.u  ı/.1  u/@u

was derived and applied for finding the three-parameter family of exact solutions   p  p 2xCt ıc1 exp 2ı 2x C ıt C c2 exp 2   p  p ; uD 2xCt ı C c3 exp.ıt/ c1 exp 2 2x C ıt C c2 exp 2

(1.74)

where c1 ; c2 and c3 are arbitrary constants. It should be noted that solution (1.74) had been found earlier in the paper [42] using an ad hoc ansatz. Properties of exact solutions of the form (1.74) essentially depends on the constants ci .i D 1; 2; 3/. In particular, setting sequentially c1 D 0, c2 D 0 and c3 D 0, one obtains three different travelling fronts (two of them are presented in Figs. 1.3 and 1.4). These travelling fronts connect different pairs of the steady-state points of the FNE (the travelling front connecting u D 0 and u D 1 is very similar to that pictured in Fig. 1.2). In the case ci > 0 .i D 1; 2; 3/, solution (1.74) describes the coalescence of two fronts, which are moving with different speeds. This effect was studied in [42]. In the case ı D 0, Eq. (1.73) becomes the Zeldovich equation (1.42) and its exact solution has the form [7]:  p p c1 exp 2xCt C 2c2 2  p uD : p 2xCt C c c1 exp .x  2t/ C c 2 3 2

(1.75)

1.6 Exact Solutions of Some Equations Arising in Biological Models

25

Fig. 1.3 Exact solution (1.74) with ı D 3=4; c1 D c3 D 1; c2 D 0

Fig. 1.4 Exact solution (1.74) with ı D 3=4; c1 D c2 D 1; c3 D 0

Setting c2 D 0 in (1.75), a travelling front is obtained, which possesses quite similar properties to that presented in Fig. 1.2. We also note that solution (1.75) with c1 c2 > 0 is not continuous in the half-plane .t; x/ 2 RC  R. Equation (1.42) is reduced to the equation ut D uxx C u.1  u/2

(1.76)

26

1 Scalar Reaction-Diffusion Equations. . .

by the transformation (here i2 D 1) t ! t; x ! ix; u ! 1  u: This equation was introduced in [43] (see formulae (13) and (16) therein) in order to generalize Fisher’s model [28] describing the spread in space of a favoured gene in a population, hence (1.76) can be called a KPPE. It can be easily noted that solution (1.75) is transformed in a complex solution of (1.76), hence the solution obtained has no biological interpretation. Consider the generalized FNE ut D uxx C uux C 3 u.u  ı/.1  u/;

3 > 0; 0 < ı < 1

(1.77)

and the generalized KPPE ut D uxx C uux  3 u.1  u/2 ;

3 < 0;

(1.78)

which are particular cases of Eq. (1.64). Obviously, both equations generalize the FNE and the cubic KPPE in order to take into account a possible convective transport with the velocity u. Both Eqs. (1.77) and (1.78) can be reduced to the form vt D vyy C vvy C 0 C 1 v  3 v 3 ;

(1.79)

where  1 D 3

 1 2 .ı C 1/  ı ; 3

  2 3 2 .ı C 1/ .ı C 1/  ı ; 0 D 3 9

(1.80)

by the local substitution v.t; y/ D u 

ıC1 ; 3

yDxC

 .ı C 1/t: 3

(1.81)

Thus, Eq. (1.79) is locally equivalent to the generalized FNE and KPPE for 0 < ı < 1 and ı D 1, respectively. Equation (1.79) with 3 6D 0 is invariant only w.r.t. the trivial Lie algebra spanned by the Lie operators Pt D @t and Py D @y [24], hence the plane wave ansatz v D .!/; ! D y  ˛t; ˛ 2 R

(1.82)

reduces one to the nonlinear ODE ! ! C .˛ C /! C 0 C 1   3  3 D 0;

(1.83)

1.6 Exact Solutions of Some Equations Arising in Biological Models

27

which is not integrable (for the arbitrary given coefficients ; 0 ; 1 ; 3 and ˛). The known particular solutions of (1.83) generate the plane wave solutions of (1.79), which will be presented later as particular cases of more general solutions. It turns out that Q-conditional symmetry of (1.79) leads to new families of solutions, which possess more sophisticated structures. In fact, according to Theorem 1.4 this equation admits two Q-conditional symmetries Q1 D @t C

 3   3    v@y C 0 C 1 v  3 v 3 @v 4 2.  /

Q2 D @t 

 3 C  3 C   v@y C 0 C 1 v  3 v 3 @v ; 4 2. C /

and

p where D 2 C 83 (they coincide if D 0). Using these operators, one can construct two non-Lie ansätze Z 2y .!/ D  C vv v3 dv   ; Z 0 1 3 (1.84) 3  ! D 0 C11v3 v3 dv  2. / t and Z .!/ D !D

Z

v 0 C1 v3 v3

dv C

2y C ;

1 0 C1 v3 v3

dv 

3 C 2. C/

(1.85) t

by solving the first-order PDEs Q1 .v/ D 0 and Q2 .v/ D 0, respectively. Since these ansätze cannot be written in an explicit form (w.r.t. the variable v), we prefer to solve directly the overdetermined systems vt D vyy C vvy C 0 C 1 v  3 v 3 ; 3  3 Q1 .v/  vt C 3  4 vvy  2. / .0 C 1 v  3 v / D 0

(1.86)

vt D vyy C vvy C 0 C 1 v  3 v 3 ; 3 C 3 Q2 .v/  vt  3 C 4 vvy  2. C/ .0 C 1 v  3 v / D 0:

(1.87)

and

This approach is equivalent to the substitution of formulae (1.84) and (1.85) into (1.79).

28

1 Scalar Reaction-Diffusion Equations. . .

Consider first the overdetermined system (1.86). Eliminating ut in the first equation using the second one, we arrive at the second-order ODE vyy C

3. C / C vvy  .0 C 1 v  3 v 3 / D 0; 4 2.  /

(1.88)

which can be reduced to the form Vy y C 3VVy C V 3 

1 0 V D0 3 3

(1.89)

by the local substitution v.t; y/ D V.t; y /;

y D

C y: 4

(1.90)

It is known that ODE (1.89) is reduced to the linear third-order ODE Wy y y 

1 1 Wy  W D 0 3 3

(1.91)

by the nonlocal substitution [41] (see item (6.38)) VD

Wy : W

(1.92)

To construct the general solution of (1.91), one needs to solve the cubic equation 0 C 1 ˛  3 ˛ 3 D 0;

(1.93)

which has three different roots ˛1 ; ˛2 ; ˛3 in the case of the generalized FNE and two different roots ˛1 and ˛2 in the case of the generalized KPPE. The relevant general solutions take the form         W t; y D 1 .t/ exp ˛1 y C 2 .t/ exp ˛2 y C 3 .t/ exp ˛3 y

(1.94)

and         W t; y D 1 .t/ exp ˛1 y C 2 .t/ exp ˛2 y C 3 .t/ y exp ˛2 y ;

(1.95)

where i .t/ .i D 1; 2; 3/ are arbitrary (at the moment) functions. Consider first solution (1.94). Taking into account formulae (1.89), (1.90) and (1.92), one easily obtains the general solution of the nonlinear second-order ODE (1.88) v .t; y/ D

˛1 1 .t/ exp . 1 y/ C ˛2 2 .t/ exp . 2 y/ C ˛3 3 .t/ exp . 3 y/ ; 1 .t/ exp . 1 y/ C 2 .t/ exp . 2 y/ C 3 .t/ exp . 3 y/

(1.96)

1.6 Exact Solutions of Some Equations Arising in Biological Models

29

where i D C ˛i ; i D 1; 2; 3. In order to obtain the general solution of the 4 overdetermined system (1.86), it is sufficient to substitute (1.96) into the second equation of this system. After the relevant calculations a cumbersomeexpression   is obtained, however, it splits into separate parts for the functions exp i C j y . j < i D 1; 2; 3/ and we arrive at the ODE system  2  ˛1  ˛22 1 2 ;  2  3 .3 / P 2 3  P3 2 D 2. / ˛2  ˛32 2 3 ;  2  3 .3 / P3 1  P 1 3 D 2. / ˛3  ˛12 1 3 P 1 2  P2 1 D

3 .3 / 2. /

(hereafter the dot over i .t/ .i D 1; 2; 3/ denotes differentiation w.r.t. the variable t). This system is integrable and its general solution has the form   1 D c1 .t/ exp ˇ1 ˛12 t ; 2 D c2 .t/ exp ˇ1 ˛22 t ; 3 D c3 .t/ exp ˇ1 ˛32 t ;

(1.97)

3 .3 / where ci .i D 1; 2; 3/ are arbitrary constants, ˇ1 D 2. / ; and .t/ is an arbitrary function. Finally, substituting (1.97) into (1.96), we obtain the three-parameter family of exact solutions of RDCE (1.79)

  ˛i ci exp i y C ˇ1 ˛i2 t   v.t; y/ D ci exp i y C ˇ1 ˛i2 t

(1.98)

(hereafter summation is assumed from 1 to 3 over the repeated index i). In a quite similar way one can solve the overdetermined system (1.87), if the algebraic equation (1.93) has three different roots ˛1 ; ˛2 ; ˛3 (in fact, the first equation can also be reduced to the form (1.89)). Finally, the following family of exact solutions of Eq. (1.79) v.t; y/ D

  ˛i ci exp i y C ˇ2 ˛i2 t   ci exp i y C ˇ2 ˛i2 t

(1.99)

p 3 .3 C/ is obtained (here i D  ˛i .i D 1; 2; 3/, ˇ2 D 2. C/ and D 2 C 83 ). 4 Let us consider now solution (1.95). Taking into account formulae (1.89), (1.90) and (1.92), the general solution of the nonlinear ODE (1.88) v.t; y/ D

˛1 1 .t/ exp . 1 y/ C ˛2 2 .t/ exp . 2 y/ C 3 .t/ .1 C 2 y/ exp . 2 y/ 1 .t/ exp . 1 y/ C 2 .t/ exp . 2 y/ C

C 3 .t/ y 4

exp . 2 y/ (1.100)

is obtained (we use here again the notation i D C 4 ˛i ; i D 1; 2). To obtain the general solution of the overdetermined system (1.86), it is sufficiently to substitute

30

1 Scalar Reaction-Diffusion Equations. . .

(1.100) into the second equation of this system. It is important to note at this step that the cubic equation (1.93) has two different roots ˛1 and ˛2 only under the condition ˛1 C 2˛2 D 0. After simplification an expression is obtained, which splits into separate parts for the functions exp .. 1 C 2 / y/ ; y exp .. 1 C 2 / y/ and exp .2 2 y/. Finally, we arrive at the ODE system    3 .3  /  3 9˛2 1 2  3˛22 1 3 ; 3˛2 P 1 2  P2 1 C P1 3  P3 1 D 2 .  / 33 .3  / 2 ˛2 1 3 ; P1 3  P 3 1 D 2 .  /

(1.101)

3 .3  / ˛2 32 : P2 3  P 3 2 D  In the case 3 D 0, the solution (1.100) is reduced to a particular case of (1.96), hence we assume 3 6D 0. The ODE system (1.101) is integrable under this restriction and its general solution has the form   1 D c1 exp 3ˇ1 ˛22 t .t/; 2 D .c2 C 2ˇ1 ˛2 t/ .t/; 3 D .t/:

(1.102)

Finally, substituting (1.102) into (1.100), we obtain the two-parameter family of exact solutions    1 C ˛2 c2 C 2 y C 2ˇ1 ˛22 t  2˛2 c1 exp 3 ˇ1 ˛22 t  2 y v.t; y/ D    (1.103) c2 C C y C 2ˇ1 ˛2 t C c1 exp 3 ˇ1 ˛22 t  2 y 4 of the nonlinear RDCE (1.79) when this equation takes the form vt D vyy C vvy C 3 .˛1  v/.v  ˛2 /2 ;

˛1 C 2˛2 D 0:

(1.104)

The overdetermined system (1.87) has been solved in a similar way and the following family of exact solutions of the nonlinear RDCE (1.104)    1 C ˛2 c2 C 2 y C 2ˇ2 ˛22 t  2˛2 c1 exp 3 ˇ2 ˛22 t  2 y v.t; y/ D    c2 C  y C 2ˇ2 ˛2 t C c1 exp 3 ˇ2 ˛22 t  2 y 4

(1.105)

has been found (the values ; ˇ2 and 2 are the same as in (1.99)). Remark 1.4 Any solution of the form (1.98), (1.99), (1.103) and (1.105) with c1 c2 6D 0 cannot be found using the Lie ansatz (1.82) so they are non-Lie solutions of the nonlinear RDCE (1.79). One observes that the exact solution families constructed above contains several plane wave solutions of the form (1.82). In fact, vanishing one of the constants

1.6 Exact Solutions of Some Equations Arising in Biological Models

31

ci .i D 1; 2; 3/ in (1.98) and (1.99), we obtain the travelling wave solutions    ij ˛i C ˛j c exp j  i y C ˇ1 t   v .t; y/ D (1.106)  ij 1 C c exp j  i y C ˇ1 t and

   ij ˛i C ˛j c exp j  i y C ˇ2 t   ; v .t; y/ D  ij 1 C c exp j  i y C ˇ2 t

(1.107)

respectively. Here any summation over i or j is not assumed, c is a positive constant and ij

ˇ1 D

  4ˇ1  4ˇ2  ij ˛i C ˛j ; ˇ2 D ˛i C ˛j ; i 6D j; i; j D 1; 2; 3: C 

Obviously, each travelling wave solution of the form (1.106)–(1.107) tends either to ij ˛i or to ˛j if ! D y C ˇk t ! ˙1; k D 1; 2 (we remind the reader that the roots ˛1 ; ˛2 ; ˛3 are called the steady-state points of the nonlinear equation (1.79)). Note ij that each travelling wave has the speed ˇk , therefore, taking into account the identity ij ji ˇk D ˇk , several (at least six) different travelling wave solutions are generated by formulae (1.106)–(1.107). Now we present families of exact solutions of the generalized FNE and KPPE, which are easily constructed using the solutions found above and formulae (1.80) and (1.81). The generalized FNE (1.77) possesses two families of exact solutions     ˛i ci exp i x C u0 i C ˇ1 ˛i2 t     u.t; x/ D u0 C ci exp i x C u0 i C ˇ1 ˛i2 t

(1.108)

    ˛i ci exp i x C u0 i C ˇ2 ˛i2 t   ;   u.t; x/ D u0 C ci exp i x C u0 i C ˇ2 ˛i2 t

(1.109)

and

where the summation is assumed over the repeated index i and ci .i D 1; 2; 3/ are arbitrary constants, while the specified constants are ıC1 ıC1 2ı  1 2ı ; ˛1 D  ; ˛2 D ; ˛3 D ; 3 3 3 3  C ˛i ; i D ˛i ; i D 1; 2; 3; i D 4 4 p 3 .3  / 3 .3 C / ; ˇ2 D ; D 2 C 83 : ˇ1 D 2.  / 2. C / u0 D

32

1 Scalar Reaction-Diffusion Equations. . .

Any solution of the form (1.108) (or (1.109)) can be reduced to the relevant solution of the classical FNE, in particular, solution (1.74) can be recovered by setting  D 0 and 3 D 1. Thus, each solution (1.108) and (1.109) describes the coalescence of two fronts, which are moving with different speeds provided ci > 0; i D 1; 2; 3: Using formulae (1.108) and (1.109), we obtain several plane wave solutions of the generalized FNE (1.77):      ij ˛i C ˛j c exp j  i x C u0 C ˇ1 t     u.t; x/ D u0 C  ij 1 C c exp j  i x C u0 C ˇ1 t

(1.110)

and      ij ˛i C ˛j c exp j  i x C u0 C ˇ2 t     u.t; x/ D u0 C  ij 1 C c exp j  i x C u0 C ˇ2 t

(1.111)

(here i 6D j .i; j D 1; 2; 3/ and any summation over i or j is not assumed). In the case c > 0, they are travelling fronts and each of them connects two steady-state points among u D 0, u D 1 and u D ı. In contrast to the travelling fronts presented in the figures above, the form of solutions (1.110) and (1.111) depends also on the parameter . In particular, its sign may change the travelling front direction and the restriction 2  83 is needed in order to avoid complex values. Using formulae (1.80) and (1.81) with ı D 1, two families of exact solutions u.t; x/ D

8 C c2 C 2. C /x C . C /2 t  C   2 c1 exp  4 x C 3 4 t C c2 C 2. C /x C . C / t

(1.112)

8 C c2 C 2.  /x C .  /2 t     c1 exp 4 x C C3 t C c2 C 2.  /x C .  /2 t 4

(1.113)

and u.t; x/ D

of the generalized KPPE (1.78) are obtained from (1.103) and (1.105) with ˛1 D  23 and ˛2 D 13 . Moreover, the travelling wave solutions (1.110) and (1.111) with u0 D 2 2 1 3 ; ˛i D  3 ; ˛j D 3 are also the solutions of Eq. (1.78). Because the generalized KPPE (1.78) contains 3 < 0, the parameter  in the convective term cannot be arbitrary but 2  83 . In particular, this means that any real-valued solutions

1.7 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations. . .

33

of KPPE (1.76) cannot be extracted from formulae (1.112) and (1.113). Finally, we note that solutions (1.112) and (1.113) cannot describe the coalescence of two fronts in contrast to those for the generalized FNE.

1.7 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations with Variable Diffusivity Among RDCEs with variable coefficients the equations with power-law diffusion and convective terms are the most common equations in real-world applications. In this section, we want to find all possible Q-conditional symmetries of the form Q D @t C .t; x; U/@x C .t; x; U/@U ;

(1.114)

where  and  are unknown functions, for two special subclasses of the RDCE class (1.29), namely: Ut D .U m Ux /x C U m Ux C C.U/;

(1.115)

Ut D .U m Ux /x C U mC1 Ux C C.U/;

(1.116)

where  and m 6D 0 are arbitrary constants while C.U/ is an arbitrary function. Now we present the main results in the form of two theorems. We remind the reader that we are looking for purely conditional symmetry operators only, which cannot be reduced to Lie symmetry operators described completely in [24, 25]. Theorem 1.5 ( [21]) Equation (1.115) is Q-conditional invariant under the operator (1.114) iff it and the relevant operator (up to equivalent representations generated by multiplying on the arbitrary smooth function M.t; x; U/) have the following forms:   (i) Ut D .U m Ux /x C U m Ux C 1 U mC1 C 2 .U m  3 / ;

(1.117)

m

(1.118) Q D @t C .1 U C 2 U / @U ; .m C 1/2 6D 0I  1  (ii) Ut D U Ux x C U 1 Ux C .1 ln U C 2 /.U  3 /; 1 6D 0; Q D @t C .1 ln U C 2 /U@U I  1  1 1 (iii) Ut D U  2 Ux C U  2 Ux C 1 U C 2 U 2 C 3 ; x   1 Q D @t C f .t; x/@x C 2 g.t; x/U C h.t; x/U 2 @U ;

(1.119) (1.120)

34

1 Scalar Reaction-Diffusion Equations. . .

where the function triplet (f ; g; h) is the general solution of the system 2ffx C ft C fg D 0; fxx  fx  2gx  fh D 0;   1 .g C 2fx / C gt D 0; g 2

(1.121)

2gh  1 h C 2fx h  2 fx C ht  gx  gxx D 0; h2 

2 3 h  3 fx C g  hx  hxx D 0: 2 2

Hereafter 1 ; 2 and 3 are arbitrary constants. It should be noted that cases (i) and (ii) with  D 0 immediately give the RDEs and the relevant symmetries obtained in [5]. Although system (1.121) contains five equations on three unknown functions, it is compatible. In fact, the system with f D g D 0; 1 D 0 is reduced to the second-order ODE hxx C hx C

2 h  h2 D 0 2

(1.122)

and hence 1

Q D @t C 2h.x/U 2 @U is the Q-conditional symmetry operator for an arbitrary nonzero solution of (1.122). Unfortunately, ODE (1.122) cannot be integrated for the arbitrary coefficients  and 2 , however, some particular solutions can be easily established. For example, setting h D 22 , a particular case of (i) with m D  12 ; 1 D 0 is obtained. Setting  D 2 D 0 in (1.122), we arrive at the known ODE hxx D h2 with the general solution h D W .0; c1 ; x C c2 /; where c1 and c2 are arbitrary constants, W is the Weierstrass function with periods 0 and c1 . Its simplest solution takes the form 1 h D 6x2 and leads to the Q-conditionalsymmetry operator Q D @t C12x2 U 2 @U of  1 the nonlinear diffusion equation Ut D U  2 Ux [5]. However the result derived x

in [5] can be generalized as follows. An arbitrary solution of (1.121) with  D 0 generates the Q-conditional symmetry operator (1.120) of RDE  1  1 Ut D U  2 Ux C  1 U C  2 U 2 C  3 : x

Theorem 1.6 ( [21]) Equation (1.116) is Q-conditional invariant under the operator (1.114) iff it and the relevant operator (up to equivalent representations generated by multiplying on the arbitrary smooth function M.t; x; U/) have the following forms: (i) Ut D .U m Ux /x C U mC1 Ux C 1 U C 2 U m ; m ¤ 1; Q D @t  U mC1 @x C .1 U C 2 U m / @U I

(1.123)

1.7 Q-Conditional Symmetry of Reaction-Diffusion-Convection Equations. . .

35

    1   1 3 1 1 1 2 2 2 2 2 (ii) Ut D U Ux C U Ux C 1 U C 2 U C 3 CU ; x 22     1 3 1 31 Q D @t C U 2 C @x C 1 U 2 C 2 U 2 C 3 @U : 2 It should be noted that cases (i) and (ii) with  D 0 immediately give the RDEs and the relevant symmetries obtained in [5]. Consider some equations, which arise as particular cases of those from Theorem 1.5 and 1.6 and are known in applications. Equation (1.117) with m D 1 contains as a subcase the equation Ut D .UUx /x C UUx C 1 U.1  U/;

(1.124)

which is a natural generalization of the Murray equation Ut D Uxx C UUx C 1 U.1  U/;

(1.125)

which was extensively studied in Sect. 1.6. On the other hand, (1.124) is nothing else but the porous-Fisher equation (see [19, 52, 75] and the references therein) with the Burgers convection term UUx (see (1.59)). In population dynamics, this equation implies that the population disperses to regions of lower density more rapidly as the population gets more crowded. So, Eq. (1.124) may be called the porous-Murray equation, i.e., Murray equation with slow diffusion. The Murray equation with fast diffusion (i.e., power-law diffusivity with a negative exponent) can be obtained from (1.117). Indeed, setting m D 2, 3 D 0 and 2 D 1 , we arrive at the equation   Ut D U 2 Ux x C U 2 Ux C 1 U.1  U/:

(1.126)

It should be noted that Eq. (1.126) with 1 D 0 is linearizable by the known integral substitution [32] while one with 1 6D 0 is not linearizable. It is interesting to note that Eq. (1.119) with 2 D 1 and 3 D 0 takes the form    1  1 1 1 (1.127) Ut D U  2 Ux C U  2 Ux C 1 U 2 1  U 2 : x

This equation is an analogue of the Murray equation with fast diffusion. Another analogue of the Murray equation with fast diffusion is Ut D

   1  1 1 1 U  2 Ux C U 2 Ux C 1 U 2 1  U 2 ; x

which is a particular case of Eq. (1.123).

(1.128)

36

1 Scalar Reaction-Diffusion Equations. . . 1

Remark 1.5 Equations (1.127) and (1.128) are reduced by the substitution U 2 ! U to the following evolution equations with constant diffusion and the logistic term UUt D Uxx C Ux C

1 U.1  U/ 2

and UUt D Uxx C U 2 Ux C

1 U.1  U/: 2

1.8 Exact Solutions of Some Equations with Power-Law Diffusivity and Their Interpretation Here we construct exact solutions using the Q-conditional symmetry operators found in Sect. 1.7, we show that they are non-Lie solutions (i.e., cannot be obtained using Lie symmetry operators) and present a biological interpretation of the solutions obtained. We start from case (i) of Theorem 1.5. Equation (1.117) and operator (1.118) are transformed by the substitution ( VD

U mC1 ; m ¤ 1; ln U; m D 1

(1.129)

to the forms   Vxx D V n Vt  Vx C 1 V C 2 .3  V n /

(1.130)

and   Q D @t C 1 V C 2 @V ; where i D i .m C 1/; i D 1; 2. The relevant ansatz is constructed using the standard procedure, i.e., we solve the linear equation Q.V/ D 0. Since its general solution depends on 1 , two ansätze are obtained: ( VD

2 t C '.x/; '.x/e

 1t



 2  1

1 D 0;

; 1 ¤ 0

(1.131)

with '.x/ being an unknown function. Substituting (1.131) with 1 D 0 into (1.130), one arrives at the linear ODE 'xx C 'x  2 3 D 0

1.8 Exact Solutions of Some Equations with Power-Law Diffusivity. . .

37

with the general solution ' D c1 C c2 ex C

2 3 x: 

Hereafter c1 and c2 are arbitrary constants. Thus, Eq. (1.130) with 1 D 0 possesses the exact solution V D 2 t C c1 C c2 ex C

2 3 x: 

Finally, applying substitution (1.129), we obtain the exact solution   1 2 3 .m C 1/ mC1 x U D 2 .m C 1/t C c1 C c2 e x C 

(1.132)

of RDCE with power nonlinearities Ut D .U m Ux /x C U m Ux C 2 U m  2 3 ; m ¤ 1:

(1.133)

Using the result of [24, 25] one establishes that Eq. (1.133) (with arbitrary coefficients) is invariant only under two-dimensional Lie algebra with the basic operators @t and @x . As shown in Sect. 1.6, this algebra leads to plane wave solutions only. Obviously, the exact solution presented above has a different structure and cannot be reduced to this form, therefore it is a non-Lie solution. Note that solution (1.132) is a plane wave solution if one additionally sets c2 D 0. In a quite similar way it can be shown that all solutions obtained below are also non-Lie solutions and may be reduced to Lie solutions only under additional constraints. Substituting (1.131) with 1 6D 0 into (1.130), one again obtains a linear secondorder ODE, which is integrable in terms of different elementary functions depending on ı D 2 C 41 3 D 2 C 41 .m C 1/3 . Dealing in a quite similar way to the case 1 D 0, we finally obtain three exact solutions  p  1  p  ı 2 mC1 l.t;x/ x  2ı x 2 c1 e  C c2 e ; ı > 0; UD e 1  1  2 mC1 ; ı D 0; U D el.t;x/ .c1 C c2 x/  1 1 ! mC1 ! p p ıx ıx  2 C c2 sin U D el.t;x/ c1 cos ;ı 0 if 1 > 0. Hence solution (1.134) takes the form s UD

  p    pı x ı exp 21 t  x c1 e 2 C c2 e 2 x ; 2

where ı D 2 C 81 . This solution unboundedly grows if t ! 1 or x ! ˙1 and its straightforward interpretation is not obvious. More interesting solutions occur in the case of (1.124) with the anti-logistic term (1 < 0). Depending on ı D 2  8 one obtains three types of solutions. In the case m D 1;  D 3; 1 D 1; 2 D 0; c1 D c2 D 3, solution (1.134) is presented in Fig. 1.5. This solution tends to zero if t ! 1 and satisfies the zero Dirichlet conditions: x D 0 W U D 0;

x D 1 W U D 0:

If  D 2, then solution (1.136) with m D 1 is valid. In the case 1 D 1; 2 D 0; c1 D 0; c2 D 1; this solution is presented in Fig. 1.6. We note that the solution is again vanishing if t ! 1 and it satisfies the zero Dirichlet conditions U D 0 on the bounded interval Œ0; . In population dynamics both solutions describe the scenario predicting the total extinction of species.

1.8 Exact Solutions of Some Equations with Power-Law Diffusivity. . .

39

Fig. 1.6 Exact solution (1.136) with m D 1;  D 2; 1 D 1; 2 D 0; c1 D 0; c2 D 1

Consider the Murray equation with fast diffusion (1.126). Since ı D 2 > 0 solution (1.134) takes the form U D .1 C exp.1 t/ .c1 C c2 exp.x///1

(1.138)

and possesses attractive properties. Assuming c1 > 0 and c2 > 0, one sees that this solution is positive and bounded for arbitrary .t; x/ 2 RC  R. Moreover, the solution tends either to zero (1 < 0) or to 1 (1 > 0) if t ! 1. Both values, U D 0 and U D 1, are steady-state points of (1.126). Solution (1.138) tends to the steadystate point U D 0 if x ! 1, while U D .1 C c1 exp.1 t//1 if x ! 1. It should also be noted that solution (1.138) with c1 D 0 is a travelling front with the same structure as one for the Murray equation (1.125) (see formula (1.71)). An example of solution (1.138) is presented in Fig. 1.7. This solution describes the scenario predicting an inhomogeneous distribution of the species population at any fixed time. Notably, there are the region with nearly zero population density (negative values of x > 0 and the intermediate region, in which the density is rapidly growing with x. The high population density (for large values of x) is allowed on a semi-infinite space interval but there are no spikes. Thus, one may claim that the population disperses to regions of lower density more rapidly as the population gets more crowded. It is exactly such behaviour that models with power-law diffusivity should describe in population dynamics. Finally, we present the following observation. If one applies substitution (1.129) to RDCE (1.137) and its solutions (1.134)–(1.136), then Eq. (54) [16] with (65)– (66) [16] and ˛.s/ D sn , and solutions (69)–(71) [16] are exactly obtained. Although the authors of that paper do not use any symmetries to construct exact

40

1 Scalar Reaction-Diffusion Equations. . .

Fig. 1.7 Exact solution (1.138) with  D 2; 1 D 1:5; c1 D 2; c2 D 1

solutions, formula (72) [16] is nothing else but the equation Q.V/ D 0, where Q is the conditional symmetry operator   Q D @t C 1 V C 2 @V of equation   Vxx D V n Vt  Vx C 1 V C 2 .3  V n /: Thus, we obtain new confirmation of the well-known hypothesis that any exact solution can be obtained by a Lie or conditional symmetry operator.

1.9 Concluding Remarks It is commonly accepted that a new approach for finding symmetries was proposed in 1969 by Bluman and Cole [9]. Although the approach is based on the classical Lie scheme [3, 8, 36, 59, 64], the resulting symmetries can be non-Lie symmetries of the equation in question, therefore they were called nonclassical symmetries. Following the Fushchych works [34, 36], we call them Q-conditional symmetries (other terminology used for this symmetry is conditional symmetry [46, 47] and reduction operator [65]). Since 1987 when the Bluman–Cole approach was rediscovered in [33, 60] and later was successfully applied to a wide range of nonlinear PDEs (see, e.g., the most recent book [12] devoted to this topic and references therein), several other definitions of non-Lie symmetries have been introduced and applied

References

41

for solving nonlinear PDEs (weak symmetry [60, 61, 67], potential symmetry [10, 11, 44], conditional symmetry [20, 34, 36], generalized conditional symmetry [31, 69, 70, 76]). A common idea that underlies all these symmetries can be described as follows: in order to find the relevant non-Lie symmetry, one needs to consider the given equation together with differential constraint(s), i.e., a system of equations. The main problem is how to define suitable constraint(s) in a such way that the overdetermined system obtained will produce new symmetries leading to new solutions of the given equation. Moreover, solving the system of DEs, which is nonlinear in the case of non-Lie symmetry, is another nontrivial problem. For example, the nonlinear system of DEs for searching nonclassical symmetries of the linear heat equation obtained in [9] was not solved therein. The most general results were obtained much later in [4, 35] and [49]. In the papers [35] and [49], it was proved that the general solution is expressed in terms of three solutions of the linear heat equation, while the authors of [4] showed how the general solution is also obtainable via the matrix Cole–Hopf transformation. In [4] and [49] the system of DEs for searching nonclassical symmetries of the Burgers equation were also solved (notably, solving this problem was initiated in [2], i.e., immediately after the appearance of the pioneering work [9]). In this chapter, all the main results on Q-conditional symmetry of RDCEs of the form (1.29) are summarized. They are presented in the form of Theorems 1.2, 1.4, 1.5 and 1.6. Although some of them were published about 25 years ago, and the others were established in the 2000s, this is the first attempt to present an extensive review regarding this matter. It should be noted that a set of RDCEs with exponential nonlinearities were studied in the recent paper [22]. The nonlinear RDCEs listed in these theorems contain several well-known equations arising in applications and their direct generalizations. In the particular case, the Murray equation (1.125), its porous analogue (1.124) and its analogue (1.126) with the fast diffusion (see also (1.127)–(1.128)) were identified. Several RDCEs with cubic nonlinearities, notably, FNE, KPPE, the Huxley equation and their natural generalizations were also constructed.

References 1. Ablowitz, M., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840 (1979) 2. Ames, W.F.: Nonlinear Partial Differential Equations in Engineering. Academic Press, New York (1972) 3. Arrigo, D.J.: Symmetry Analysis of Differential Equations. Wiley, Hoboken, NJ (2015) 4. Arrigo, D.J., Hickling, F.: On the determining equations for the nonclassical reductions of the heat and Burgers’ equation. J. Math. Anal. Appl. 270, 582–589 (2002) 5. Arrigo, D.J., Hill, J.M.: Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math. 94, 21–39 (1995)

42

1 Scalar Reaction-Diffusion Equations. . .

6. Arrigo, D.J., Broadbridge, P., Hill, J.M.: Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal. J. Math. Phys. 34, 4692–4703 (1993) 7. Arrigo, D.J., Hill, J.M., Broadbridge, P.: Nonclassical symmetries reductions of the linear diffusion equation with a nonlinear source. IMA J. Appl. Math. 52, 1–24 (1994) 8. Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002) 9. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969) 10. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989) 11. Bluman, G.W., Reid, G.J., Kumei, S.: New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988) 12. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010) 13. Britton, N.F.: Essential Mathematical Biology. Springer, Berlin (2003) 14. Burgers, J.M.: Correlation problems in a one-dimensional model of turbulence I. Proc. Acad. Sci. Amsterdam 53, 247–260 (1950) 15. Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974) 16. Carini, M., Fusco, D., Manganaro, N.: Wave-like solutions for a class of parabolic models. Nonlinear Dyn. 32, 211–222 (2003) 17. Cherniha, R.: New symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. In: Proceedings of the International Workshop “Similarity Methods”, Universität Stuttgart, Stuttgart, pp. 323–336 (1998) 18. Cherniha, R.: New Q-conditional symmetries and exact solutions of some reaction-diffusionconvection equations arising in mathematical biology. J. Math. Anal. Appl. 2, 783–799 (2007) 19. Cherniha, R., Dutka, V.: Exact and numerical solutions of the generalized Fisher Equation. Rep. Math. Phys. 47, 393–411 (2001) 20. Cherniha, R., Henkel, M.: On nonlinear partial differential equations with an infinitedimensional conditional symmetry. J. Math. Anal. Appl. 298, 487–500 (2004) 21. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations. J. Phys. A Math. Theor. 40, 10049–10070 (2007) 22. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of reaction– diffusion–convection equations with exponential nonlinearities. J. Math. Anal. Appl. 403, 23–37 (2013) 23. Cherniha, R.M., Serov, M.I.: Lie and non-lie symmetries of nonlinear diffusion equations with convection term. In: Proceedings of the 2nd International Conference “Symmetry in Nonlinear Mathematical Physics”, pp. 444–449. Institute of Mathematics, Kyiv, Ukraine (1997) 24. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear secondorder evolution equations with convection term. Eur. J. Appl. Math. 9, 527–542 (1998) 25. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear secondorder evolution equations with convection term II. Eur. J. Appl. Math. 17, 597–605 (2006) 26. Clarkson, P.A., Mansfield, E.L.: Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica D 70, 250–288 (1994) 27. Danilov, V.G., Maslov, V.P., Volosov, K.A.: The flow around a flat plate. In: Mathematical Modelling of Heat and Mass Transfer Processes, pp. 254–294. Springer, Dordrecht (1995) 28. Fisher, R.A.: The Genetical Theory of Natural Selection. University Press, Oxford (1930) 29. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369 (1937) 30. Fitzhugh, R.: Impulse and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961) 31. Fokas, A.S., Liu, Q.M.: Generalized conditional symmetries and exact solutions of nonintegrable equations. Theor. Math. Phys. 99, 571–582 (1994) 32. Fokas, A.S., Yortsos, Y.C.: On the exactly solvable equation St D Œ.ˇS C /2 Sx x C ˛.ˇS C

/1 Sx occurring in two-phase flow in porous media. SIAM J. Appl. Math. 42, 318–332 (1982) 33. Fushchych, W., Tsyfra, I.: On a reduction and solutions of nonlinear wave equations with broken symmetry. J. Phys. A Math. Gen. 20, L45–L48 (1987)

References

43

34. Fushchych, V.I., Serov, M.I., Chopyk, V.I.: Conditional invariance and nonlinear heat equations (in Russian). Proc. Acad. Sci. Ukr. 9, 17–21 (1988) 35. Fushchych, W.I., Shtelen, W.M., Serov, M.I., Popovych, R.O.: Q-conditional symmetry of the linear heat equation. Proc. Acad. Sci. Ukr. 12, 28–33 (1992) 36. Fushchych, W., Shtelen, W., Serov, M.: Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1993) 37. Gandarias, M.L.: New symmetries for a model of fast diffusion. Phys. Lett. A 286, 153–160 (2001) 38. Gandarias, M.L., Romero, J.L., Diaz, J.M.: Nonclassical symmetry reductions of a porous medium equation with convection. J. Phys. A Math. Gen. 32, 1461–1473 (1999) 39. Gilding, B.H., Kersner, R.: Travelling Waves in Nonlinear Reaction–Convection–Diffusion. Birkhauser Verlag, Basel (2004) 40. Ibragimov, N.H., Meleshko, S.V.: Linearization of third-order ordinary differential equations by point and contact transformations. J. Math. Anal. Appl. 308, 266–289 (2005) 41. Kamke, E.: Differentialgleichungen. Lösungmethoden and Lösungen (in German), 6th edn. B. G. Teubner, Leipzig (1959) 42. Kawahara, T., Tanaka, M.: Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation. Phys. Lett. A 97, 311–314 (1983) 43. Kolmogorov, A., Petrovsky, I., Piskounov, N.: Investigation of an equation of diffusion coupled with increasing of matter amount, and its application to a biological problem (in Russian). Mosc. Univ. Math. Bull. 1(6), 1–26 (1937) 44. Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 15, 161–209 (1989) 45. Kuang, Y., Nagy, J. D., Eikenberry S. E.: Introduction to mathematical oncology. In: Chapman & Hall/CRC Mathematical and Computational Biology Series. CRC Press, Boca Raton, FL (2016) 46. Levi, D., Winternitz, P.: Non-classical symmetry reduction: example of the Boussinesq equation. J. Phys. A Math. Gen. 22, 2915–2924 (1989) 47. Levi, D., Winternitz, P.: Symmetries and conditional symmetries of differential-difference equations. J. Math. Phys. 34, 3713–3730 (1993) 48. Liu, Q.M., Fokas, A.S.: Exact interaction of solitary waves for certain nonintegrable equations. J. Math. Phys. 37, 324–345 (1996) 49. Mansfield, E.L.: The nonclassical group analysis of the heat equation. J. Math. Anal. Appl. 231, 526–542 (1999) 50. Murray, J.D.: Nonlinear Differential Equation Models in Biology. Clarendon Press, Oxford (1977) 51. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) 52. Murray, J.D.: An Introduction I: Models and Biomedical Applications. Springer, Berlin (2003) 53. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003) 54. Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2071 (1962) 55. Nucci, M.C.: Symmetries of linear, C-integrable, S-integrable and nonintegrable equations and dynamical systems. In: Nonlinear Evolution Equations and Dynamical Systems, pp. 374–381. World Scientific Publishing, River Edge (1992) 56. Nucci, M.C.: Iterations of the non-classical symmetries method and conditional Lie-Bäcklund symmetries. J. Phys. A Math. Gen. 29, 8117–8122 (1996) 57. Nucci, M.C., Clarkson, P.A.: The nonclassical method is more general than the direct method for symmetry reductions. an example of the Fitzhugh–Nagumo equation. Phys. Lett. A 164, 49–56 (1992) 58. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems. Modern Perspectives, 2nd edn. Springer, Berlin (2001) 59. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1986)

44

1 Scalar Reaction-Diffusion Equations. . .

60. Olver, P., Rosenau, P.: Group-invariant solutions of differential equations. SIAM J. Appl. Math. 47, 263–278 (1987) 61. Olver, P.J., Vorob’ev, E.M.: Nonclassical and conditional symmetries. In: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, pp. 291–328. CRC Press, Boca Raton (1996) 62. Oron, A., Rosenau, P.: Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A 118, 172–176 (1986) 63. Ovsiannikov, L.V.: Group properties of nonlinear heat equation (in Russian). Dokl. AN SSSR 125, 492–495 (1959) 64. Ovsiannikov, L.V.: The Group Analysis of Differential Equations. Academic, New York (1980) 65. Popovych, R.O.: Reduction operators of linear second-order parabolic equations. J. Phys. A Math. Theor. 41, 185202 (31pp) (2008) 66. Pucci, E.: Similarity reductions of partial differential equations. J. Phys. A Math. Gen. 25, 2631–2640 (1992) 67. Pucci, E., Saccomandi, G.: On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598 (1992) 68. Pucci, E., Saccomandi, G.: Evolution equations, invariant surface conditions and functional separation of variables. Physica D. 139, 28–47 (2000) 69. Qu, C.: Group classification and generalized conditional symmetry reduction of the nonlinear diffusion–convection equation with a nonlinear source. Stud. Appl. Math. 99, 107–136 (1997) 70. Qu, C., Estévez, P.G.: On nonlinear diffusion equations with x-dependent convection and absorption. Nonlin. Anal. TMA 57, 549–577 (2004) 71. Saccomandi, G.: A personal overview on the reduction methods for partial differential equations. Note Mat. 23, 217–248 (2005) 72. Serov, M.I.: Conditional invariance and exact solutions of the nonlinear equation. Ukr. Math. J. 42, 1216–1222 (1990) 73. Waniewski, J.: Theoretical Foundations for Modeling of Membrane Transport in Medicine and Biomedical Engineering. Institute of Computer Science Polish Academy of Sciences, Warsaw (2015) 74. Webb, G.M.: PainlevKe analysis of a coupled Burgers’ heat equation system, and nonclassical similarity reductions of the heat equation. Physica D 41, 208–218 (1990) 75. Witelski, T.P.: Merging traveling waves for the porous-Fisher’s equation. Appl. Math. Lett. 8, 57–62 (1995) 76. Zhdanov, R.Z.: Conditional Lie-Backlund symmetry and reduction of evolution equations. J. Phys. A Math. Gen. 28, 3841–3850 (1995) 77. Zhdanov, R.Z., Lahno, V.I.: Conditional symmetry of a porous medium equation. Physica D 122, 178–186 (1998) 78. Zhdanov, R.Z., Tsyfra, I.M., Popovych, R.O.: A precise definition of reduction of partial differential equations. J. Math. Anal. Appl. 238, 101–123 (1999)

Chapter 2

Q-Conditional Symmetries of Reaction-Diffusion Systems

Abstract A recently developed theoretical background for searching Q-conditional (nonclassical) symmetries of systems of evolution partial differential equations is presented. We generalize the standard definition of Q-conditional symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of symmetry of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some special properties, which distinguish them from the standard conditional symmetry. The general class of two-component nonlinear reaction-diffusion systems is examined in order to find the Q-conditional symmetry operators. The relevant systems of so-called determining equations are solved under additional restrictions. As a result, several reaction-diffusion systems possessing conditional symmetry are constructed. In particular, it is shown that the diffusive Lotka–Volterra system, the Belousov–Zhabotinskii system (with the correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry.

2.1 Reaction-Diffusion Systems and Their Applications In 1952, Turing published a remarkable paper [56], in which a revolutionary idea about the mechanism of morphogenesis (the development of structures in an organism during its life) has been proposed. From the mathematical point of view Turing’s idea immediately leads to the construction of reaction-diffusion systems (RDSs) (not single equations!) exhibiting so-called Turing instability (see, e.g., [41, Sect. 14.3]). Nowadays nonlinear RDSs are basic equations for many well-known nonlinear models used to describe a wide range of processes in physics, biology, medicine, chemistry, ecology, etc. This chapter is mostly devoted to the investigation of two-component RDSs of the form   Ut D D1 .U/Ux x C F.U; V/;   2 (2.1) Vt D D .V/Vx x C G.U; V/;

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6_2

45

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2 Q-Conditional Symmetries of Reaction-Diffusion Systems

where U D U.t; x/ and V D V.t; x/ are two unknown functions representing the densities of populations (cells, tumours, chemicals), F.U; V/ and G.U; V/ are the given smooth functions describing the interaction between them and the environment, the functions D1 .U/ and D2 .V/ are the relevant diffusivities (hereafter they are positive smooth functions) and the subscripts t and x denote differentiation with respect to (w.r.t.) these variables. The class of RDSs (2.1) generalizes a number of nonlinear models describing various processes in biology, medicine and ecology (see, e.g., the well-known books [10, 41, 43, 45]). Usually the diffusivities Dk .k D 1; 2/ are taken to be positive constants dk . An important subclass of RDSs of the form (2.1) consists of those, which satisfy the well-known requirements leading to Turing instability, hence they can be used for description of the chemical basis of morphogenesis [10, Chap. 7], [43, Chap. 2]. The classical examples of such RDSs are the Gierer–Meinhardt system, the Schnakenberg system, etc. The two-component diffusive Lotka–Volterra system (DLVS) Ut D d1 Uxx C U.a1 C b1 U C c1 V/; Vt D d2 Vxx C V.a2 C b2 U C c2 V/

(2.2)

is another common RDS of the form (2.1) [10, 41, 43, 45]. System (2.2) is the standard generalization of the classical Lotka–Volterra system that takes into account the diffusion process for interacting species (see the terms d1 Uxx and d2 Vxx ). Although the classical Lotka–Volterra system was independently introduced by Lotka and Volterra about 90 years ago, its different generalizations are widely studied at present because of their importance for mathematical modelling of a wide range of processes in biology, ecology, economics, etc. It is well known that DLVS (2.2) models several types of interaction between two populations of species. Three common types are predator–prey interaction, the competition (for food, space, etc.) of species and mutualism. Each type of species interaction is defined by the signs of coefficients in DLVS (2.2). For example, the coefficients ak > 0; bk  0; ck  0; k D 1; 2 are used in order to describe the competition, while the cases c1 b2 < 0 and c1 > 0; b2 > 0 model predator–prey interaction and mutualism, respectively (see [42, Chap. 3] for details). A separate subclass of the RDS class (2.1) is formed by so-called   ! systems, which possess spiral wave solutions. Spiral waves occur naturally in a wide variety of biological, physiological and chemical effects (see [41, Chap. 12] and the references therein). Typically the   ! systems have a complicated structure involving the nonlinearities .U 2 C V 2 / and !.U 2 C V 2 /, hence their analysis is rather complicated. The most widely known are spiral waves occurring in the

2.1 Reaction-Diffusion Systems and Their Applications

47

Belousov–Zhabotinskii reaction [41, Sect. 13.3]. In contrast to the   ! systems, the corresponding mathematical model is simple and it is the Belousov–Zhabotinskii system Ut D d1 Uxx C U.a1  b1 U  c1 V/ C rV; Vt D d2 Vxx  V.a2 C b2 U/;

(2.3)

where all the parameters should be nonnegative. As noted above, typically the diffusivities Dk .k D 1; 2/ in RDSs of the form (2.1) are taken to be positive constant, however, in certain insect dispersal models they depend on the densities U and V. For example, a power dependence is adopted in diffusion models, when there is an increase in diffusion due to population pressure [29], [41, Sect. 11.4], [42, Sect. 13.4] (see also the application to modelling flows of thin films of viscous fluid [23]). Probably the simplest nonlinear RDS with variable diffusivities follows as a particular case from the seminal work [54] and takes the form ut D ..d1 C d11 u/u/xx C u.a1  b1 u  c1 v/; vt D ..d2 C d22 v/v/xx C v.a2  b2 u  c2 v/;

(2.4)

where the diffusivities D1 D d1 C 2d11 u and D2 D d2 C 2d22 v are linear functions. Systems of the form (2.4) are used in order to model the competition in a heterogeneous environment (see, e.g., [44] and references cited therein). Nonlinear multi-component RDSs are an important tool for mathematical modelling of a wide range of processes involving several kinds of species (cells, chemicals, etc.). Such systems can possess some properties that are not common for the relevant two-component systems. Thus, it is time to extend the results obtained for two-component RDSs to the multi-component systems. It turns out that this is a highly nontrivial problem and there are not many rigorous studies in this direction. To the best of our knowledge, these studies are mostly focused on investigation of the multi-component DLVS (see [39, 57, 58] and the papers cited therein). It should be noted that the multi-component systems describe much more complicated interactions between n populations than DLVS (2.2). The case n D 3 is studied in Chap. 3. During recent decades nonlinear RDSs have been extensively studied by means of different mathematical methods and techniques. In this chapter and Chaps. 3 and 4, we restrict ourselves to the application of symmetry-based methods (another terminology is group-theoretical methods) in order to construct subclasses of RDSs with nontrivial conditional symmetry, to identify and to study in detail those, which are used in biological applications.

48

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

2.2 Q-Conditional Symmetry for Systems of Partial Differential Equations Although finding Lie symmetries of two-component RD systems of the form (2.1) was initiated about 30 years ago [6, 31, 32, 61], the complete Lie symmetry classification problem was completed only in the 2000s in papers [12, 21, 22] (for constant diffusivities) and [23, 24, 36] (for nonconstant diffusivities). The time is therefore ripe for a complete description of non-Lie symmetries for the class of RDSs (2.1). However, it seems to be an extremely difficult task because, firstly, several definitions of non-Lie symmetries have been introduced (nonclassical symmetry [3, 8], conditional symmetry [20, 34, 35], generalized conditional symmetry [30, 37, 59], etc.), secondly, an exhaustive description of nonLie symmetries needs to solve the corresponding systems of determining equations (DEs), which are nonlinear and can fully be solved only in exceptional cases. Hereafter we use the most common notion among non-Lie symmetries, nonclassical symmetry, which we call Q-conditional symmetry following the well-known book [35] and our previous papers. It is well known that the notion of Qconditional symmetry plays an important role in investigation of nonlinear partial differential equations (PDEs) because, having such symmetries in an explicit form, one may construct new exact solutions, which are not obtainable by the classical Lie algorithm. However, for an exhaustive description of such symmetries, one needs to solve the corresponding nonlinear systems of DEs and this is a very difficult task. To the best of our knowledge, only a few papers devoted to the search for Q-conditional symmetries for systems of PDEs were published before 2010 [2, 5, 25, 28, 40]. The majority of such papers were published during the current decade [4, 13, 15–19, 55]. Generally speaking, in order to solve the Q-conditional symmetry classification problem for the RDS class (2.1), one should look for new constructive approaches allowing to solve the relevant nonlinear system of DEs. A possible approach based on new definitions of Q-conditional symmetry was proposed in [13] and is presented in this section. Consider a system of m evolution equations (m  2) with two independent .t; x/ and m dependent u D .u1 ; u2 ; : : : ; um / variables. Let us assume that the kth-order (k  2) equations of evolution type   ; i D 1; 2; : : : ; m uit D F i t; x; u; ux ; : : : ; u.k/ x

(2.5)

are defined in a domain ˝ R2 of the variables t and x. Hereafter, F i are the smooth functions of the corresponding variables, the subscripts differentiation  j 1 t and xj mdenote  i . j/ @ ju @ u @ u w.r.t. these variables, uit D @u and u  D ; : : : ; ; j D 1; 2; : : : ; k. x @t @x j @x j @x j It is well known (see, e.g., [7, Sect. 4.3]) that in order to find Lie invariance operators, one needs to consider system (2.5) as the manifold M D fS1 D 0; S2 D 0; : : : ; Sm D 0g ;

2.2 Q-Conditional Symmetry for Systems of Partial Differential Equations

49

where   .i D 1; 2; : : : ; m/; Si  uit  F i t; x; u; ux ; : : : ; u.k/ x in the prolonged space of the variables t; x; u; u; : : : ; u: 1

k

Here, the symbol u ( j D 1; 2; : : : ; k) denotes totalities of the jth-order derivatives j

w.r.t. the variables t and x. According to the definition, system (2.5) is invariant (in the Lie sense!) under the transformations generated by the infinitesimal operator Q D  0 .t; x; u/@t C  1 .t; x; u/@x C 1 .t; x; u/@u1 C    C m .t; x; u/@um ;

(2.6)

if the following invariance criterion is satisfied: ˇ ˇ Q .Si /ˇ k

M

D 0 .i D 1; 2; : : : ; m/:

(2.7)

The operator Q is the kth-order prolongation of the operator Q and its coefficients k

are expressed via the functions  0 ;  1 ; 1 ; : : : ; m by well-known formulae (see, e.g., [46, 49]), which will be specified in the next section for k D 2. The crucial idea, which is used for introducing the notion of Q-conditional symmetry (nonclassical symmetry) is to change the manifold M I in particular, the operator Q is used for such a purpose. It was noted only recently [13] that there are several different possibilities to realize this idea in the case of PDE systems. Definition 2.1 ([13]) Operator (2.6) is called Q-conditional symmetry (nonclassical symmetry) for an evolution system of the form (2.5) if the following invariance criterion is satisfied: ˇ ˇ Q .Si /ˇ D 0; i D 1; 2; : : : ; m; (2.8) k

Mm

where the manifold Mm is ˚

 S1 D 0; S2 D 0; : : : ; Sm D 0; Q.u1 / D 0; : : : ; Q.um / D 0 ;

where Q.ui / D  0 uit C  1 uix  i .i D 1; 2; : : : ; m/. Definition 2.2 ( [13]) Operator (2.6) is called Q-conditional symmetry of the first type for an evolution system of the form (2.5) if the following invariance criterion is satisfied: ˇ ˇ Q .Si /ˇ D 0; i D 1; 2; : : : ; m; k

M1

50

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

where the manifold M1 is ˚  S1 D 0; S2 D 0; : : : ; Sm D 0; Q.ui1 / D 0 with a fixed number i1 .1  i1  m/. Definition 2.3 ( [13]) Operator (2.6) is called Q-conditional symmetry of the p-th type for an evolution system of the form (2.5) if the following invariance criterion is satisfied: ˇ ˇ Q .Si /ˇ D 0; i D 1; 2; : : : ; m; k

Mp

where the manifold Mp is ˚

S1 D 0; S2 D 0; : : : ; Sm D 0; Q.ui1 / D 0; : : : ; Q.uip / D 0



with any given numbers i1 ; : : : ; ip .1  p  ip  m/. Obviously, these three definitions coincide in the case of m D 1, i.e., a single evolution equation. If m > 1, then one obtains a hierarchy of conditional symmetry operators. It can easily be seen that Mm Mp M1 M ; hence, each Lie symmetry is automatically a Q-conditional symmetry of the first and p-th types, while Q-conditional symmetry of the first type is that of the p-th type. From the formal point of view, it is enough to find all the Q-conditional symmetry (nonclassical symmetry) operators. Having the full list of Q-conditional symmetries one may simply check, which of them is the Lie symmetry or/and Q-conditional symmetry of the p-th type. On the other hand, to construct Q-conditional symmetry of the p-th type for a system of PDEs, one needs to solve another nonlinear system, the system of DEs. It turns out that the system of DEs in the case p D m (i.e., for searching Q-conditional symmetry) is much more complicated than in the case p < m, in particular p D 1 (i.e., for search of Q-conditional symmetry of the first type). As a result, examples of Q-conditional symmetry can be only found using particular solutions of the relevant system of DEs, while a complete classification of Q-conditional symmetries of the first type can be derived for many classes of PDE systems, including the RDS class (2.1). Hereafter we assume that  0 6D 0, i.e., the so-called no-go case when  0 D 0 is not taken into account. The natural reason to avoid examination of the no-go case follows the well-known statement (firstly proved in [60]) that exhaustive description of Q-conditional symmetries with  0 D 0 for scalar evolution equations is equivalent to solving the equation in question. This statement can be easily extended to systems of evolution equations using Definition 2.1. However, very recently (see [19] for details) we have shown that the no-go case can be completely

2.2 Q-Conditional Symmetry for Systems of Partial Differential Equations

51

examined (at least for some subclasses of class (2.1)) using the notion of Qconditional symmetry of the first type. Using the definition of Q-conditional symmetry of the p-th type, one may prove that Properties 1.2 and 1.3 (see Chap. 1) are still valid, however Property 1.1 is no longer valid provided p < m [13]. On the other hand, a new property can be formulated as follows. m P Theorem 2.1 Let us assume that X D l .t; x; u/@ul (with a fixed number l¤i1 ; lD1

i1 ; 1  i1  m) is a Lie symmetry operator of system (2.5) while Q is a Qconditional symmetry of the first type, which was found using the manifold ˚  M1 D S1 D 0; S2 D 0; : : : ; Sm D 0; Q.ui1 / D 0 : Then any linear combination C1 X C C2 Q (hereafter C1 and C2 are arbitrary constants) produces another Q-conditional symmetry of the first type. Proof In order to prove this theorem, one needs to show that the operator Z D C1 X C C2 Q satisfies Definition 2.2 on the manifold ˚  M1 D S1 D 0; S2 D 0; : : : ; Sm D 0; Z.ui1 / D 0 : This means that we need to prove that ˇ ˇ Z .Si /ˇ k

M1

D 0; i D 1; 2; : : : ; m:

(2.9)

Firstly we note that the manifold M1 coincides with M1 in this case because Z.ui1 / D .C1 X C C2 Q/.ui1 / D C2 Q.ui1 /. So one may write the following equalities ˇ ˇ Z .Si /ˇ k

ˇ ˇ   ˇ Q .Si /ˇˇ X D .S / D C C C Z ˇ i 1 2 k M1 M1 ˇk ˇ k ˇ M1 ˇ ˇ ˇ D C1 X .Si /ˇ C C2 Q.Si /ˇ D C1 X .Si /ˇ ; i D 1; 2; : : : ; m: k

M1

M1

k

k

(2.10)

M1

On the other hand ˇ ˇ X .Si /ˇ k

M

D 0; i D 1; 2; : : : ; m

(here M D fS1 D 0; S2 D 0; : : : ; Sm D 0g) because the Lie symmetry X of system (2.5) must satisfy criterion (2.7). Finally, one easily realizes that M1 M , so that the above equalities produce ˇ ˇ X .Si /ˇ k

M1

D 0; i D 1; 2; : : : ; m:

(2.11)

52

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

This means that Z is a Q-conditional symmetry of the first type because (2.9) immediately follows from (2.10) and (2.11). The proof is now complete. t u It should be stressed that Theorem 2.1 is not valid for arbitrary given Qconditional symmetry but only for that of the first type. However, this theorem can be easily generalized on Q-conditional symmetry of the p-th type (p < m) using the relevant modification of the Lie symmetry operator. m P Theorem 2.2 Let us assume that X D l .t; x; u/@ul (with the fixed set of l2A; lD1  ˚ numbers A D i1 ; : : : ; ip j 1  ip  m ; p < m) is a Lie symmetry operator of system (2.5) while Q is a˚ Q-conditional symmetry of the p-th type, which was found  using the manifold S1 D 0; S2 D 0; : : : ; Sm D 0; Q.ui1 / D 0; : : : ; Q.uip / D 0 . Then any linear combination C1 X CC2 Q produces another Q-conditional symmetry of the p-th type of the evolution system (2.5).

2.3 Systems of Determining Equations In this section, we construct the system of DEs for finding Q-conditional symmetries for the class of RDSs (2.1) and present its preliminary analysis. From the very beginning, one notes that RDS (2.1) can be simplified by applying the Kirchhoff substitution Z Z u D D1 .U/dU; v D D2 .V/dV; (2.12) where u.t; x/ and v.t; x/ are new unknown functions and Dk 6D 0; k D 1; 2 (in the case of nonconstant diffusivities, we assume that they have a finite number of roots). We remind the reader that the diffusivity coefficients must be nonnegative, hence there exist unique inverse functions to those arising in the right-hand sides of (2.12). Substituting (2.12) into (2.1), one obtains uxx D d1 .u/ut C C1 .u; v/; vxx D d2 .v/vt C C2 .u; v/;

(2.13)

where the functions d1 ; d2 and C1 ; C2 are uniquely defined via D1 ; D2 and F; G, respectively. In fact, (2.1) and (2.13) are related by the formulae d 1 .u/ D D11.U/ ; d2 .v/ D D21.V/ ; C .u; v/ D F.U; V/; C2 .u; v/ D G.U; V/; 1

(2.14)

2.3 Systems of Determining Equations

53

R 1 1 R 2 1 where U D D1 .u/  D .u/du ; V D D2 .v/  D .v/dv (the superscripts 1 mean inverse functions). Hereafter we construct conditional symmetries for the class of RDSs (2.13) instead of (2.1). Having a conditional symmetry operator and a system of the form (2.13), one may easily transform them into the relevant operator and RDS of the form (2.1) provided the inverse functions in (2.14) are known. Let us apply Definition 2.1 to construct the system of DEs for finding the Qconditional symmetry operator of the form Q D @t C .t; x; u; v/@x C 1 .t; x; u; v/@u C 2 .t; x; u; v/@v :

(2.15)

Conditions (2.8) for system (2.13) take the form   ˇˇ Q uxx  d1 .u/ut  C1 .u; v/ ˇ D 0; 2   ˇˇM2 Q vxx  d2 .v/vt  C2 .u; v/ ˇ D 0; 2

(2.16)

M2

where the manifold M2 is ˚

 uxx D d1 .u/ut  C1 .u; v/; vxx D d2 .v/vt C C2 .u; v/; Q.u/ D 0; Q.v/ D 0 :

One can apply the second prolongation of the operator Q @ @ @ @ C t2 C x1 C x2 @ut @vt @ux @vx @ @ @ @ 1 @ 2 @ Ctx1 C tx2 C tt1 C tt2 C xx C xx @utx @vtx @utt @vtt @uxx @vxx

Q D Q C t1 2

to each equations of (2.13). Here the coefficients k and  k with the relevant indices are calculated by the well-known formulae (see, e.g., [46, 49]) and are presented below. Since system (2.13) should be considered as a manifold in the prolonged space of independent variables t; x; u; v; ut ; vt ; ux ; vx ; utx ; vxt ; utt ; vtt ; uxx ; vxx ; we arrive at the second-order PDEs 1 ; 1 du1 ut C 1 Cu1 C 2 Cv1 C t1 d1 D xx 2 2 2 2 1 2 2 2 2  dv vt C  Cv C  Cu C t d D xx

(2.17)

54

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

1 2 with t1 ; t2 ; xx and xx defined in (2.19). To obtain the system of DEs in an explicit form, one needs to take into account not only system (2.13) (it will lead only to the system of DEs for Lie symmetry operators!) but also two additional conditions

ut C ux D 1 ; vt C vx D 2

(2.18)

generated by operator (2.15). Thus, inserting into (2.17) the explicit expression for k and  k : t1 D 1t C 1u ut C 1v vt  ux .t C u ut C v vt / ; t2 D 2t C 2u ut C 2v vt  vx .t C u ut C v vt / ; 1 xx D 1xx C 21xu ux C 21xv vx C 1uu u2x C 1vv vx2 C 21uv ux vx C 1u uxx   C 1v vxx  ux xx C 2xu ux C 2xv vx C uu u2x C vv vx2 C 2uv ux vx

(2.19)

 ux .u uxx C v vxx /  2uxx .x C u ux C v vx / ; 2 xx D 2xx C 22xu ux C 22xv vx C 2uu u2x C 2vv vx2 C 22uv ux vx C 2u uxx   C 2v vxx  ux xx C 2xu ux C 2xv vx C uu u2x C vv vx2 C 2uv ux vx

 ux .u uxx C v vxx /  2vxx .x C u ux C v vx / and excluding four derivatives ut ; vt ; uxx ; vxx using (2.13) and (2.18), one arrives at two cumbersome equations of the form   d1 1t C 1u .1  ux / C 1v .2  vx /  ux t C u .1  ux /  Cv .2  vx / C 1 du1 .1  ux / C 1 Cu1 C 2 Cv1 D 1xx C 21xu ux C 21xv vx C 1uu u2x C 1vv vx2 C 21uv ux vx    ux xx C 2xu ux C 2xv vx C uu u2x C vv vx2 C 2uv ux vx   C .1  ux /d1 C C1 .1u  2x  3u ux  2v vx /   C .2  vx /d2 C C2 .1v  v ux /;   d2 2t C 2u .1  ux / C 2v .2  vx /  vx t C u .1  ux /  Cv .2  vx / C 2 dv2 .2  vx / C 1 Cu2 C 2 Cv2 D 2xx C 22xu ux C 22xv vx C 2uu u2x C 2vv vx2 C 22uv ux vx  vx .xx C 2xu ux C 2xv vx C uu u2x C vv vx2 C 2uv ux vx /   C .2  vx /d2 C C2 .2u  2x  3v vx  2u ux /   C .1  ux /d1 C C1 .2u  u vx /:

(2.20)

2.3 Systems of Determining Equations

55

The next step is to take into account that the unknown functions 1 ; 2 and  do not depend on the derivatives ux and vx and therefore we split two equations arising in (2.20) w.r.t. u3x ; ux vx2 ; vx u2x ; ux vx ; u2x ; vx2 ; vx ; ux and vx3 ; vx u2x ; ux vx2 ; ux vx ; u2x ; vx2 ; vx ; ux ; respectively. As a result, we obtain the nonlinear system of DEs .1/ uu D vv D uv D 0; .2/ 1vv D 0; .3/ 2uu D 0; .4/ 2u d1 C 1uu  2xu D 0; .5/ 2v d2 C 2vv  2xv D 0; .6/ v .d1 C d2 / C 21uv  2xv D 0; .7/ u .d1 C d2 / C 22uv  2xu D 0; .8/ 1v .d1  d2 / C 21xv  2v C1  2v 1 d1 D 0; .9/ 2u .d2  d1 / C 22xu  2u C2  2u 2 d2 D 0; .10/  1 du1 C .2u 1  t  v 2  2x /d1 Cv 2 d2 C 3u C1 C v C2  21xu C xx D 0; .11/  2 dv2 C .2v 2  t  u 1  2x /d2 Cu 1 d1 C 3v C2 C u C1  22xv C xx D 0; .12/ .1 /2 du1 C .1t C 2 1v C 2x 1 /d1  2 1v d2 C1 Cu1 C 2 Cv1  1u C1 C 2x C1  1v C2  1xx D 0; .13/ .2 /2 dv2 C .2t C 1 2u C 2x 2 /d2  1 2u d1 C1 Cu2 C 2 Cv2  2u C1 C 2x C2  2v C2  2xx D 0:

(2.21)

The system of DEs (2.21) is very complicated and it seems to be unrealistic that its general solution can be derived for arbitrary given functions d1 .u/; C1 .u; v/; d2 .v/ and C2 .u; v/. This means that the conditional symmetry classification can be done only under additional restrictions (see Sects. 2.4 and 2.5). However, if one applies Definition 2.2 to search for Q-conditional symmetries of the first type, then the system of DEs obtained is simpler and the conditional symmetry classification problem can be completely solved (Chap. 4 is devoted to this topic). From the point of view of qualitative PDE theory, system (2.21) is an overdetermined nonlinear system of PDEs with seven unknown functions ; 1 ; 2 ; d1 .u/; C1 .u; v/; d2 .v/ and C2 .u; v/. An overview of possible approaches in an attempt to create a general algorithm of integrating overdetermined systems is presented in [53] (see also discussion in [11]). However, to the best of our knowledge, there is no constructive algorithm of integration of such systems at present. In order to solve a given nonlinear overdetermined system, one should develop a separate algorithm, adapted to the system in question. We study system (2.21) in Sects. 2.4 and 2.5. In order to construct its solutions, some additional restrictions will be used. In conclusion of this section, which (together with Sect. 2.2) contains a theoretical background for Chaps. 2–4, we present the following observation. According

56

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

to the definition of conditional symmetry proposed in [9, Chap. 5], the differential consequences of (2.18) should be used, hence one may reformulate criterion (2.16) in a such way that the manifold M2 D fuxx D d1 .u/ut  C1 .u; v/; vxx D d2 .v/vt C C2 .u; v/; Q.u/ D 0; Q.v/ D 0;

@ Q.u/ @t

D 0;

@ Q.u/ @x

D 0;

@ Q.v/ @t

D 0;

@ Q.v/ @x

D 0g

will be used instead of M2 . It turns out that the definition obtained does not lead to any new conditional symmetries of system (2.13) because (2.13) is a system of evolution equations [13]. Here we present a sketch of the proof (the detailed proof is presented in [50]). Let us calculate the differential consequences of the equations Q.u/ D 0 and Q.v/ D 0 (see (2.18)) w.r.t. the variables t and x: utt D 1t C 1u ut C 1v vt  t ux  u ut ux  v vt ux  uxt ;

(2.22)

utx D 1x C 1u ux C 1v vx  x ux  u ux ux  v vx ux  uxx ;

(2.23)

vtt D

2t

C

2u ut

C

2v vt

 t vx  u ut vx  v vt vx  vxt ;

vtx D 2x C 2u ux C 2v vx  x vx  u ux vx  v vx vx  vxx :

(2.24) (2.25)

Obviously, the derivatives utt ; utx ; vtt and vtx can be easily found from (2.22)–(2.25). However, the expressions obtained do not affect the algorithm presented above because the governing equations (2.18) and (2.19) do not involve these derivatives, hence one again arrives at the system of DEs (2.21). Other possibilities are to find the first-order derivatives from (2.22)–(2.25) and substitute into (2.18) and (2.19). However, the resulting system is again (2.21).

2.4 Conditional Symmetries of Reaction-Diffusion Systems with Constant Diffusivities As we noted above, the diffusivity coefficients in RDSs are usually taken to be positive constants. Let us consider system (2.13) in the case d 1 D 1 and d2 D 2 : uxx D 1 ut C C1 .u; v/; vxx D 2 vt C C2 .u; v/;

(2.26)

2.4 Conditional Symmetries of Reaction-Diffusion Systems with Constant. . .

57

where 1 and 2 are positive constants. So, the system of DEs (2.21) for finding coefficients of operator (2.15) takes the form .1/ uu D vv D uv D 0; .2/ 1vv D 0; .3/ 2uu D 0; .4/ 21 u C 1uu  2xu D 0; .5/ 22 v C 2vv  2xv D 0; .6/ .1 C 2 /v C 21uv  2xv D 0; .7/ .1 C 2 /u C 22uv  2xu D 0; .8/ .1  2 /1v C 21xv  2v C1  21 v 1 D 0; .9/ .2  1 /2u C 22xu  2u C2  22 u 2 D 0; .10/ 1 .2u 1  t  v 2  2x / C 2 v 2 C3u C1 C v C2  21xu C xx D 0; .11/ 2 .2v 2  t  u 1  2x / C 1 u 1 C3v C2 C u C1  22xv C xx D 0; .12/ 1 .1t C 2 1v C 2x 1 /  2 2 1v C 1 Cu1 C 2 Cv1 1u C1 C 2x C1  1v C2  1xx D 0; .13/ 2 .2t C 1 2u C 2x 2 /  1 1 2u C 1 Cu2 C 2 Cv2 2u C1 C 2x C2  2v C2  2xx D 0:

(2.27)

As pointed out in the previous section, the construction of the general solution of such systems is a difficult task. Here we solve system (2.27) under the additional restrictions  D .u; v/; i D i .u; v/; i D 1; 2

(2.28)

in order to construct Q-conditionally invariant RDSs with constant diffusivities. Solving Eqs. (1)–(3) of system (2.27), we obtain  D au C bv C c; 1 D p1 .u/v C q1 .u/; 2 D p2 .v/u C q2 .v/;

(2.29)

where a; b and c are arbitrary constants, and p1 ; p2 ; q1 and q2 are arbitrary smooth functions. Substituting (2.29) into Eqs. (6) and (7) from (2.27) and splitting the equations obtained w.r.t. the powers of u and v, we arrive at the system a2 .1 C 2 / D 0; b2 .1 C 2 / D 0; .1 C 2 /a.bv C c/ C 2p2v D 0; .1 C 2 /b.au C c/ C 2p1u D 0:

(2.30)

Obviously, solutions of the first pair of Eq. (2.30) are a D b D 0 because 1 and 2 are positive, hence  D c. Solving Eqs. (4)–(7) of system (2.27), we obtain pi D const D ˛i .i D 1; 2/; q1 D ˇ1 u C 1 ; q2 D ˇ2 v C 2 ;

58

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

where ˇi and i .i D 1; 2/ are arbitrary constants. Thus, expressions (2.29) take the form  D c; 1 D ˛1 v C ˇ1 u C 1 ; 2 D ˛2 u C ˇ2 v C 2 :

(2.31)

Substituting (2.31) into Eqs. (8) and (9) of system (2.27), we arrive at c˛1 .1  2 / D 0; c˛2 .1  2 / D 0:

(2.32)

Solving the system of algebraic equations (2.32), we derive three solutions 1 D 2 ; ˛1 D ˛2 D 0 and c D 0. The general solution of the remaining Eqs. (10)–(13) essentially depends on the above solutions. Examination of the first two solutions leads to the following theorem. Theorem 2.3 ([51]) The system of DEs for finding Q-conditional symmetry operators of the form (2.15) (under restrictions (2.28)) for system (2.26) coincide with the system of DEs for finding of Lie symmetry operators provided 1 D 2 or 1v D 2u D 0. Proof Substituting (2.31), with 1 D 2 , into system (2.27), we obtain that Eqs. (1)– (11) are transformed into identities, while Eqs. (12) and (13) take the form 1 Cu1 C 2 Cv1  1u C1  1v C2 D 0; 1 Cu2 C 2 Cv2  2u C1  2v C2 D 0:

(2.33)

In [22] the DEs for finding Lie symmetries with condition 1 D 2 were written down in the explicit form. Substituting conditions (2.28) into these equations, we see that the equations obtained in this way are identical to Eq. (2.33). Substituting (2.31), with ˛1 D ˛2 D 0; into system (2.27), we see, that Eqs. (1)– (11) also transform into identities, and Eqs. (12) and (13) take the form 1 Cu1 C 2 Cv1  1u C1 D 0; 1 Cu2 C 2 Cv2  2v C2 D 0:

(2.34)

Comparing Eq. (2.34) with equations, which are obtained for finding Lie symmetries of system (2.26) with conditions (2.28) from [21], we see that they are identical. The proof is now complete. t u Thus, to find Q-conditional symmetry operators, which are inequivalent to Lie symmetry operators, we must assume that 1 ¤ 2 ; ˛12 C ˛22 ¤ 0. Now one needs to set c D 0 (see Eq. (2.32)). In this case expressions (2.31) take the form  D 0; 1 D ˛1 v C ˇ1 u C 1 ; 2 D ˛2 u C ˇ2 v C 2 :

(2.35)

2.4 Conditional Symmetries of Reaction-Diffusion Systems with Constant. . .

59

So, Eqs. (1)–(11) of system (2.27) are satisfied identically by expressions (2.35), while Eqs. (12) and (13) take the form .˛1 v C ˇ1 u C 1 /Cu1 C .˛2 u C ˇ2 v C 2 /Cv1  ˇ1 C1  ˛1 C2 D ˛1 .2  1 /.˛2 u C ˇ2 v C 2 /; .˛1 v C ˇ1 u C 1 /Cu2 C .˛2 u C ˇ2 v C 2 /Cv2  ˛2 C1  ˇ2 C2 D ˛2 .1  2 /.˛1 v C ˇ1 u C 1 /:

(2.36)

Thus, we can formulate the following theorem. Theorem 2.4 ([51]) Nonlinear RDS (2.26) is Q-conditionally invariant under operator (2.15) with coefficients (2.35) if and only if (iff) the nonlinearities C1 ; C2 are the solutions of the linear first-order system (2.36). To find the general solution of system (2.36) one needs to analyse the two cases ˛2 D 0 and ˛2 ¤ 0. The case ˛2 ¤ 0; i.e., 2u ¤ 0 (then automatically 1v ¤ 0) is much more complicated and needs a separate examination (see a particular result in [51]). In the case ˛2 D 0, system (2.36) contains an autonomous equation and has the form .˛1 v C ˇ1 u C 1 /Cu1 C .ˇ2 v C 2 /Cv1 D ˇ1 C1 C ˛1 C2 C ˛1 .2  1 /.ˇ2 v C 2 /; .˛1 v C ˇ1 u C 1 /Cu2 C .ˇ2 v C 2 /Cv2 D ˇ2 C2 : (2.37) Since ˛1 ¤ 0, renaming C1 ! ˛1 C1 ; u ! ˛1 u; 1 ! ˛1 1 , and taking into account that we can get rid of the parameter 1 using linear substitutions w.r.t. u and v, system (2.37) can be reduced to the form .v C ˇ1 u/Cu1 C .ˇ2 v C 2 /Cv1 D ˇ1 C1 C C2 C .2  1 /.ˇ2 v C 2 /; .v C ˇ1 u/Cu2 C .ˇ2 v C 2 /Cv2 D ˇ2 C2 :

(2.38)

One notes the particular solution of system (2.38) 1 Cpart D

1 1 2 .2  1 /.˛1 v C ˇ1 u C 1 /; Cpart D .1  2 /.ˇ2 v C 2 /: 2 2

In order to construct the general solution of (2.38), we need to solve the corresponding homogeneous system, that is .v C ˇ1 u/Cu1 C .ˇ2 v C 2 /Cv1 D ˇ1 C1 C C2 ; .v C ˇ1 u/Cu2 C .ˇ2 v C 2 /Cv2 D ˇ2 C2 :

(2.39)

The general solution of (2.39) depends essentially on the parameters ˇ1 ; ˇ2 and 2 . As a result, the following theorem is proved.

60

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

Table 2.1 Q-conditional symmetry operators of RDS (2.26) with 1 ¤ 2

4

f .!/Cg.!/vC 12 .2 1 /v

5

f .!/v C g.!/v ln.v/C 12 .2  1 /.v C ˇ1 u/

C2 .u; v/ vg.v/ g.v/ g.!/ C 12 .1  2 / ! D 2u  v 2 ˇ2 g.!/ .v C 2 / C 12 ˇ2 .1  2 /.v C 2 / ! D ˇ2 u  vC 2 ln .v C 2 / ˇ1 g.!/v C 12 .1  2 /ˇ1 v   ! D v 1 exp ˇv1 u

6

f .!/v ˇ2 C g.!/vC 1 .2  1 /.v C ˇ1 u/ 2

.ˇ2  ˇ1 /g.!/vC 12 .1  2 /ˇ2 v

f .!/ exp.ˇ1 v/  2 g.!/C 1 .2  1 /ˇ1 .u C 2 v/ 2

! D v ˇ2 ..ˇ1  ˇ2 /u C v/ g.!/ C 12 .1  2 / ! D exp.ˇ1 v/  

1 2 3

7

C1 .u; v/ f .v/ C ug.v/ .v C u/f .v/  g.v/ f .!/ C g.!/vC 1 .2  1 /v 2

ˇ1



ˇ1

u C 2 v C

2 ˇ1

Q @t C v@u @t C ˇ1 .v C u/@u ; ˇ1 ¤ 0 @t C v@u C @v

@t C .v C ˇ1 u/@u C ˇ2 v@v ; ˇ1 ˇ2 .ˇ1  ˇ2 / ¤ 0

@t C v@u C ˇ2 .v C 2 /@v ; ˇ2 ¤ 0 @t C .v C ˇ1 u/@u C ˇ1 v@v ; ˇ1 ¤ 0

@t C ˇ1 .u C 2 v/@u C @v ; ˇ1 2 ¤ 0

Theorem 2.5 ([51]) RDS (2.26) with 1 ¤ 2 is Q-conditionally invariant under operator (2.15) under restrictions (2.28) and 2u D 0 iff the system and corresponding operator have one of the seven forms listed in Table 2.1. Any other system of the form (2.26) admitting operator (2.15) with the above restrictions is reduced to one of those from Table 2.1 by the linear transformation u ! c1 u C c2 ; v ! c3 v C c4 with correctly specified constants c1 ¤ 0; c2 ¤ 0; c3 and c4 . Table 2.1 presents seven subclasses of RDSs with the constant diffusivities, which admit Q-conditional symmetry. Each subclass involves arbitrary smooth functions f and g of the relevant arguments. Depending on the form of f and g, one may extract RDSs arising in applications and construct exact solutions for them using the symmetry operators obtained. This approach is realized for several nonlinear RDSs in Chaps. 3 and 4. Here we present an interesting example only. Let us consider case 4 from Table 2.1. Assuming that f D c1 ! 2  a1 ! (c1 6D 0) and 1 g D c1 !  a2 C 2  , one may extract the nonlinear RDS 2 uxx D 1 ut C ˇ2 u.a1 C ˇ2 c1 u  c1 v/ C rv; vxx D 2 vt C ˇ2 v.a2 C ˇ2 c1 u  c1 v/;

(2.40)

where all the coefficients are arbitrary constants, while a2 D a1  r C 2  1 . Making the discrete transformation v ! v and setting ˇ2 D 1 (for simplicity), system (2.40) and its symmetry operator are transformed into 1 ut D uxx C u.a1  c1 u  c1 v/ C rv; 2 vt D vxx C v.a2  c1 u  c1 v/

(2.41)

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

61

and @t  v@u C v@v : Now one realizes that system (2.41) with r D 0 is the DLVS describing, for example, the competition of two populations (provided all parameters are nonnegative), while (2.41) with r 6D 0 and a2 < 0 is the Belousov–Zhabotinskii type system.

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law Diffusivities In this section, we find Q-conditional symmetry operators of the form Q D @t C .t; x; U; V/@x C 1 .t; x; U; V/@U C 2 .t; x; U; V/@V

(2.42)

of two-component RDSs with the power-law diffusivities Ut D .U k Ux /x C F.U; V/; Vt D .V l Vx /x C G.U; V/:

(2.43)

As pointed out in Sect. 2.1, a power dependence of the diffusion coefficients D1 .U/ and D2 .V/ is typically adopted in models with variable diffusivities. Hence, RDSs of the form (2.43) form the most important class of such systems, if one wants to apply the results obtained for some real-world models. Note that RDSs with the diffusivities D1 D d1 U k and D2 D d2 V l (d1 and d2 are arbitrary positive constants) are reduced to the form (2.43) via scale transformations [23]. First of all we apply the local substitution u D U kC1 ; k ¤ 1; v D V lC1 ; l ¤ 1

(2.44)

(this is a particular case of the Kirchhoff substitution (2.12)) in order to simplify the further computations. Of course, the cases k D l D 1 and k D 1; l 6D 1 (l D 1; k 6D 1 is symmetric) are special and need separate investigation. 1 Substitution (2.44) reduces operator (2.42) to the form (2.15) with @u D kC1 U k @U ; 1 l @v D lC1 V @V , while system (2.43) takes the form uxx D um ut C C1 .u; v/; vxx D v n vt C C2 .u; v/;

(2.45)

  1 1 k l where m D  kC1 ¤ 1; n D  lC1 ¤ 1; C1 .u; v/ D .k C 1/F u kC1 ; v lC1   1 1 and C2 .u; v/ D .l C 1/G u kC1 ; v lC1 .

62

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

Thus, the system of DEs (2.21) corresponding to the RDS system (2.45) takes the form .1/ uu D vv D uv D 0; .2/ 1vv D 0; .3/ 2uu D 0; .4/ 2u um C 1uu  2xu D 0; .5/ 2v v n C 2vv  2xv D 0; .6/ v .um C v n / C 21uv  2xv D 0; .7/ u .um C v n / C 22uv  2xu D 0; .8/ 1v .um  v n / C 21xv  2v C1  2v 1 um D 0; .9/ 2u .v n  um / C 22xu  2u C2  2u 2 v n D 0; .10/  m1 um1 C .2u 1  t  v 2  2x /um Cv 2 v n C 3u C1 C v C2  21xu C xx D 0; .11/  n2 v n1 C .2v 2  t  u 1  2x /v n Cu 1 um C 3v C2 C u C1  22xv C xx D 0; .12/ m.1 /2 um1 C .1t C 2 1v C 2x 1 /um  2 1v v n C1 Cu1 C 2 Cv1  1u C1 C 2x C1  1v C2  1xx D 0; .13/ n.2 /2 v n1 C .2t C 1 2u C 2x 2 /v n  1 2u um C1 Cu2 C 2 Cv2  2u C1 C 2x C2  2v C2  2xx D 0:

(2.46)

Equations (1) from system (2.46) are easily integrated and lead to  D a.t; x/u C b.t; x/v C c.t; x/;

(2.47)

where a; b and c are arbitrary (at the moment) smooth functions. Substituting (2.47) into Eqs. (6) and (7) of (2.46) and taking into account the second and third equations of (2.46), one arrives at the requirement a D b D 0. Thus, Eqs. (2)–(7) of system (2.46) can be straightforwardly integrated and their general solution takes the form  D .t; x/; 1 D q1 .t/v C r1 .t; x/u C p1 .t; x/;

(2.48)

2 D q2 .t/u C r2 .t; x/v C p2 .t; x/; where the functions in the right-hand sides are arbitrary. The remaining Eqs. (8)–(13) of system (2.46) involving the functions C1 and C2 are the classification equations. To solve them one should consider three different cases depending on the functions q1 .t/; q2 .t/ and .t; x/ arising in (2.48): (a) q1 .t/ D q2 .t/ D 0; .t; x/ ¤ 0I (b) q1 .t/ D q2 .t/ D 0; .t; x/ D 0I (c) q1 .t/2 C q2 .t/2 ¤ 0; .t; x/ D 0:

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

63

Remark 2.1 The fourth possible case q1 .t/2 C q2 .t/2 ¤ 0; .t; x/ 6D 0 arises only under the restriction m D n D 0, which follows from Eqs. (8)–(9) of system (2.46). Hereafter we assume m2 C n2 6D 0; because RDSs with constant diffusivities were examined in Sect. 2.4. It turns out that case (a) does not lead to any Q-conditional symmetry operators of the form (2.42). Theorem 2.6 ([50]) RDS (2.43) with k2 C l2 ¤ 0 and .k C 1/.l C 1/ 6D 0 admits only such operators of the form (2.42) with  ¤ 0 and 1V D 2U D 0, which are equivalent to the Lie symmetry operators. Proof Here we present only a sketch of the proof. In order to prove Theorem 2.6 one should solve the system of DEs (2.46) under conditions q1 .t/ D q2 .t/ D 0; .t; x/ ¤ 0 and (2.48). In particular, DEs (10)–(13) from (2.46) take the form  t C 2x C mr1 um C mp1 um1 C 2rx1  xx D 0;   t C 2x C nr2 v n C np2 v n1 C 2rx2  xx D 0;  1       2 1 r u C p1 Cu1 C r2 v C p2 Cv1 C 2x  r1 C1  p1xx  rxx u C m p1 um1     C p1t C 2mh1 p1 C 2x p1 um C rt1 C m.r1 /2 C 2x r1 umC1 D 0; (2.49)  1       2 2 r u C p1 Cu2 C r2 v C p2 Cv2 C 2x  r2 C2  p2xx  rxx v C n p2 v n1      2 C p2t C 2nh2 p2 C 2x p2 v n C rt2 C n r2 C 2x r2 v nC1 D 0: 

Now one notes that the third and fourth equations of system (2.49) are linear firstorder PDEs w.r.t. C1 and C2 . According to the standard technique of solving such equations, one needs to find variable !, using the following ordinary differential equation (ODE) du dv D 2 : 1 Cp r u C p2

r1 u

Obviously, its solution essentially depends on r1 ; p1 ; r2 and p2 . As a result, one needs to examine six different cases (1) r1 D p1 D r2 D p2 D 0; (2) r1 D p1 D r2 D 0; p2 ¤ 0; (3) r1 D p1 D 0; r2 ¤ 0; (4) r1 D r2 D 0; p1 ¤ 0; p2 ¤ 0; (5) r1 D 0; p1 ¤ 0; r2 ¤ 0; (6) r1 ¤ 0; r2 ¤ 0:

64

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

Note that three additional cases r2 D p2 D r1 D 0; p1 ¤ 0; r2 D p2 D 0; r1 ¤ 0; r1 ¤ 0; r2 D 0; p2 ¤ 0 can be excluded from the examination because each of them can be obtained from those above by renaming u ! v, v ! u. Here we consider in detail only case (1). In this case, the third and fourth equations of system (2.49) take the form x C1 D 0; x C2 D 0; hence two subcases, x D 0 and C1 D C2 D 0, arise. The latter simply means that F 1 .U; V/ D F 2 .U; V/ D 0, i.e., RDS (2.43) reduces to two independent diffusion equations. The first subcase implies that t D 0 (see the first and second equations in (2.49)), hence  D const. Thus, we conclude that the nonlinear RDS uxx D um ut C C1 .u; v/; vxx D v n vt C C2 .u; v/

(2.50)

admits only Q-conditional symmetry operators of the form Q D @t C @x ; D const. On the other hand, system (2.50) is invariant w.r.t. the Lie symmetry operators Pt D @t and Px D @x ; hence the above operator Q is nothing else but the Lie symmetry operator. Cases (2)–(6) can be studied in a quite similar way. Finally, the detailed examination leads exactly to the Lie symmetry operators, which are listed in Table 1 [24], in each case. The sketch is now complete. t u In contrast to case (a), examination of case (b) leads to new results. Theorem 2.7 ([25, 50]) RDS (2.43) with k2 C l2 ¤ 0 and .k C 1/.l C 1/ 6D 0 is Q-conditional invariant under the operator (2.42) with  D 0 and 1V D 2U D 0 iff it and the relevant operator have the forms listed in Table 2.2 (in the table, f and g are arbitrary smooth functions of the relevant argument, while j . j D 1; 2; 3; 4/ are arbitrary constants). Proof To prove the theorem one needs to construct the general solution of subsystem (8)–(13) of system (2.46) having the general solution (2.48) of subsystem (1)–(7) and applying the restrictions  D 0 and 1v D 2u D 0. Obviously, Eqs. (8)

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

65

Table 2.2 Q-conditional symmetries of RDS (2.43) RDSs of the form (2.43) 1 Ut D .U k Ux /x C f .U kC1 / 1

Vt D V  2 Vx

Q-conditional operators 1

@t C 2p.x/V 2 @V

1

x

Restrictions pxx D p2 C p; p ¤ 0

2V 2 Cg.U kC1 /

2 Ut D .U k Ux /x C 1 U k C f .U kC1  ˛V lC1 / Vt D .V l Vx /x C 2 V l C g.U kC1  ˛V lC1 /   1 3 U D U  12 U  2U 2 C t x x   1 1 f U2 V2  1  1 Vt D V  2 Vx  2V 2 C x  1  1 g U2 V2

@t C 1 U k @U C 2 V l @V

4 Ut D .U k Ux /x C 1 U k C f .!/ Vt D .V l Vx /x C .V lC1  3 /.g.!/ C 2 V l /

@t C 1 U k @U C 2 .V  3 V l /@V

5 Ut D .U k Ux /x C .U kC1  1 /. f .!/ C 2 U k / Vt D .V l Vx /x C .V lC1  3 /.g.!/ C 4 V l /

 1 @t C 2p.x/ U 2 @U C  1 V 2 @V

 .kC1/

˛ D 12 .lC1/ ; 2 ¤ 0; 21 C l2 ¤ 0 pxx D p2 C p; p ¤ 0

!D

exp.2 .lC1/U kC1 /

 .kC1/ ; .V lC1 3 / 1 2 ¤ 0; either 21 C 23 ¤ 0 or 23 Ck2 ¤ 0 or 21 Cl2 ¤ 0  .lC1/ k @t C 2 .U  1 U /@U C .U kC1 1 / 4 ! D lC1 2 .kC1/ ; l 4 .V  3 V /@V .V 3 / 2 4 ¤ 0; either 21 C 23 ¤ 0; or 23 Ck2 ¤ 0 or 21 Cl2 ¤ 0

and (9) are automatically satisfied, while Eqs. (10) and (11) are reduced to the form 1xu D 0 and 2xv D 0; respectively, i.e.: r1 D r1 .t/;

r2 D r2 .t/:

So, the remaining Eqs. (12) and (13) take the form  1    r u C p1 Cu1 C r2 v C p2 Cv1  r1 C1      2 C rt1 C m.r1 /2 umC1 C p1t C 2mr1 p1 um C m p1 um1  p1xx D 0;     1 r u C p1 Cu2 C r2 v C p2 Cv2  r2 C2    2   2 C rt C n.r2 /2 v nC1 C p2t C 2nr2 p2 v n C n p2 v n1  p2xx D 0:

(2.51)

System (2.51) consists of two independent first-order linear PDEs w.r.t. the unknown functions C1 .u; v/ and C2 .u; v/, therefore its general solution can be straightforwardly constructed, however we should remember that the coefficients in (2.51) are functions of t and x. To construct all possible solutions of (2.51) one needs to consider the cases (1)–(6) as above (up to renaming u ! v and v ! u): In case (1), operator (2.42) immediately takes the form Q D @t ; which is, of course, the Lie symmetry operator. A similar situation occurs in case (3) because all the operators obtained are equivalent to the relevant Lie symmetry operators listed in [23]. The most interesting cases are (2) and (4)–(6).

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2 Q-Conditional Symmetries of Reaction-Diffusion Systems

Consider case (2) in detail. In this case system (2.51) takes the form p2 Cv1 D 0; p2 Cv2 C p2t v n C n. p2 /2 v n1  p2xx D 0

(2.52)

and its formal integration leads to the solution C1 D f .u/ R  p2xx C2 D  p2

p2t n v p2

  np2 v n1 dv C g.u/;

(2.53)

where f and g are arbitrary smooth functions. Since the function C2 does not depend on t and x, three subcases should be separately examined: n D 0, n D 1 and n 6D 0I 1. p2 p2 The first subcase immediately gives C2 D xxp2 t v C g.u/; so that p2xx  p2t D ; p2 where  is an arbitrary constant. So, the system uxx D um ut C f .u/; vxx D vt C v C g.u/

(2.54)

admits the Q-conditional symmetry operator Q D @t C p2 .t; x/@v ;

(2.55)

where p2 .t; x/ is the general solution of the linear PDE p2t D p2xx  p2 . However, if one now applies substitution (2.44) to (2.54) and (2.55), then the RDS and the Lie symmetry listed in [23] (see case 5 in Table 1) are obtained. So, subcase n D 0 does not lead to any Q-conditional symmetries. In the subcase n D 1 the general solution of (2.52) takes the form C1 D f .u/; where  D

p2xx p2

C2 D v C g.u/;

 p2 . So, the system uxx D um ut C f .u/; vxx D vvt C v C g.u/

(2.56)

admits the Q-conditional symmetry operator Q D @t C p2 .x/@v ;

(2.57)

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

67

where the function p2 .x/ is the general solution of the nonlinear ODE p2xx D . p2 /2 C p2 :

(2.58)

Applying now substitution (2.44) to (2.56)–(2.57) and introducing the relevant notations, one arrives at the system and the Q-conditional symmetry operator listed in case 1 of Table 2.2. Considering the subcase n ¤ 0I 1; we immediately obtain p2 D  D const (see (2.53)) and this leads to the system uxx D um ut C f .u/; vxx D v n vt  v n C g.u/

(2.59)

Q D @t C @v :

(2.60)

and the operator

 V l @V with  6D 0 by using Operator (2.60) is reduced to the form Q D @t C lC1 substitution (2.44). On the other hand, system (2.59) and operator (2.60) correspond to a particular case at 1 D ˛ D 0 of those listed in case 2 of Table 2.2. Thus, case (2) is completely investigated. Case (4) can be examined in a quite similar way and the system

uxx D um ut  ˛um C f .u  ˛v/; vxx D v n vt  v n C g.u  ˛v/

(2.61)

Q D @t C .˛@u C @v /

(2.62)

and the operator

are obtained, where ˛ ¤ 0 is an arbitrary constant. It is easily seen that systems and operators (2.59)–(2.62) can be united, i.e., the restriction ˛ ¤ 0 is not essential. Applying now substitution (2.44) to (2.61)–(2.62) and introducing the relevant notations, one arrives at the system and the Q-conditional symmetry operator listed in case 2 of Table 2.2. It turns out that the power m D n D 1 leads to an additional symmetry in this case. In fact, the system uxx D uut C u C f .u  v/; vxx D vvt C v C g.u  v/

(2.63)

is conditionally invariant w.r.t. the operator Q D @t C p2 .x/.@u C @v /;

(2.64)

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2 Q-Conditional Symmetries of Reaction-Diffusion Systems

where p2 .x/ is the general solution of the nonlinear ODE (2.58). Formulae (2.63)– (2.64) together with the substitution (2.44) generate the system and the operator listed in case 3 of Table 2.2. Those listed in cases 4 and 5 of the table can be similarly obtained by examination of cases (5) and (6). Finally, we note that all operators arising in Table 2.2 are not Lie symmetry operators because any Lie symmetry operator of RDS (2.43) must be linear on U and V [23]. The proof is now complete. t u Remark 2.2 Restrictions on the coefficients k arising in the last column of Table 2.2 guarantee that the relevant operators do not coincide with Lie symmetries. Of course, those operators are still Q-conditional symmetry operators if some k vanish, however, they are equivalent to the relevant Lie symmetry operators obtained in [23]. Case (c) is the most complicated. In order to examine this case, one needs to consider separately three subcases (c1) p1x D 0; p2x ¤ 0; (c2) p1x ¤ 0; p2x ¤ 0; (c3) p1x D p2x D 0: We note that the subcase p2x D 0; p1x ¤ 0 is reduced to (c1). A complete analysis of subcase (c1) is done in [50] and the result can be formulated as follows. Theorem 2.8 RDS (2.43) with k2 C l2 ¤ 0 and .k C 1/.l C 1/ 6D 0 is Q-conditional invariant under the operator (2.42) with p1x D 0; p2x ¤ 0 iff it and the relevant operator have 16 forms listed below (F.U/ is an arbitrary smooth function, ˛; ˇ;

and j . j D 1; 2; 3; 4/ are arbitrary constants). Ut D .U k Ux /x C F 1 ; Vt D Vxx C F 2 ; k ¤ 0: 1. F 1 D 1 U kC1 C 2 ; F 2 D 3 V C F.U/;  Q D @t C exp ..1 .k C 1/ C 3 / t/ U kC1 C p @V ; pt D pxx C 3 p C 2 exp ..1 .k C 1/ C 3 / t/ ; ¤ 0. kC1 2. F 1 D F.U/; F 2 D 3 V  2 .k C 1/F.U/  C 2 3 U ; kC1 Q D @t C 2 U C V C p @V ; pt D pxx C 3 p; 2 ¤ 0. 3. F 1 D 1 C 2 U k ; F 2 D 3 V C 4 U 2.kC1/ C 5U kC1 ;  kC1 U C p @V ; Q D @t C 2 U k @U C e3 t  22 4 .kC1/    3 2 .kC1/ pt D pxx C 3 p  .k C 1/ .1 C 2 / e3 t C 2 3 4 C 2 5 ; 2 3 ¤ 0.

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

69

4. F 1 D 1 C 2 U k ; F 2 D 3U 2.kC1/ C 4 U kC1 ;  Q D @t C 2 U k @U C .22 3 .k C 1/t C / U kC1 C p @V ; pt D pxx  .22 3 t C /.2 C 1 /.k C 1/ C 2 4 ; 2 ¤ 0. kC1 5. F 1 D 1 C 2 U k ; F 2 D  C 5exp ˛U kC1 ; 3 V C 4 U

 kC1 U

e3 t C ˛2 4 .kC1/ C˛ .k C 1/V C p @V ; 2  3 C 2 4 .k C 1/; pt D pxx  3 p  .1 C 2 /.k C 1/ e3 t C ˛2 4 .kC1/ 3 ˛2 3 ¤ 0.   6. F 1 D 1 C 2 U k ; F 2 D 3 U kC1 C 4 exp ˛U kC1 ;  Q D @t C 2 U k @U C

Q D @t C2 U k @U C .  ˛2 3 .k C 1/t/ U kC1 C˛2 .kC1/V Cp @V ; pt D pxx   .1 C 2 /.k C 1/.  ˛2 3 t/ C 2 3 .k C 1/; ˛2 ¤ 0. 2 1 7. F D 1  kC1 U kC1 C 2 U;

F 2 D 1 .k C 1/V  3 .k C 1/U C 3 U kC1 ; 2 Q D @t C 1˛ exp. U@U 2 kt/   3 ..2 ˇ/ exp.2 kt/C˛ˇ/ kC1 C U C ˇV C p @V ; 2 ˛.kC1/ pt D pxx C 1 .k C 1/p; 2 ˛ ¤ 0. 8. F 1 D .1  2 /U kC1 C 2 .k C 1/U; kC1 F 2 D 1 .k C 1/V  3 .k C 1/U C 3 U C 4 .k C 1/ ln U;   2 .kC1/ 1 Q D @t C 1 exp.2 k.kC1/t/ U@U C 3 exp.2 k.k C 1/t/U kC1 C p @V ; 2

2 4 .kC1/ ; 2 ¤ 0. pt D pxx C 1 .k C 1/p  1 exp. 2 k.kC1/t/     1 kC1 k 9. F D U ; C ˛ 1 C 2 U  kC1  ˇ  2 C ˛ C 4 U kC1 C ˛ ; F D .1 C 2 /.k C 1/V C 3 U Q D @t C 2 U C ˛U k @U  C .  2 4 .ˇ  1/.k C 1/t/ U kC1 C ˇ2 .k C 1/V C p @V ; pt D pxx C .1 C 2 /.k C 1/p  ˛2 4 .k C 1/.ˇ  1/  ˛.1 C2 /.k C 1/.24 .k C 1/.ˇ  1/t C /; 2 ˇ.ˇ  1/ ¤ 0. 10. F 1 D U kC1 C ˛ 1 C 2 U k ;   kC1 ˇ  F 2 D 3 V C 4 U kC1 C ˛ C 5 U C˛ ; Q D @t C 2 .U C ˛U k /@U C exp..1  2 .k C 1/ C 3 /t/   5 .ˇ1/.kC1/ kC1 U C 12  C ˇ .k C 1/V C p @V ; 2 2 .kC1/C3  pt D pxx C 3 p C ˛ .1  2 .k C 1// exp ..1  2 .k C 1/ C 3 /t/  3 5 .ˇ1/.kC1/ ; 2 .1  2 .k C 1/ C 3 /ˇ.ˇ  1/ ¤ 0.  2 1  .kC1/C 2 3     ˛ 1 C2 U k ; 11. F 1 D U kC1 C   F 2 D 3 V C U kC1 C ˛ 4 ln.U kC1  C ˛/ C 5 ;   Q D @t C 2 U C ˛U k @U C exp ..1  2 .k C 1/ C 3 /t/   2 4 .kC1/ kC1 U  1  C  .k C 1/V C p @V ; 2 2 .kC1/C3

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pt D pxx C 3 p  ˛.1 C 2 /.k C 1/ exp ..1  2 .k C 1/ C 3 /t/

2 3 4 .kC1/ C ˛ ; 1  2 .k C 1/ C 3 ¤ 0. 1 2 .kC1/3   kC1  1 C ˛ 1 C 2 U k ; 12. F D U   F 2 D .1 C 2 /.k C 1/V C U kC1 C ˛ 3 ln.U kC1 C ˛/ C 4 ;

Q D @t C 2 .U C ˛U k /@U C .2 3 .k C 1/t C / U kC1  C2 .k C 1/V C p @V ;   pt D pxx C .1 C 2 /.k C 1/ p  ˛.2 3 .k C 1/t C / C ˛2 3 .k C 1/.    13. F 1 D U kC1 C ˛ 1 C 2 U k ;  kC1   F 2 D .1 C 2 /.k C 1/V C 3 ln U kC1 U C ˛ C  C ˛ ; 4   k kC1 C p @V ; Q D @t C 2 .U C ˛U /@U C .2 4 .k C 1/t C / U  pt D pxx C .k C 1/ .1 C 2 / .p  ˛.2 4 .k C 1/t C //  C2 .˛4 C 3 / . 





; 14. F 1 D U kC1 C ˛ 1 C 2 U k   2 kC1

 F D 3 V C 4 ln U C ˛ C 5 U kC1 C ˛ ;   Q D @t C 2 U C ˛U k @U    2 5 U kC1 C p @V ; C exp ..3  .1 C 2 /.k C 1//t/ C .1 C2/.kC1/ 3 pt D pxx C 3 p  ˛.1 C 2 /.k C 1/ exp ..3  .1 C 2 /.k C 1///   3 5 ; 3  .1 C 2 /.k C 1/ ¤ 0: C  C2 3 .˛ 4 1 C2 /.kC1/

15. F 1 D 1 U C 2 U k C 3 U k V  3 V C 4 U kC1 C 5 ; F 2 D .4  1 /.k   C 1/V;   Q D @t C 3 V  1 U kC1 C 2 U k @U C ˛U kC1 C ˇV C p @V ; pt D pxx C .k C 1/ ..4  1 /p  ˛.5 C 2 // ; 3 ¤ 0. Ut D .U k Ux /x C F 1 ; 1

Vt D .V  2 Vx /x C F 2 : 

1



16. F 1 D 1 V  21 V 2 C ˛ U k C 3 U kC1  21 2 U C 4 ;

 1  F 2 D 23 .k C 1/ V 2 C ˛  22 .k C 1/.1 V C 4 /;  1  1   1 Q D @t 21 V 2 C 2 U kC1 C ˛ U k @U C2 p.x/  1 2 .k C 1/V 2 V 2 @V ; p00 D p2  3 .k C 1/p C 1 2 .k C 1/2 .2 4  ˛3 /; 1 ¤ 0.

The detailed proof of Theorem 2.8 is based on the construction of the general solution of the system of DEs (2.46) provided the given restrictions on the operator Q take place. We omit here the relevant cumbersome calculations. Remark 2.3 Each Q-conditional symmetry presented in cases 1–15 involves the function p.t; x/, which is an arbitrary solution of the linear diffusion equation pt D pxx C pR1 .t; x/ C R0 .t; x/, where R1 .t; x/ and R0 .t; x/ are the correctly specified

2.5 Conditional Symmetries of Reaction-Diffusion Systems with Power-Law. . .

71

functions. The Q-conditional symmetry arising in case 16 contains the function p.x/, which is an arbitrary solution of the integrable ODE. Remark 2.4 In case 16, we have corrected inexactnesses arising in [50]. Remark 2.5 Each system arising in Theorem 2.8 is semi-coupled, i.e., contains autonomous equations. It is unlikely that such systems can reflect any general physical or biological laws. However, they may be governing equations for some specific models describing real-world processes. For example, case 1 with 1 D 2 D 3 D F.U/ D 0 generates a system of two autonomous diffusion equations admitting the Q-conditional symmetry Q D @t C . U kC1 C p/@V . These are governing equations for the classical Stefan type problem modelling melting and evaporation of metals (see, e.g., [1, 14]). In contrast to (c1), examination of case (c2) leads to a trivial result: RDS (2.43) is not invariant under the operator (2.42) provided p1x ¤ 0 and p2x ¤ 0. The proof of this statement can be derived by solving the system of DEs (2.46) under the restrictions q1 .t; x/2 C q2 .t; x/2 ¤ 0;  D 0; p1x ¤ 0; p2x ¤ 0: Finally, examination of case (c3) leads to the system .q1 v C r1 u C p1 /Cu1 C .q2 u C r2 v C p2 /Cv1  r1 C1  q1 C2 C.q1 .q2 u C r2 v C p2 / C q1t v C rt1 u C p1t /um Cm.q1 v C r1 u C p1 /2 um1  q1 .q2 u C r2 v C p2 /v n D 0; 1 .q v C r1 u C p1 /Cu2 C .q2 u C r2 v C p2 /Cv2  r2 C2  q2 C1 C.q2 .q1 v C r1 u C p1 / C q2t u C rt2 v C p2t /v n Cn.q2 u C r2 v C p2 /2 v n1  q2 .q1 v C r1 u C p1 /um D 0

(2.65)

(here qi D qi .t/; ri D ri .t/; pi D pi .t/; i D 1; 2), which should be solved w.r.t. the functions Ci D Ci .u; v/; i D 1; 2. Although system (2.65) is linear, the algorithm for the construction of its general solution is cumbersome because the functions qi ; ri and pi .i D 1; 2/ are not specified. Of course, the general solution can be easily constructed in the case of correctly specified coefficients. For example, if q1 D 0, then the first equation is autonomous and system (2.65) can be solved in a similar way as it was done in [51]. As examples, we examined the systems uut D uxx C u.a1 C b1 u C c1 v/; vvt D vxx C v.a2 C b2 u C c2 v/

(2.66)

uut D uxx C u.a1  b1 u  c1 v/ C rv; vvt D vxx  v.a2 C b2 u/:

(2.67)

and

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Obviously, systems (2.66) and (2.67) are natural generalizations of DLVS (2.2) and of the Belousov–Zhabotinskii system (2.3), respectively. It turns out that both systems admit Q-conditional symmetry operators, which can be constructed using particular solutions of system (2.65) with m D n D 1 and the functions Ck .k D 1; 2/ taken from (2.66) and (2.67). Finally, we conclude that the generalized DLVS (2.66) admits Q-conditional symmetry of the form Q D @t C .a1 C b1 u C c1 v/@u C .a2 C b2 u C c2 v/@v (the restriction c21 C b22 ¤ 0 guarantees that it is a non-Lie symmetry) and the generalized Belousov–Zhabotinskii system (2.67) with c1 D r.2b1  b2 /.r C a1 C a2 /.a1 C a2 /2 and a1 C a2 6D 0 is conditionally invariant w.r.t. the operator Q D .a1 C a2 /2 @t  r2 .2b1  b2 /v@u C r.a1 C a2 /.2b1  b2 /v@v (the restriction r.2b1  b2 / ¤ 0 guarantees that it is a non-Lie symmetry).

2.6 Concluding Remarks A novel way to find new type of symmetries for PDEs was proposed in 1969 [8]. In the same paper, the idea was realized in the form of an algorithm for finding new symmetries of the linear heat (diffusion) equation. Although the algorithm is based on the classical Lie scheme [46, 49], the resulting symmetries can be nonLie symmetries of the equation in question, therefore they were called nonclassical symmetries. Following [27, 35], we call them Q-conditional symmetries in order to distinguish from other types of symmetries (weak symmetry [47, 48, 52], conditional symmetry [20, 34, 35], generalized conditional symmetry [30, 59]) because each non-Lie symmetry can be called nonclassical. From the applicability point of view, the algorithm for finding Q-conditional symmetry of a given PDE is highly nontrivial (each time a nonlinear system of PDEs must be integrated) and this was a reason why nontrivial examples of Q-conditional symmetries were not found for a long time. In 1987 the Bluman–Cole algorithm was rediscovered in [33, 47] and later successfully applied to a wide range of nonlinear PDEs (see, e.g., [9, 26] and the references therein), especially reaction-diffusion-convection equations (see Chap. 1 for details). It turns out that the problem of finding Q-conditional symmetry becomes much more complicated in the case of (multi-component) nonlinear systems of PDEs. To the best of our knowledge, there are very few papers devoted to the search for Q-conditional (nonclassical) symmetries of systems of evolution equations, which

References

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were published before 2005 [5, 28, 38]. A majority of such papers were published during the last 10 years [4, 13, 15–19, 25, 40, 55]. In this chapter, the recently developed theoretical background for searching for Q-conditional symmetries of evolution systems of PDEs is presented. We generalize the standard notation of Q-conditional (nonclassical) symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of Q-conditional symmetry of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some properties, which distinguish it from nonclassical symmetry. The class of two-component nonlinear RDSs (2.1) is examined in order to find Qconditional symmetry operators. The relevant system of DEs was derived and solved under additional restrictions, so that several RDSs of the form (2.1) possessing conditional symmetry were obtained. In particular, it was shown that the DLVS and the Belousov–Zhabotinskii system (with correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry operators. Finally, it is worth highlighting the following remarks about Q-conditional symmetry of the class of RDSs (2.1). 1. The system of DEs (2.21) is very complicated and we believe that its general solution cannot be derived without additional restrictions on the symmetry operators in question. In the case of scalar reaction-diffusion equations (RDEs), the relevant system of DEs is essentially simpler, hence its general solution can be constructed (see Chap. 1). 2. Particular solutions of system (2.21) usually lead to RDSs involving arbitrary function(s) in reaction terms (see Tables 2.1 and 2.2). Scalar RDEs with Qconditional symmetry do not involve any arbitrary functions, but arbitrary constants only (see Chap. 1). 3. The definition of Q-conditional symmetry of the first type should be applied in order to obtain a system of DEs, which can be integrated without any restrictions (in contrast to (2.21)). Thus, a complete classification of such symmetries could be derived (see Chap. 4).

References 1. Alexiades, V., Solomon, A.: Mathematical Modeling of Melting and Freezing Processes. Hemisphere Publishing Corporation, Washington (1993) 2. Allassia, F., Nucci, M.C.: Symmetries and heir equations for the laminar boundary layer model. J. Math. Anal. Appl. 201, 911–42 (1996) 3. Ames, W.F.: Nonlinear Partial Differential Equations in Engineering. Academic, New York (1972) 4. Arrigo, D.J., Ekrut, D.A., Fliss, J.R., Le, L.: Nonclassical symmetries of a class of Burgers’ systems. J. Math. Anal. Appl. 371, 813–820 (2010) 5. Barannyk, T.: Symmetry and exact solutions for systems of nonlinear reaction-diffusion equations. Proc. Inst. Math. Nat. Acad. Sci. Ukr. 43, 80–85 (2002)

74

2 Q-Conditional Symmetries of Reaction-Diffusion Systems

6. Berman, V.S., Danilov, Yu. A.: Group properties of the generalized Landau-Ginzburg equation. Sov. Phys. Dokl. 26, 484–486 (1981) 7. Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002) 8. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969) 9. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2010) 10. Britton, N.F.: Essential Mathematical Biology. Springer, Berlin (2003) 11. Carini, M., Fusco, D., Manganaro, N.: Wave-like solutions for a class of parabolic models. Nonlinear Dyn. 32, 211–222 (2003) 12. Cherniha, R.: Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Rep. Math. Phys. 46, 63–76 (2000) 13. Cherniha, R.: Conditional symmetries for systems of PDEs: new definition and their application for reaction-diffusion systems. J. Phys. A Math. Theor. 43, 405207 (19 pp) (2010) 14. Cherniha, R., Cherniha, N.: Exact solutions of a class of nonlinear boundary value problems with moving boundaries. J. Phys. A Math. Gen. 26, L935–L940 (1993) 15. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system. Math. Comput. Model. 54, 1238–1251 (2011) 16. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities. Commun. Nonlinear Sci. Numer. Simul. 17, 3177–3188 (2012) 17. Cherniha, R., Davydovych, V.: Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system. J. Phys. A Math. Theor. 46, 185204 (14 pp) (2013) 18. Cherniha, R., Davydovych, V.: Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations. In: Algebra, Geometry and Mathematical Physics, vol. 85, pp. 533–553. Springer, Berlin/Heidelberg (2014) 19. Cherniha, R., Davydovych, V.: Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case. Appl. Math. Comput. 268, 23–34 (2015) 20. Cherniha, R., Henkel, M.: On nonlinear partial differential equations with an infinitedimensional conditional symmetry. J. Math. Anal. Appl. 298, 487–500 (2004) 21. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. J. Phys. A Math. Gen. 33, 267–282, 7839–7841 (2000) 22. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II. J. Phys. A Math. Gen. 36, 405–425 (2003) 23. Cherniha, R., King, J.R.: Nonlinear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. J. Math. Anal. Appl. 308, 11–35 (2005) 24. Cherniha, R., King, J.R.: Lie symmetries and conservation laws of nonlinear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math. 71, 391–408 (2006) 25. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of reaction– diffusion systems with power diffusivities. J. Phys. A Math. Theor. 41, 185208–185222 (2008) 26. Cherniha, R., Pliukhin, O.: New conditional symmetries and exact solutions of reaction– diffusion–convection equations with exponential nonlinearities. J. Math. Anal. Appl. 403, 23–37 (2013) 27. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear secondorder evolution equations with convection term. Eur. J. Appl. Math. 9, 527–542 (1998) 28. Cherniha, R.M., Serov, M.I.: Nonlinear systems of the Burgers-type equations: Lie and Qconditional symmetries, ansatze and solutions. J. Math. Anal. Appl. 282, 305–328 (2003) 29. Feireisl, E., Hilhorst, D., Mimura, M., Weidenfeld, R.: On a nonlinear diffusion system with resource-consumer interaction. Hiroshima Math. J. 33, 253–295 (2003) 30. Fokas, A.S., Liu, Q.M.: Generalized conditional symmetries and exact solutions of nonintegrable equations. Theor. Math. Phys. 99, 571–582 (1994) 31. Fushchych, W.I., Cherniha, R.M.: The Galilean relativistic principle and nonlinear partial differential equations. J. Phys. A Math. Gen. 18, 3491–3503 (1985)

References

75

32. Fushchych, W.I., Cherniha, R.M.: Exact solutions of two multi-dimensional nonlinear Schrödinger type equations (in Russian). Inst. Math. Acad. Sci. of USSR, Kyiv (1986, Preprint 86.85) 33. Fushchych, W., Tsyfra, I.: On a reduction and solutions of nonlinear wave equations with broken symmetry. J. Phys. A Math. Gen. 20, L45–L48 (1987) 34. Fushchych, W.I., Serov, M.I., Chopyk, V.I.: Conditional invariance and nonlinear heat equations (in Russian). Proc. Acad. of Sci. Ukr. 9, 17–21 (1988) 35. Fushchych, W., Shtelen, W., Serov, M.: Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics. Kluwer Academic Publishers, Dordrecht (1993) 36. Knyazeva, I.V., Popov, M.D.: A system of two diffusion equations. In: CRC Handbook of Lie group Analysis of Differential Equations, vol. 1, pp. 171–176. CRC Press, Boca Raton (1994) 37. Liu, Q.M., Fokas, A.S.: Exact interaction of solitary waves for certain nonintegrable equations. J. Math. Phys. 37, 324–345 (1996) 38. Lou, S.Y., Ruan H.Y.: Non-classical analysis and Painlevé property for the Kuperschmidt equations. J. Phys. A Math. Gen. 26, 4679–4688 (1993) 39. Martinez, S.: The effect of diffusion for the multispecies Lotka–Volterra competition model. Nonlinear Anal. Real World Appl. 4, 409–436 (2003) 40. Murata, S.: Non-classical symmetry and Riemann invariants. Int. J. Non Linear Mech. 41, 242–246 (2006) 41. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) 42. Murray, J.D.: An Introduction I: Models and Biomedical Applications. Springer, Berlin (2003) 43. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003) 44. Namba, T.: Competition for space in a heterogeneous environment. J. Math. Biol. 27, 1–16 (1989) 45. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems. Modern Perspectives, 2nd edn. Springer, Berlin (2001) 46. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1986) 47. Olver, P., Rosenau, P.: Group-invariant solutions of differential equations. SIAM J. Appl. Math. 47, 263–278 (1987) 48. Olver, P.J., Vorob’ev, E.M.: Nonclassical and conditional symmetries. In: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, pp. 291–328. CRC Press, Boca Raton (1996) 49. Ovsiannikov, L.V.: The Group Analysis of Differential Equations. Academic, New York (1980) 50. Pliukhin, O.H.: Conditional symmetries and exact solutions of reaction-diffusion systems with power coefficients of diffusion (in Ukrainian). PhD thesis, Institute of Mathematics of NAS of Ukraine, Kyiv (2009) 51. Pliukhin, O.: Q-conditional symmetries and exact solutions of nonlinear reaction–diffusion systems. Symmetry 7, 1841–1855 (2015) 52. Pucci, E., Saccomandi, G.: On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598 (1992) 53. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: The Method of Differential Constraints and Its Applications in Gas Dynamics (in Russian). Nauka, Novosibirsk (1984) 54. Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979) 55. Torrisi, M., Tracinà, R.: Exact solutions of a reaction–diffusion system for Proteus mirabilis bacterial colonies. Nonlinear Anal. Real World Appl. 12, 1865–1874 (2011) 56. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952) 57. van Vuuren, J.H.: The existence of travelling plane waves in a general class of competitiondiffusion systems. IMA J. Appl. Math. 55, 135–148 (1995) 58. Wong, J.: The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions. J. Comput. Appl. Math. 223, 421– 437 (2009)

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59. Zhdanov, R.Z.: Conditional Lie-Backlund symmetry and reduction of evolution equations. J. Phys. A Math. Gen. 28, 3841–3850 (1995) 60. Zhdanov, R.Z., Lahno, V.I.: Conditional symmetry of a porous medium equation. Physica D 122, 178–186 (1998) 61. Zulehner, W., Ames, W.F.: Group analysis of a semilinear vector diffusion equation. Nonlinear Anal. 7, 945–69 (1983)

Chapter 3

Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Abstract Two- and three-component diffusive Lotka–Volterra systems are examined in order to find Q-conditional symmetries, to construct exact solutions and to provide their biological interpretation. An exhaustive description of Q-conditional symmetries of the first type (a special subset of nonclassical symmetries) of these nonlinear systems is derived. An essential part of this chapter is devoted to the construction of exact solutions of the systems in question using the symmetries obtained. Starting from examples of travelling fronts (finding such solutions is important from the applicability point of view), we concentrate mostly on finding exact solutions with a more complicated structure. As a result, a wide range of exact solutions are constructed for the two-component diffusive Lotka–Volterra system and some examples are presented for the three-component diffusive Lotka–Volterra system. Moreover, a realistic interpretation for two and three competing species is provided for some exact solutions.

3.1 The Lotka–Volterra System and Its Application It is widely accepted that Lotka [24] and Volterra [34] were the prominent investigators who created the mathematical background of ecology. In the 1920s they developed independently a mathematical model for describing a prey–predator interaction. The model consists of two nonlinear ordinary differential equations (ODEs) of the form du D u.a  bv/; dt dv D v.c C du/; dt where the functions u.t/ and v.t/ describe the time evolution of the numbers of prey and predators, respectively, a; b; c and d are positive parameters with

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6_3

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well-known interpretation. In words, one may formulate the Lotka and Volterra model as follows [5]: [rate of change of u] = [net rate of growth of u without predation]  [net rate of loss of u due to predation], [rate of change of v] =  [net rate of loss of v without prey] C [net rate of growth of v due to predation]. It was soon noted that two ODEs with quadratic nonlinearities can also describe other types of interaction (competition, mutualism), hence the classical Lotka– Volterra model is usually presented in the form ut D u.a1 C b1 u C c1 v/; vt D v.a2 C b2 u C c2 v/:

(3.1)

Depending on the signs of coefficients in (3.1), three common types of interaction between two populations arise, namely: predator–prey interaction, competition and mutualism. A natural generalization of (3.1) follows if one considers the interaction of several species, say m instead of 2, and takes into account their diffusion in space. As a result, the diffusive m-component Lotka–Volterra system is obtained 0 uit D di ui C ui @ai C

m X

1 bij u j A ;

i D 1; : : : ; m;

jD1

where u1 .t; x/; u2 .t; x/; : : : ; um .t; x/ are unknown concentrations of species, di  0; ai and bij are arbitrary constants (i; j D 1; : : : ; m/; while x D .x1 ; x2 ; : : : ; xn / and @2 @2  is the Laplace operator @x 2 C    C @x2 . Nowadays the diffusive Lotka–Volterra n 1 system (DLVS) is used as the basic model for a variety of processes in ecology, biology, medicine, chemistry, economics, etc. [5, 27–29]. In those processes, the functions uj . j D 1; : : : ; m/ can be concentrations/densities of species, cells (in tissue, tumour, bones, etc.), drugs or chemicals. On the other hand, the DLVS is a much more complicated model (compared with that without diffusion); its extensive rigorous study started about 40 years ago (see, e.g., [15, 18, 22, 32]). In the case of the two-component DLVS, existence and construction of plane wave solutions were examined [10, 19, 21, 23, 31]. In the case of multi-component DLVSs, the progress in this direction has been rather modest (see some results in [6, 17, 20, 26]). Of course, there is a large number of studies, in which numerical methods are applied, however, we concentrate on analytical (especially symmetry-based) methods and techniques.

3.2 The Two-Component Diffusive Lotka–Volterra System

79

3.2 The Two-Component Diffusive Lotka–Volterra System Here we examine the two-component DLVS 1 ut D uxx C u.a1 C b1 u C c1 v/; 2 vt D vxx C v.a2 C b2 u C c2 v/;

(3.2)

where k > 0; ak ; bk and ck are arbitrary constants (k D 1; 2), u D u.t; x/ and v D v.t; x/ are unknown functions presenting, for example, the population concentrations. Hereinafter we assume that (3.2) is nonlinear and b22 C c21 ¤ 0, i.e., the system cannot consist of two independent equations. In this section, Q-conditional symmetries of DLVS (3.2) are completely described. All possible Q-conditional symmetries of the first type in an explicit form are constructed. The relevant non-Lie ansätze reducing the DLVS with correctly specified coefficients to ODE systems and examples of exact solutions are found. A possible biological interpretation of some exact solutions is presented. In particular, those describing different scenarios of the competition of two populations are discussed.

3.2.1 Determining Equations Lie symmetries of (3.2) have been completely described in [10]. Here we present briefly this result in order to distinguish Lie and Q-conditional symmetries. It is obvious that DLVS (3.2) with arbitrary coefficients is invariant under the two-dimensional Lie algebra generated by the following operators of translation with respect to (w.r.t.) the independent variables t and x: Pt D @t ;

Px D @x :

(3.3)

This algebra is usually called the principal (trivial) algebra. From the applicability point of view this means that one may look for the plane wave solutions of DLVS (3.2) of the form u D .!/; v D

.!/; ! D x  ˛t; ˛ 2 R:

(3.4)

In papers [10, 19, 21, 23, 31], ansatz (3.4) was used for constructing some plane wave solutions in an explicit form (see Sect. 3.2.3). It turns out that there are several cases when this nonlinear system admits a nontrivial Lie symmetry, i.e., three- and higher-dimensional Lie algebra. Theorem 3.1 ([10]) DLVS (3.2) is invariant w.r.t. three- and higher-dimensional Lie algebra if and only if (iff) its reaction terms and the corresponding symmetry operator(s) have the forms listed in Table 3.1. If DLVS (3.2) with other reaction

80

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Table 3.1 Lie symmetries of DLVS (3.2) Reaction terms 1 2 3 4 5 6 7

u.b1 u C c1 u/ v.b2 u C c2 v/ b1 u2 b2 uv b1 u2 0 u.a1 C b1 u/ b2 uv u.a1 C b1 u/ 0 u.a1 C b1 u/ v.a1 C b1 u/ b1 u2 b1 uv

Restriction

Lie symmetries extending algebra (3.3) D D 2tPt C xPx  2.u@u C v@v / D; v@v D; v@v ; X 1 D P.t; x/@v v@v v@v ; X 1 D P.t; x/@v

1 D 2

v@v ; u@v

1 D 2

v@v ; u@v ; D R D b1 tu@u C @v

The function P.t; x/ is an arbitrary solution of the linear diffusion equation 1 Pt D Pxx and, hence, the operator X 1 generates an infinite-dimensional Lie algebra, which is typical for all linear partial differential equations

terms admits a nontrivial Lie algebra, then it is reduced to one of the forms presented in Table 3.1 by a local substitution from the set u ! c11 exp.c10 t/u C c12 ; v ! c21 C c22 exp.c20 t/v (here cki (k D 1; 2; i D 0; 1; 2) are some correctly specified constants). It can be seen from Table 3.1 that DLVSs possessing nontrivial Lie symmetry are semi-coupled (see cases 2–7), except for case 1. From the point of view of real-world applications, DLVS (3.2) with a1 D a2 D 0 only is important and can be examined in order to find exact solutions using the operator of scale transformations D. It turns out that DLVS (3.2) also admits Q-conditional symmetry under the relevant coefficient restrictions, which do not affect the biological sense of the system. Definition 2.1 should be applied in order to construct the system of determining equations (DEs) for finding the Q-conditional symmetry operator of the form Q D @t C .t; x; u; v/@x C 1 .t; x; u; v/@u C 2 .t; x; u; v/@v :

(3.5)

However, we have done this already for the general reaction-diffusion system (RDS) in Chap. 2. Thus, one needs only to substitute dk D k ; C1 D u.a1 C b1 u C c1 v/; C2 D v.a2 C b2 u C c2 v/

3.2 The Two-Component Diffusive Lotka–Volterra System

81

into (2.21). As a result, the system of DEs takes the form uu D vv D uv D 1vv D 2uu D 0; 21 u C

1uu

(3.6)

 2xu D 0;

(3.7)

22 v C 2vv  2xv D 0;

(3.8)

.1 C 2 /v C

21uv

 2xv D 0;

(3.9)

.1 C 2 /u C 22uv  2xu D 0; .1 

2 /1v

C

21xv

(3.10) 1

C 2u.a1 C b1 u C c1 v/v  21 v  D 0;

(3.11)

.2  1 /2u C 22xu C 2v.a2 C b2 u C c2 v/u  22 u 2 D 0;   1 2u 1  t  v 2  2x C 2 v 2  3u u.a1 C b1 u C c1 v/

(3.12)

 v v.a2 C b2 u C c2 v/  21xu C xx D 0;  2 2v 2  t  u 1  2x C 1 u 1  3v v.a2 C b2 u C c2 v/

(3.13)



 u u.a1 C b1 u C c1 v/  22xv C xx D 0; (3.14)  1  2 1 1 2 1 1 t C  v C 2x   2  v  1 .a1 C 2b1 u C c1 v/  c1 2 u C u.a1 C b1 u C c1 v/.1u  2x / C 1v v.a2 C b2 u C c2 v/ D 1xx ; (3.15)  2  2 2 2 1 2 2 t C  u C 2x   1  u  2 .a2 C b2 u C 2c2 v/  b2 1 v C 2u u.a1 C b1 u C c1 v/ C v.a2 C b2 u C c2 v/.2v  2x / D 2xx :

(3.16)

It turns out that the functions 1 and 2 can be at maximum linear functions w.r.t. u and v. In fact, the differential consequences of (3.9) and (3.10) w.r.t. these variables lead to the expressions .1 C 2 /v2 D 0;

.1 C 2 /u2 D 0

so that u D v D 0. Having  D .t; x/, Eqs. (3.6)–(3.10) can be easily solved and one arrives at 1 D q1 .t; x/v C r1 .t; x/u C p1 .t; x/; 2 D q2 .t; x/u C r2 .t; x/v C p2 .t; x/;

(3.17)

hence, the most general form of operator (3.5) for system (3.2) is     Q D @t C @x C q1 v C r1 u C p1 @u C q2 u C r2 v C p2 @v ;

(3.18)

where the functions qk ; rk ; pk .k D 1; 2/ should be found from the remaining Eqs. (3.11)–(3.16).

82

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Substituting (3.17) and u D v D 0 into Eqs. (3.11)–(3.16), they can be split w.r.t. u; v; u2 ; v 2 ; uv. Finally, one obtains the system .c1  c2 /q1 D 0;

(3.19)

2

.b1  b2 /q D 0;

(3.20)

c1 q2 C b1 .r1 C 2x / D 0;

(3.21)

1

2

b2 q C c2 .r C 2x / D 0;

(3.22)

.2b1  b2 /q1 C c1 .r2 C 2x / D 0;

(3.23)

2

1

.2c2  c1 /q C b2 .r C 2x / D 0;

(3.24)

.1  2 /q1 C 2q1x D 0;

(3.25)

2

(3.26)

.2  1 /q C

2q2x

D 0;

1 .t C 2x / C 2rx1  xx D 0; 2 .t C 2x / C

2rx2

(3.27)

 xx D 0;

(3.28)

1 1 .rt1 C 2r1 x / C .1  2 /q1 q2  c1 p2  2b1 p1  2a1 x  rxx D 0; (3.29) 2 D 0; (3.30) 2 .rt2 C 2r2 x / C .2  1 /q1 q2  b2 p1  2c2 p2  2a2 x  rxx

1 .q1t C 2q1 x / C .1  2 /q1 r2  .a1  a2 /q1  c1 p1  q1xx D 0;

(3.31)

2 .q2t

(3.32)

2

2 1

2

2

C 2q x / C .2  1 /q r C .a1  a2 /q  b2 p 

q2xx

D 0;

1 . p1t C 2p1 x / C .1  2 /q1 p2  a1 p1  p1xx D 0;

(3.33)

2 . p2t

(3.34)

2

2 1

2

C 2p x / C .2  1 /q p  a2 p 

p2xx

D 0;

which does not involve the unknown functions u and v.

3.2.2 Q-Conditional Symmetry Analysis of the overdetermined system (3.19)–(3.34) shows that its solutions essentially depends on the relation between 1 and 2 . Theorem 3.2 ([7]) In the case 1 ¤ 2 , DLVS (3.2) is Q-conditionally invariant under operator (3.18) iff b1 D b2 D b; c1 D c2 D c. If bc D 0; b2 C c2 ¤ 0, then system (3.2) and the Q-conditional symme tries (up to local transformations u ! bu; v ! exp a22 t v; b ¤ 0 and

3.2 The Two-Component Diffusive Lotka–Volterra System

u ! exp



a1 1 t



83

v; cv ! u; c ¤ 0) have the forms 1 ut D uxx C u.a1 C u/; 2 vt D vxx C vu;

(3.35)

1 Q D @t C 12˛ @  2 x    C '.t/ exp.˛1 x/u C exp.˛1 x/ 2 ' 0 .t/ C a1 '.t/  ˛12 '.t/ C ˛2 v @v ; (3.36)

where the function '.t/ ¤ 0 is the general solution of the linear ODE 22 ' 00 C 2 .a1  2˛12 /' 0 C ˛12 .˛12  a1 /' D 0:

(3.37)

If bc ¤ 0 and the additional restrictions q1x D q2x D 0

(3.38)

hold, then exactly three cases (up to local transformations u ! bu; v ! cv and u ! v; v ! u) exist when system (3.2) admits Q-conditional symmetry operators. They are listed as follows: (i)

1 ut D uxx C u.a1 C u C v/; 2 vt D vxx C v.a2 C u C v/; a1 ¤ a2 ;

(3.39)

Q1 D .1  2 /@t  .a1 v C a2 u C a1 a2 /.@u  @v /; a1 a2 ¤ 0;

(3.40)

Q2 D .1  2 /@t C .a1  a2 /u.@u  @v /;

(3.41)

Q3 D .1  2 /@t  .a1  a2 /v.@u  @v /I

(3.42)

(ii)

1 ut D uxx C u.a C u C v/; 2 vt D vxx C v.a C u C v/;

(3.43)

Q1 D .1  2 /@t  a.v C u C a/.@u  @v /; a ¤ 0;

(3.44)

Q2 D .1  2 /t@t  .1 v C 2 u/.@u  @v /I

(3.45)

(iii)

1 ut D uxx C u.a1 C u C v/; 2 vt D vxx C v.a2 C u C v/;

a ¤ 0;

(3.46)

84

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Q4 D .e

Q1 D .1  2 /@t  a.1 v C 2 u C a1 2 /.@u  @v /;

(3.47)

Q2 D @t C au.@u  @v /;

(3.48)

Q3 D @t  av.@u  @v /;

(3.49)

at

 ˛.1  2 //@t C a˛.1 v C 2 u C a1 2 /.@u  @v /; ˛ ¤ 0: (3.50)

Proof Using Eqs. (3.19)–(3.20) from the system of DEs, one notes that three cases can take place: (a) b1 6D b2 and/or c1 6D c2 I (b) b1 D b2 D b 6D 0; c1 D c2 D 0I (c) b1 D b2 D b 6D 0; c1 D c2 D c 6D 0: Note that the fourth possible case b1 D b2 D 0; c1 D c2 D c ¤ 0 is reduced to the second case by renaming u ! v; v ! u. It turns out that case (a) produces the restrictions q1 D 0;

q2 D 0;

(3.51)

which lead only to Lie symmetry operators of system (3.2). Let us show this. If b1 ¤ b2 and c1 ¤ c2 , then restrictions (3.51) immediately follow from (3.19)– (3.20). If b1 D b2 D b and c1 ¤ c2 , then q1 D 0 follows from (3.19); furthermore, Eqs. (3.21) and (3.24) lead to .c2 c1 /q2 D 0 , q2 D 0 (the subcase b1 ¤ b2 ; c1 D c2 D c leads to the same result). Having restrictions (3.51), we immediately obtain c1 p1 D 0; b2 p2 D 0 from (3.31) and (3.32). Obviously, if c1 b2 ¤ 0, then p1 D p2 D 0:

(3.52)

If c1 b2 D 0, say, c1 D 0; b2 ¤ 0 (subcase c1 ¤ 0; b2 D 0 can be treated in the same way) then p2 D 0. Simultaneously system (3.2) is reduced to one with an autonomous equation, which is nothing else but the Fisher equation 1 ut D uxx C u.a1 C b1 u/:

(3.53)

Now we assume that such a system is Q-conditionally invariant under an operator of the form (3.18). However, the Fisher equation does not admit any Q-conditional symmetry [3, 14, 33] but only Lie symmetry Q D ˛1 @t C ˛2 @x

(3.54)

3.2 The Two-Component Diffusive Lotka–Volterra System

85

if a1 is an arbitrary constant, and Q D .2˛0 t C ˛1 /@t C .˛0 x C ˛2 /@x  2˛0 u@u (here ˛i are arbitrary constants, i D 0; 1; 2) if a1 D 0. So, p1 D 0 and the restrictions (3.52) are again obtained. Restrictions (3.51) and (3.52) essentially simplify the system of DEs (3.19)– (3.34), which takes the form b1 .r1 C 2x / D 0;

(3.55)

b2 .r1 C 2x / D 0;

(3.56)

2

c1 .r C 2x / D 0;

(3.57)

c2 .r2 C 2x / D 0;

(3.58)

1 .t C 2x / C

2rx1

 xx D 0;

(3.59)

2 .t C 2x / C 2rx2  xx D 0;

(3.60)

1 .rt1

1

C 2r x /  2a1 x 

1 rxx

D 0;

(3.61)

2 2 .rt2 C 2r2 x /  2a2 x  rxx D 0:

(3.62)

Now one may easily check that any solution of system (3.55)–(3.62) leads to a Qconditional symmetry operator, which will be equivalent to the corresponding Lie symmetry operator listed in Table 3.1. Consider, for example, the most general case b2 c1 ¤ 0. Obviously, (3.56) and (3.57) with b2 c1 ¤ 0 lead to r1 D r2 D 2x :

(3.63)

So, having (3.63) and 1 ¤ 2 ; we obtain the system t C 2x D 0; xx D 0;

(3.64)

from (3.59), (3.60) and (3.63). Substituting the general solution of (3.64) .t; x/ D

˛0 x C ˛2 2˛0 t C ˛1

into (3.63), we can solve Eqs. (3.61)–(3.62). Finally, the operator @t C

˛0 x C ˛2 2˛0 @x  .u@u C v@v / 2˛0 t C ˛1 2˛0 t C ˛1

(3.65)

is obtained if a1 D a2 D 0. However, operator (3.65) is nothing else but a linear combination of Lie symmetry operators of (3.2) (see Table 3.1, case 1) multiplied by 2˛0 t C ˛1 . If a21 C a22 ¤ 0, then operator (3.54) occurs, which is again the Lie symmetry operator. Thus, case (a) is completely examined.

86

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Consider case (b).  system (3.2) can be reduced to (3.35) by the substitution  Here u ! bu; v ! exp a22 t v; because c1 D c2 D 0. Since the first equation of (3.35) is the Fisher equation (3.53) the same approach can be used as above. Thus, using operator (3.54), one can obtain the restrictions q1 D p1 D r1 D 0 and operator (3.18) takes the form Q D @t C ˛@x C .q2 u C r2 v C p2 /@v ; ˛ D const: Simultaneously, the system of DEs (3.19)–(3.34) reduces to rx2 D rt2 D 0; .2  1 /˛ q2 C 2q2x D 0;

2 p2t  p2xx D 0; 2 q2t C a1 q2  p2  q2xx D 0:

The general solution of this linear system can be straightforwardly constructed and it reads as follows r2 D ˛2 ;

q2 D '.t/ exp.˛1 x/;

  p2 D exp.˛1 x/ 2 ' 0 .t/ C a1 '.t/  ˛12 '.t/ ;

where ˛1 D 12 ˛.1  2 / and ˛2 are arbitrary constants, while the function '.t/ is the general solution of the ODE (3.37). Thus, system (3.35) and the Q-conditional symmetry (3.36) are found. Consider case (c). Here the DLVS (3.2) is reduced to system (3.39), i.e., (3.2) with b D c D 1, by the substitution u ! bu; v ! cv. Now we take into account the restrictions q1x D q2x D 0; hence, the system of DEs reads as follows .1  2 /q1 D 0; .2  1 /q2 D 0; 1

2

2

(3.66)

1

q C r C 2x D 0; q C r C 2x D 0;

(3.67)

1 .t C 2x / C 2rx1  xx D 0; 2 .t C 2x / C 2rx2  xx D 0;

(3.68)

1 .rt1

D 0;

(3.69)

2 2 .rt2 C 2r2 x / C .2  1 /q1 q2  p1  2p2  2a2 x  rxx D 0;

(3.70)

1

1 2

2

1

C 2r x / C .1  2 /q q  p  2p  2a1 x 

1 .q1t

1

1 2

1

1 rxx

1

C 2q x / C .1  2 /q r  .a1  a2 /q  p D 0;

(3.71)

2 .q2t C 2q2 x / C .2  1 /q2 r1 C .a1  a2 /q2  p2 D 0;

(3.72)

1 . p1t

1

1 2

1

C 2p x / C .1  2 /q p  a1 p 

p1xx

D 0;

(3.73)

2 . p2t C 2p2 x / C .2  1 /q2 p1  a2 p2  p2xx D 0:

(3.74)

If q1 D q2 D 0, then we again obtain Lie symmetry operators only (see case (a)). So, nontrivial results are obtainable only under the restriction .q1 /2 C .q2 /2 ¤ 0. Equations (3.66) under this restrictions produce  D 0, hence, we obtain q1 D r2 D  .t/; q2 D r1 D '.t/

(3.75)

3.2 The Two-Component Diffusive Lotka–Volterra System

87

from (3.67) and (3.68) (here '.t/ and .t/ are arbitrary smooth functions at the moment). Substituting (3.75) into (3.71)–(3.72), we find p1 D .a1  a2 / .t/ C .2  1 / 2 .t/  1 0 .t/; p2 D .a2  a1 /'.t/ C .1  2 /' 2 .t/  2 ' 0 .t/:

(3.76)

Having (3.75) and (3.76), Eqs. (3.69)–(3.70) can be rewritten as ODEs for the functions '.t/ and .t/: 1 2 /.'C '.t/ P D  a2 a1 C. .1 2 /2

P .t/ D

/

..31  2 / ' C 22 / ;

(3.77)

a2 a1 C.1 2 /.'C / .1 2 /2

.21 ' C .32  1 / / :

(3.78)

Finally, using formulae (3.75)–(3.78), the last two Eqs. (3.73)–(3.74) can be rewritten as two algebraic equations for finding '.t/ and .t/. The difference of those leads to the classification equation  .1 ' C 2 / a1  a2  .1  2 /.' C

/



a1 .41 C 52 /

 a2 .51 C 42 /  4.1  2 /..21 C 2 /' C .1 C 22 / / D 0: (3.79) Thus, three subcases follow from (3.79): a1  a2 I 1  2

(c1)

'D

C

(c2)

'D

2 I 1

(c3)

' D ..4a1  5a2 /1 C .5a1 C 4a2 /2 4.1  2 /.1 C 22 / / =4.1  2 /.21 C 2 /:

In subcase (c1), Eqs. (3.73) and (3.74) are equivalent to the equation .a1  .1  2 / /

.a1  a2  .1  2 / / D 0:

1 2 If D 1a ; then ' D  1a and using (3.75) and (3.76) we arrive at 2 2 operator (3.40). 2 If D 0, then ' D a11 a 2 , so that operator (3.41) is obtained (the restriction a1 ¤ a2 guarantees that it is not a Lie symmetry operator). 2 If D a11 a 2 , then ' D 0, hence operator (3.42) and the same restriction a1 ¤ a2 are obtained. Thus, the proof of item (i) is complete. In subcase (c2), Eqs. (3.73) and (3.74) are equivalent to the equation

.t/.a1  a2 /.a1 2  a2 1 / D 0:

88

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Since .t/ ¤ 0 (otherwise one arrives at the Lie symmetry operator Q D @t ) two possibilities occur: a1 D a2 and a1 D 12 a2 . If a1 D a2 , then we obtain .t/ D 1 .˛ C .1  2 /t/1 ; ˛ D const from (3.78). Note that we set ˛ D 0 without loss of generality. Now the functions p1 ; p2 ; can be found from (3.76), hence, we obtain operator (3.45). Operator (3.44) follows from (3.40) if one sets a1 D a2 D a. Thus, the proof of item (ii) is complete. If a1 D 12 a2 (here a2 6D 0 otherwise item (ii) is obtained), then Eq. (3.78) produces   1 a2 .t/ D ˛0 a2 1 ˛0 .1  2 /2  exp  t ; 2 where ˛0 is a nonvanishing constant. Thus, using notations a2 D a2 ; 2 ˛0 D ˛ and Eq. (3.76), we obtain the most complicated operator (3.50). Finally, operators (3.47), (3.48) and (3.49) are nothing else but (3.40), (3.41) and (3.42) with a1 D a1 ; a2 D a2 , respectively. Thus, system (3.46) is obtained and all operators arising in item (iii) are constructed. It turns out that the detailed analysis of subcase (c3) does not lead to any new operators. The proof is now complete. t u Theorem 3.3 ([7]) In the case 1 D 2 , DLVS (3.2) admits only such operators of the form (3.18), which are equivalent to the Lie symmetry operators. The proof of the theorem is rather simple because Eqs. (3.25)–(3.26) immediately produce restrictions (3.38) and the next steps are the standard routine. Finally, we note that the solution of (3.19)–(3.34) without restrictions (3.38) is still unknown. This means that one cannot claim that all possible Q-conditional symmetries (3.5) of DLVS (3.2) are known. Now we turn to finding Q-conditional symmetries of the first type. In order to find such symmetries one should apply Definition 2.2 to DLVS (3.2). The system of DEs obtained is simpler than (3.19)–(3.34) and can be integrated without any restrictions (see details in [7]). Here we present only the final result. Theorem 3.4 In the case 1 ¤ 2 , DLVS (3.2) is invariant under Q-conditional operators of the first type only in two cases: b1 D b2 D b; c1 D c2 D c; bc D 0; b2 C c2 ¤ 0 and b1 D b2 D b; c1 D c2 D c; bc ¤ 0. The corresponding systems and Q-conditional symmetries (up to the point transformations bu ! u; v !  a2 exp 2 t v; c D 0I u ! exp a11 t v; cv ! u; b D 0I bu ! u; cv ! v; bc ¤ 0)

3.2 The Two-Component Diffusive Lotka–Volterra System

89

have the forms 1 ut D uxx C u.a1 C u/; 2 vt D vxx C vu; Q D @t C 12˛1 2 @x C .'.t/ exp.˛1 x/u    C exp.˛1 x/ 2 ' 0 .t/ C a1 '.t/  ˛12 '.t/ C ˛2 v @v ;

(3.80) (3.81)

where the function '.t/ ¤ 0 is the general solution of the linear ODE (3.38). 1 ut D uxx C u.a1 C u C v/; 2 vt D vxx C v.a2 C u C v/;

a1 ¤ a2 ;

(3.82)

Q1 D .1  2 /@t C .a1  a2 /u.@u  @v /;

(3.83)

Q2 D .1  2 /@t  .a1  a2 /v.@u  @v /:

(3.84)

There are no other Q-conditional operators of the first type. In the case 1 D 2 , DLVS (3.2) is invariant only under such Q-conditional operators of the first type, which coincide with Lie symmetry operators. It can be noted that Theorem 3.4 presents a particular result of Theorem 3.2 (formulae (3.80)–(3.84) coincide with the relevant formulae in Theorem 3.2), and this is in agreement with the general theory (see Sect. 2.2). However, the theorem gives an exhaustive list of Q-conditional symmetries of the first type in explicit form and this is important from the mathematical point of view.

3.2.3 Reductions to Systems of Ordinary Differential Equations and Exact Solutions First of all, we note that DLVS (3.2) is invariant under time and space translations, hence, its arbitrary solution u0 .t; x/; v0 .t; x/ generates a two-parameter family of solutions of the form u.t  t0 ; x  x0 /; v.t  t0 ; x  x0 /. Having this in mind, we always apply these translations in order to get t0 D x0 D 0 in the solutions obtained below. Now we look for the plane wave solutions of DLVS (3.2) in the case when the system models the competition between two populations. Introducing new notation k ! 1=k ; ak ! ak =k ; bk ! bk =k , ck ! ck =k we rewrite (3.2) in the typical form ut D 1 uxx C u.a1  b1 u  c1 v/; vt D 2 vxx C v.a2  b2 u  c2 v/;

(3.85)

90

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

where all the coefficients are positive. Using ansatz (3.4), the competition model (3.85) is reduced to ODEs 1 '! ! C ˛'! C '.a1  b1 '  c1 / D 0; 2 ! ! C ˛ ! C '.a2  b2 '  c2 / D 0:

(3.86)

Since it is impossible to construct the general solution of this nonlinear ODE system with arbitrary coefficients, we consider a special case. Let us assume that the following condition is satisfied: D ˇ0 C ˇ1 ';

(3.87)

where ˇ0 and ˇ1 are certain constants to be found below. Substituting (3.87) into (3.86), we arrive at an overdetermined system, which possesses nonconstant solutions only under the condition 1 D 2 D . So, without loss of generality we set  D 1 below. As a result, the single second-order ODE '! ! C ˛'! C '.a  b'/ D 0

(3.88)

is obtained. Here the parameters a and b depend essentially on the additional parameter ˇ0 , namely: ( aD

a1 D a2 ; ˇ0 D 0; a1  a2 cc12 ; ˇ0 D ac22 ; ( ˇ1 D

b1 b2 c2 c1 ;  aa21bc11 ;

( bD

c1 b2 b1 c2 ; c1 c2

ˇ0 D 0; b1 C c1 ˇ1 ; ˇ0 D ac22 ;

c1 6D c2 ; b1 6D b2 ; c1 D c2 ; b 1 D b 2 :

(3.89)

(3.90)

ODE (3.88) is well known because it is the reduced equation of the Fisher equation (see Chap. 1 for details). In particular, ODE (3.88) possesses the exact solution [1] '.!/ D

  r 2 a a 1 C c exp ˙ ! ; b 6

(3.91)

p where ˛ D p5 a and c is an arbitrary constant. 6 If c > 0, then taking into account formulae (3.87) and (3.4) and fixing the upper sign in (3.91), the solution of DLVS (3.85)  r 2 a 5a a uD 1  tanh x t ; 4b 24 12 v D ˇ0 C ˇ1 u

(3.92)

is obtained. Here the coefficients a, b, ˇ0 and ˇ1 are defined by (3.89) and (3.90).

3.2 The Two-Component Diffusive Lotka–Volterra System

91

It turns out that solution (3.92), which is a typical travelling front, has essentially different properties depending on the value of the constant ˇ0 , therefore one simulates different types of population interaction (see below). If c < 0, then solution (3.91) generates the following solution of DLVS (3.85):  r 2 a 5a a uD 1  coth x t ; 4b 24 12 v D ˇ0 C ˇ1 u: In to (3.92), this solution blows up at all points .t; x/ of the plane p acontrast 5a x  t D 0. We assume that such a solution may describe a pathologic type of 24 12 interaction when both populations grow unboundedly. Now we apply solution (3.92) for solving the Neumann boundary value problem (BVP) for the nonlinear DLVS (3.85). Theorem 3.5 ([10]) A bounded exact solution of the nonlinear BVP, which consists of DLVS (3.85) (with 1 D 2 D 1), the initial conditions  r 2 a a 1  tanh x uD  u0 .x/; 4b 24 v D ˇ0 C ˇ1 u0 .x/

(3.93)

and the Neumann conditions at infinity ux .t; 1/ D ux .t; C1/ D vx .t; 1/ D vx .t; C1/ D 0

(3.94)

in the domain ˝ D f.t; x/ 2 Œ0; C1/  .1; C1/g has the form (3.92). In formulae (3.93) and (3.92), the coefficients a, b, ˇ0 and ˇ1 are defined by (3.89) and (3.90). The proof is reduced to the verification that the boundary conditions (3.94) and the initial conditions (3.93) are satisfied by the exact solution (3.92). In order to provide some biological interpretation of this theorem, we note that two essentially different cases occur, namely: ˇ0 6D 0 and ˇ0 D 0. If ˇ0 6D 0, then solution (3.92) possesses the asymptotical behaviour  .u; v/ !

a1 ;0 b1

 as t ! 1;

(3.95)

provided the following condition is satisfied: A > maxfB; Cg;

(3.96)

where A D aa12 ; B D bb12 ; C D cc12 (note that the condition A.B1/ D B.C1/ follows from (3.92) and (3.95)). In population dynamics, this means the uncompromising

92

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

competition between two populations of species u and v. An increase in population u leads to a decrease in species v. Eventually, the complete disappearance of species v takes place. In the case of the opposite condition A < minfB; Cg

(3.97)

the competition, in fact, has the same character. In this case, species u eventually dies, while species v dominates. Note that there are no other relations between A; B; and C (except (3.96) and (3.97)) provided ˇ0 6D 0. If ˇ0 D 0 (in this case, the restriction a1 D a2 D a follows from (3.89)), then solution (3.92) possesses the property  .u; v/ !

 a.C  1/ a.1  B/ ; ; b2 .C  B/ c2 .C  B/

t ! 1:

(3.98)

2 The restriction ˇ1 D bc12 b > 0 must also be satisfied (see (3.90)), which guarantees c1 that solution (3.92) is nonnegative. In terms of A; B and C; formula (3.98) implies either the relation

B>AD1>C

(3.99)

C > A D 1 > B:

(3.100)

or the relation

Solution (3.92) with property (3.98) describes the case of a ‘soft’ competition between two populations, which allows an arbitrarily long (in time) coexistence of species u and v. It should be stressed that theorems on the existence of solutions of the Neumann problem for DLVS (3.85) with properties (3.95) and (3.98) have been known for a long time (see, e.g., [25] and the bibliography therein); in particular, it has been established that relations (3.96), (3.97), (3.99) and (3.100) play a key role. However, to the best of our knowledge, there are no papers presenting such solutions explicitly except for a few studies [10, 16, 21, 31] devoted to the construction of appropriate plane wave solutions. In particular, paper [16] is devoted to a special case of the Belousov–Zhabotinskii system, namely DLVS (3.85) with a2 D c2 D 0. The exact solution constructed in [16] can easily be obtained from (3.92) (for details see [10]). The main part of this subsection is devoted to the construction of exact solutions with more complicated structures than the plane wave solutions found above. It is well known that by using Q-conditional symmetries one can reduce the given two-dimensional PDE (system of PDEs) to an ODE (system of ODEs) via the same procedure as for classical Lie symmetries. Thus, to construct an ansatz corresponding to the operator Q, the system of the linear (quasi-linear) first-order PDEs Q.u/ D 0; Q.v/ D 0

(3.101)

3.2 The Two-Component Diffusive Lotka–Volterra System

93

should be solved. Substituting the ansatz obtained into the DLVS with correctly specified coefficients, one obtains a system of ODEs, i.e., the reduced system of equations. Solving the ODE system by applying the standard techniques and inserting its solution(s) into the ansatz, we immediately obtain exact solutions for the DLVS in question. Theorem 3.2 gives several possibilities for finding exact solutions of DLVS with the correctly specified coefficients. Here we apply Q-conditional symmetry operators (3.40) and (3.44) in order to find exact solutions of systems (3.39) and (3.43). Of course, all other operators can be applied in a quite similar fashion. In the case of operator (3.40), system (3.101) takes the form .1  2 /ut D .a1 v C a2 u C a1 a2 /; .1  2 /vt D a1 v C a2 u C a1 a2 :

(3.102)

In order to solve (3.102) we immediately note that ut D vt , hence, u.t; x/ D v.t; x/ C '1 .x/:

(3.103)

Substituting (3.103) into the second equation of (3.102), we arrive at the linear equation .1  2 /vt D .a1  a2 /v C a2 '1 .x/ C a1 a2 : If a1 ¤ a2 ; then this equation has the general solution vD

1 a1  a2

    a1  a2 exp t '2 .x/  a2 '1 .x/  a1 a2 ; 1  2

therefore the ansatz uD vD

1 a1 a2 1 a1 a2

    2  exp a11 a ; t ' .x/ C a ' .x/ C a a 2 1 1 1 2  2    a1 a2 exp 1 2 t '2 .x/  a2 '1 .x/  a1 a2

(3.104)

is obtained. Here '1 and '2 are functions to be found. If a1 D a2 D a; then operator (3.43) is applied, therefore the ansatz a u D '1 .x/  '2 .x/  1  .'1 .x/ C a/ t; 2 a v D '2 .x/ C 1 2 .'1 .x/ C a/ t

(3.105)

is obtained. In order to construct the reduced system, we substitute ansatz (3.104) into (3.39). This means that we simply calculate the derivatives ut ; vt ; uxx ; vxx ; and insert them

94

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

into (3.39). After the relevant simplifications one arrives at the ODE system '100 C '12 C .a1 C a2 /'1 C a1 a2 D 0; 1 2 '200 C a2 11 a 2 '2 C '1 '2 D 0

(3.106)

to find the functions '1 and '2 . Similarly, ansatz (3.105) leads to the reduced system '100 C .a C '1 /2 D 0;   '200 C '2  1a2 2 .a C '1 / D 0: Now we solve the ODE systems obtained above in order to construct exact solutions in an explicit form. Let us consider system (3.106). Since the general solution of this nonlinear ODE system cannot be found in an explicit form, we look for particular solutions. Setting '1 D ˛ D const; we find ˛ 2 C .a1 C a2 /˛ C a1 a2 D 0 ) ˛1 D a1 ; ˛2 D a2 from the first equation of system (3.106). Now we take '1 D a1 (the case '1 D a2 leads to the solution with the same structure) and substitute into the second equation of system (3.106): '200  ˇ1 '2 D 0;

(3.107)

2 6D 0. Depending on the sign of the parameter ˇ the linear ODE where ˇ D a11 a 2 (3.107) generates two families of general solutions. Using those solutions and ansatz (3.104), we obtain the following families of exact solutions of DLVS (3.39):

 p   p  1 C u D a1 C a2 a exp ˇ x C C exp  ˇ x eˇt ; 1 1 2 1 1  p   p  1 v D a1 a C1 exp ˇ1 x C C2 exp  ˇ1 x eˇt ; 2

(3.108)

if ˇ > 0; and  p  p  1 u D a1 C a2 a C cos ˇ x C C sin ˇ x eˇt ; 1 1 2 1 1   p   p 1 v D a1 a ˇ1 x C C2 sin ˇ1 x eˇt ; C1 cos 2

(3.109)

if ˇ < 0 (hereafter C1 and C2 are arbitrary constants). Let us construct solutions of (3.106) with some restrictions on 1 and 2 . Firstly, we note that the substitution '1 D '  a 1

(3.110)

3.2 The Two-Component Diffusive Lotka–Volterra System

95

simplifies the first equation of (3.106) to the form ' 00 C ' 2 C .a2  a1 /' D 0:

(3.111)

Of course, (3.111) can be reduced to the first-order ODE 

d' dx

2

2 D  ' 3 C .a1  a2 /' 2 C C 3

with the general solution involving the Weierstrass function [4]. To avoid cumbersome formulae, we set C D 0, hence, the general solution is   p a1  a2 3.a1  a2 / 1  tanh2 x ; 2 2

(3.112)

  p a2  a1 3.a1  a2 / 2 'D 1 C tan x ; 2 2

(3.113)

'D if a1 > a2 , and

if a1 < a2 . Thus, we can apply formulae (3.112) and (3.113) to solve the second ODE of (3.106). In the case of solution (3.112), this ODE takes the form '200 C '2 .a1  a2 /



3 1  32  tanh2 2.1  2 / 2

 p a1  a2 x D 0: 2

(3.114)

The general solution of (3.114) with some restrictions on 1 and 2 can be found [30]:  Z '2 D f1 .x/ C1 C C2

 1 dx ; f12 .x/

(3.115)

 Z '2 D f2 .x/ C1 C C2

 1 dx ; f22 .x/

(3.116)

if 1 D 95 2 , and

if 1 D 43 2 , where f1 .x/ D cosh3

p

a1 a2 2

 x ;

f2 .x/ D sinh

p

a1 a2 2

p   a1 a2 x cosh3 x : 2

96

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Thus, substituting the functions '1 .x/ and '2 .x/ given by formulae (3.110), (3.112) and (3.115) into ansatz (3.104), we find the exact solution uD

a1 2



3a1 2 2tanh

p

a1 a2 2

Z

 x



  1 a2 / C dx exp 5.a4 t ; f1 .x/ 2  p  a a 1 2 x v D  3a22 1  tanh2   Z 2   1 a2 / t Cf1 .x/ C1 C C2 f 21.x/ dx exp 5.a4 2 C1

C2

1 f12 .x/

(3.117)

1

1 Ci are still arbitrary constants) of DLVS (3.39) with 1 D (hereafter Ci D a1 a 2 9  and a > a . 1 2 5 2 Similarly, solution (3.116) leads to the exact solution

uD

a1 2



3a1 tanh2 2

p

a1 a2 2

 x

 Z   2/ f2 .x/ C1 C C2 f 21.x/ dx exp 3.a1a t ; 2  p 2  a1 a2 v D  3a22 1  tanh2 x   Z 2   1   2/ t Cf2 .x/ C1 C C2 f 2 .x/ dx exp 3.a1a 2

(3.118)

2

of DLVS (3.39) with 1 D 43 2 and a1 > a2 . In a quite similar way solution (3.113) can also be used to construct new solutions of DLVS (3.39). Omitting straightforward calculations we present only the result: uD

a1 2

C

3a1 2 2 tan

p

a2 a1 2

 x

 Z   1 a2 / t ; g1 .x/ C1 C C2 g21.x/ dx exp 5.a4 2   p 1  a2 a1 v D  3a22 1 C tan2 x   Z2   1   1 a2 / t ; Cg1 .x/ C1 C C2 g2 .x/ dx exp 5.a4 2

(3.119)

1

if a1 < a2 and 1 D 95 2 I uD

a1 2

C

3a1 2 2 tan

p

a2 a1 2

Z

 x



  2/ C dx exp 3.a1a t ; g2 .x/ 2   a a 2 1 v D  3a22 1 C tan2 x   Z2   2/ t ; Cg1 .x/ C1 C C2 g21.x/ dx exp 3.a1a 2 C1

C2 p

1 g22 .x/

2

(3.120)

3.2 The Two-Component Diffusive Lotka–Volterra System

97

if a1 < a2 and 1 D 43 2 ; where g1 .x/ D cos3

p

a2 a1 2

 x ;

g2 .x/ D sin

p

a2 a1 2

p   a2 a1 x cos3 x : 2

It should be noted that all the solutions obtained above cannot be constructed using Lie symmetries. In fact DLVS (3.39) does not admit any nontrivial Lie symmetry, hence, plane wave solutions of the form (3.4) can only be found by Lie symmetry reductions. Now we present an example that demonstrates remarkable properties of some solutions presented above. Consider solution (3.109) with C1 D 0. Using the substitution u D bU; v D cV .b > 0; c > 0/; one transforms DLVS (3.39) to the system describing the competition of two species 1 Ut D Uxx C U.a1  bU  cV/; 2 Vt D Vxx C V.a2  bU  cV/

(3.121)

and solution (3.109) to the form a1 b

V.t; x/ D

1 .a2 a1 /c

C

p  C2 sin ˇ1 x eˇt ; p  C2 sin ˇ1 x eˇt ;

1 .a1 a2 /b

U.t; x/ D

(3.122)

2 where the coefficient restrictions ˇ  a11 a 2 < 0; a1 > 0; a2 > 0 are assumed. Using this solution one may formulate the following theorem giving the classical solution of a nonlinear BVP with the constant Dirichlet conditions on the boundaries.

Theorem 3.6 The classical solution of the nonlinear BVP formed by the competition system (3.121), the initial profile U.0; x/ D

a1 b

V.0; x/ D

C

p  C2 sin ˇ1 x ; p  C2 sin ˇ1 x

1 .a1 a2 /b

1 .a2 a1 /c

and the boundary conditions a1 ; V D 0; b

a1 xD p W UD ; VD0 b ˇ1 xD0W UD

n  o

in the domain ˝ D .t; x/ 2 .0; C1/  0; pˇ is given by formulae (3.122). 1

98

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Fig. 3.1 Solution (3.122) of system (3.121) with a1 D 5; a2 D 2; 1 D 1; 2 D 3; b D 0:9; c D 0:05; C2 D 0:5; ˇ D 1:5

The solution (3.122) with ˇ < 0 has the time asymptotic .U; V/ !

a

1

b

 ; 0 ;

t ! C1:

Thus, this solution describes the competition between two species when species U eventually dominates while species V dies. An example of this competition with the correctly specified coefficients is presented in Fig. 3.1.

3.3 The Three-Component Diffusive Lotka–Volterra System Here we study the three-component DLVS 1 ut D uxx C u.a1 C b1 u C c1 v C e1 w/; 2 vt D vxx C v.a2 C b2 u C c2 v C e2 w/;

(3.123)

3 wt D wxx C w.a3 C b3 u C c3 v C e3 w/; where u.t; x/; v.t; x/; w.t; x/ are unknown concentrations, ak ; bk ; ck ; ek and k > 0 are arbitrary constants (hereafter k D 1; 2; 3). Note that we want to exclude the semicoupled systems, i.e., those containing an autonomous equation, hence, hereafter the restrictions c21 C e21 ¤ 0; b22 C e22 ¤ 0; b23 C c23 ¤ 0

(3.124)

3.3 The Three-Component Diffusive Lotka–Volterra System

99

are assumed. Notably, system (3.123) describes much more complicated interactions between three populations than the two-component system (2.2). For example, it can be the predator–prey–competition interaction, if one sets ak > 0; bk  0; ck  0; ek < 0; k D 1; 2; a3  0; c3 > 0; e3 > 0: Such interaction assumes that there are two competing species u and v, while the species w are predators eating the species u and v. Section 3.3.1 is devoted to the search for Lie symmetry operators in order to distinguish Lie and conditional symmetry of this nonlinear system. Section 3.3.2 is devoted to finding the system of DEs in order to find Q-conditional symmetries of the first type. Because DLVS (3.123) belongs to the general class of threecomponent RDSs 1 ut D uxx C C1 .u; v; w/; 2 vt D vxx C C2 .u; v; w/;

(3.125)

3 wt D wxx C C3 .u; v; w/; where Ck .u; v; w/ are arbitrary smooth functions, we obtain the system of DEs directly for the class of PDEs (3.125). In Sect. 3.3.3, all possible Q-conditional symmetries of the first type are constructed via solving the system of DEs. Finally, examples of reductions of DLVS (3.123) to ODE systems via conditional symmetry operators are presented. Moreover, particular exact solutions of DLVS (3.123) are derived and their biological interpretation is proposed.

3.3.1 Lie Symmetry In contrast to the two-component DLVS, the Lie symmetry of the three-component DLVS was found very recently in [8]. Obviously DLVS (3.123) admits the Lie algebra with the basic operators: Pt D @t ; Px D @x :

(3.126)

It can be easily shown that (3.126) is the principal (trivial) algebra of DLVS (3.123), i.e., this is the maximal invariance algebra of this system with arbitrary coefficients ak ; bk ; ck ; ek and k > 0. In order to find all possible extensions of the principal algebra one needs to apply the invariance criterion (2.7), to solve the obtained system of DEs and find all possible values of these coefficients leading to extensions of the principal algebra (3.126). Because all the coefficients of DLVS (3.123) are constants (otherwise the symmetry classification problem occurs, which will be discussed in Chap. 4 in the case of two-component systems), this problem can be solved by standard calculations, hence, we present the result only.

100

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Table 3.2 Lie symmetry operators of DLVS (3.123)

1

2

3

4

5

6

7

8

Reaction terms u.b1 u C c1 v C e1 w/ v.b2 u C c2 v C e2 w/ w.b3 u C c3 v C e3 w/ u.c1 v C e1 w/ v.a2 C c2 v C w/ w.a3 C v C e3 w/ u.c1 v C e1 w/ v.c2 v C w/ w.v C e3 w/ u.a1 C bu C v/ v.a2 C u C cv/ w.u C cv/ u.bu C v/ v.u C cv/ w.u C cv/ u.a1 C u C v/ v.a2 C u C v/ w.u C v/ u.a C u C v/ v.u C v/ w.u C v/ u.bu C v/ v.u C cv/ w.bu C cv/

Restrictions

Lie symmetries extending algebra (3.126) D D 2t@t Cx@x 2.u@u Cv@v Cw@w /

u@u

u@u ; D 2 D 3 D 1

exp.a2 t/v@w ; w@w

2 D 3 D 1

v@w ; w@w ; D

1 D 2 D 3 D 1; a1 a2 .a1  a2 / ¤ 0

exp.a1 t/u@w ; w@w ; exp.a2 t/v@w ; .a2 .u C a1 / C a1 v/@w

1 D 2 D 3 D 1; a¤0

exp.at/u@w ; w@w ; v@w ; .u C a C avt/@w

1 D 2 D 3 D 1; .b  1/2 C .c  1/2 ¤ 0

w@w ; ..b  1/u C .1  c/v/ @w ; D

Theorem 3.7 ([8]) DLVS (3.123) admits a nontrivial Lie algebra of symmetries iff it and the corresponding symmetry operators have the forms listed in Table 3.2. Any other DLVS admitting three- and higher-order Lie algebra is reduced to one of those from Table 3.2 by the local transformations: u ! c11 exp.c10 t/u C c12 v C c13 w; v ! c21 exp.c20 t/v C c22 u C c23 w; w ! c31 exp.c30 t/w C c32 u C c33 v; t ! c40 t C c41 ; x ! c50 x C c51 ;

(3.127)

where cij (i D 1; : : : ; 5, j D 0; : : : ; 3) are correctly specified constants (many of them vanish), which are defined by the DLVS in question. It can be seen from Table 3.2 that DLVS (3.2) admits a nontrivial Lie symmetry provided at least three coefficients vanish. It is not plausible that such systems are important for real-world applications. Of course, some of them can be an approximation of relevant models, e.g., the DLVS with a1 D a2 D a3 D 0 (case 1) prescribes the zero natural birth/death rate for all species.

3.3 The Three-Component Diffusive Lotka–Volterra System

101

3.3.2 Determining Equations In order to find all possible Q-conditional symmetry of the first type for DLVS (3.123), one needs to apply Definition 2.2 and to derive the relevant system of DEs, which should be solved at the final step. Let us consider the operator of the general form Q D  0 .t; x; u; v; w/@t C  1 .t; x; u; v; w/@x C 1 .t; x; u; v; w/@u C2 .t; x; u; v; w/@v C 3 .t; x; u; v; w/@w :

(3.128)

This operator is Q-conditional symmetry of the first type of system (3.123) provided the following criterion is fulfilled ˇ ˇ  Q .S1 /ˇˇ  Q 1 ut  uxx  C1 .u; v; w/ ˇˇ D 0; M M1 2 2 ˇ 1   ˇˇ ˇ 2 Q .S2 /ˇ  Q 2 vt  vxx  C .u; v; w/ ˇ D 0; M M1 2 2 ˇ 1  ˇˇ  ˇ 3 Q .S3 /ˇ  Q 3 wt  wxx  C .u; v; w/ ˇ D 0; 2

M1

(3.129)

M1

2

where the manifold M1 is either M11 D fS1 D 0; S2 D 0; S3 D 0; Q.u/ D 0g, or M12 D fS1 D 0; S2 D 0; S3 D 0; Q.v/ D 0g, or M13 D fS1 D 0; S2 D 0; S3 D 0; Q.w/ D 0g. Omitting rather standard calculations we present the system of DEs corresponding to the manifold M11 : x0 D u0 D v0 D w0 D u1 D v1 D w1 D 0;

(3.130)

kuu D kuv D kvv D kww D kuw D kvw D 0; k D 1; 2; 3;

(3.131)

1xv

D

1xw

D

2xw

D

3xv

D 0;

(3.132)

.1  2 /1v D .2  3 /3v D 0;

(3.133)

.1  3 /1w D .2  3 /2w D 0;

(3.134)

2x1  t0 D 0;

(3.135)

2 0 2xu C .2  1 / 1 2u D 0;

(3.136)

2 0 3xu C .3  1 / 1 3u D 0;

(3.137)

21xu C 1 t1 D 0; 22xv C 2 t1 D 0; 23xw C 3 t1 D 0;

(3.138)

1 Cu1 C 2 Cv1 C 3 Cw1 C 1xx  1 1t C.2x1  1u /C1  1v C2  1w C3 D 0;

(3.139)

102

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems 1

1 Cu2 C 2 Cv2 C 3 Cw2 C .1  2 /  0 2u C2xx  2 2t C .2x1  2v /C2  2u C1  2w C3 D 0;

(3.140)

1

1 Cu3 C 2 Cv3 C 3 Cw3 C .1  3 /  0 3u C3xx  3 3t C .2x1  3w /C3  3u C1  3v C2 D 0:

(3.141)

It turns out that the functions  0 ;  1 and k .k D 1; 2; 3/ can be partly defined independently on the functions Ck .k D 1; 2; 3/. In fact, solving subsystem (3.130)– (3.132), one obtains 0 1 2 3

D  0 .t/;  1 D  1 .t; x/; D r1 .t; x/u C q1 .t/v C h1 .t/w C p1 .t; x/; D r2 .t; x/v C q2 .t; x/u C h2 .t/w C p2 .t; x/; D r3 .t; x/w C q3 .t; x/u C h3 .t/v C p3 .t; x/;

(3.142)

where  0 ;  1 ; rk ; qk ; hk and pk are unknown functions at the moment. Because the functions Ck .u; v; w/ are defined (up to arbitrary constants) in system (3.123), the system of DEs (3.130)–(3.141) can be solved. However, solving this system in the general case is a very complicated problem and we make no attempt to solve it here.

3.3.3 Q-Conditional Symmetry of the First Type Theorem 3.8 ([8]) DLVS (3.123) is invariant under Q-conditional operators of the first type (3.128) (with  0 ¤ 0) iff it and the corresponding operators have the forms listed in Table 3.3. Any other DLVS admitting a Q-conditional operator of the first type is reduced to one of those from Table 3.3 by a local transformation from the set (3.127). Simultaneously this Q-conditional operator is transformed to the corresponding operator listed in Table 3.3 (up to equivalent representations generated by adding a Lie symmetry operator of the form h2 .t; x; u; v; w/@v C h3 .t; x; u; v; w/@w , see Theorem 2.1). DLVS (3.123) with 1 D 2 D 3 admits only such Q-conditional operators of the first type, which are equivalent to the Lie symmetries listed in Table 3.2. In Table 3.3, the following designations are introduced: Q2i D Q4i with ˛ D 0; i D 1; : : : ; 6I a1  a2 u.@u  @v / C ˛u.@v  @w /; Q41 D @t C 1  2 a1  a2 v.@v  @u / C ˛v.@u  @w /; Q42 D @t C 1  2

3.3 The Three-Component Diffusive Lotka–Volterra System

103

Table 3.3 Q-conditional symmetries of the first type of DLVS (3.123)

1

2

3

4

5

6

7

8

9

Reaction terms u.a1 C bu C bv C ew/ v.a2 C bu C bv C ew/ w.a3 C u C v C e3 w/ u.a1 C u C v C w/ v.a2 C u C v C w/ w.a3 C u C v C w/ u.a1 C u C v C w/ v.a2 C u C v C w/ w.a3 C u C v C w/ u.a1 C u C v C w/ v.a2 C u C v C w/ w.a3 C u C v C w/ u.a1 C bu C v/ v.a2 C u C cv/ w.bu C v/ u.a1 C u C v/ v.a2 C u C v/ w.u C v/ u.a1 C bu C cv/ v.a2 C u C v/ w.bu C v/ u.a C bu C cv/ v.a C u C v/ w.bu C v/ u.a1 C u C v/ v.a2 C u C v/ w.u C v/

Restrictions .b  1/2 C .e  e3 /2 ¤ 0; a1 ¤ a2

Q-conditional symmetry operators 2 @t C a11 a u.@u  @v /; 2 a1 a2 @t C 1 2 v.@v  @u /

.a1  a2 /2 C .a1  a3 /2 ¤ 0

Q2i ; i D 1; : : : ; 6

.2  3 /a1  2 a3 C3 a2 D 0; a2 ¤ a3 ; ˇ ¤ 0

Q2i ; i D 1; : : : ; 6;  3 @t C ˇ exp a22 a t u.@v  @w / 3

.2  3 /a1  .1  3 /a2 C.1  2 /a3 D 0; .a1  a2 /2 C ˛ 2 ¤ 0 .b  1/2 C .c  1/2 ¤ 0

Q4i ; i D 1; : : : ; 6 Q51 Q51 ; Q6i ; i D 1; : : : ; 4

2 D 3 D 1; b ¤ 1; c ¤ 1; a1 .1  b/ D a2 b.1  c/

@t C ..1  b/u C .1  c/v Ca2 .1  c/ / @w

2 D 3 D 1; b ¤ 1; c ¤ 1; b.2  c/ D 1

@t C .1  c/@w C ..1  b/u C.1  c/v / '4 .t/@w

2 D 3 D 1

Q9i ; i D 1; : : : ; 5

The coefficients k > 0; k D 1; 2; 3 are assumed to be different in cases 1–6

a1  a3 u.@u  @w / C ˛u.@v  @w /; 1  3 a1  a3 w.@w  @u / C ˛w.@u  @v /; Q44 D @t C 1  3 a2  a3 v.@v  @w / C ˛v.@u  @w /; Q45 D @t C 2  3 a2  a3 w.@w  @v / C ˛w.@u  @v /I Q46 D @t C 2  3    t .1  3 /2 2 1  3 ˛1  a1 ˛1 x u@w I C Q51 D @t C ˛1 @x C exp 4 3 2    t .2  3 /2 2 2  3 ˛1  a2 ˛1 x v@w ; C Q61 D @t C ˛1 @x C exp 4 3 2 Q43 D @t C

104

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

  a1  a2 .1  3 /a2  .2  3 /a1 t u@w ; D @t C u.@u  @v / C ˇ exp 1  2 3 .2  1 /   a1  a2 .2  3 /a1  .1  3 /a2 t v@w ; Q63 D @t C v.@v  @u / C ˇ exp 1  2 3 .1  2 /   a 2 1  a 1 2 .3  2 /a1  .3  1 /a2 w@w C exp t w.@u  @v /I Q64 D @t C 3 .2  1 / 3 .1  2 / Q62

Q91 D Q51 with 3 D 1; Q92 D Q64 with 2 D 3 D 1; a1  a2 u.@u  @v / C .'1 .t/u C '2 .t/v C ˇ1 / @w ; Q93 D @t C 1  1 a1  a2 v.@v  @u /I Q94 D @t C .'3 .t/u C '2 .t/v C ˇ1 / @w ; Q95 D @t C 1  1 where the functions 'i .t/ .i D 1; : : : ; 4/ are as follows: ( '1 .t/ D ( '2 .t/ D

ˇ1 t C ˇ2 ; if a2 D 0; ˇ2 exp.a2 t/ C ˇa21 ; if a2 ¤ 0;

(3.143)

ˇ1 t; if a2 D 0; t C ˇ; if a D 0; '4 .t/ D ˇ1 ; if a2 ¤ 0; ˇ exp.at/ C 1a ; if a ¤ 0; a2 ( ˇ1 t C ˇ2 ; if a1 D 0; '3 .t/ D (3.144) ˇ2 exp.a1 t/ C ˇa11 ; if a1 ¤ 0;

while ˛ and ˇ (with and without subscripts 1 and 2) are arbitrary constants. Proof In order to prove the theorem, one needs to solve the system of DEs (3.130)– (3.141) with the functions C1 D u.a1 C b1 u C c1 v C e1 w/; C2 D v.a2 C b2 u C c2 v C e2 w/; C3 D w.a3 C b3 u C c3 v C e3 w/

(3.145)

and restrictions (3.124). It is worth noting that we also exempt from examination all the cases, which lead to the Lie symmetry operators listed in Table 3.2. The proof algorithm consists of three main steps. First we substitute the functions Ck ;  0 ;  1 and k (see formulae (3.145) and (3.142)) into (3.133)–(3.141). The equations obtained can be split w.r.t. the variables u; v; w; u2 ; v 2 ; w2 ; uv; uw and vw. Thus, an overdetermined system of PDEs to find the functions  0 ;  1 ; rk ; qk ; hk and pk (k D 1; 2; 3) is obtained. Second we examine the overdetermined system in order to derive all possible values of the coefficients ak ; bk ; ck ; ek and k leading to different conditional symmetries. It turns out that the diffusion coefficients play the most important role

3.3 The Three-Component Diffusive Lotka–Volterra System

105

(see formulae (3.133)–(3.134)), hence the following four cases must be examined: (1) k > 0; k D 1; 2; 3 are different, (2) 1 D 2 (the case 1 D 3 is equivalent to 1 D 2 up to the discrete transformations v $ w), (3) 2 D 3 and (4) 1 D 2 D 3 . The final step is to apply the conditional symmetry criterion (3.129) with the manifolds M12 and M13 (instead of M11 ) to the systems obtained in cases (1)– (4). Note that there is no need to apply the conditional symmetry criteria three times independently because DLVS (3.123) consists of equations having the same structure. Here we consider in detail the most general case (1) leading to six different systems listed in cases 1–6 of Table 3.3. Because k .k D 1; 2; 3/ are different constants and formulae (3.142) take place, Eqs. (3.133) and (3.134) immediately produce q1 D hk D 0; k D 1; 2; 3. Thus, the system of DEs has the form c1 p1 D e1 p1 D e2 p2 D c3 p3 D 0; .b1  b2 /q2 D 0; 3

.b1  b3 /q D 0;

.e1  e2 /q2 D 0;

(3.147)

3

.c1  c3 /q D 0;

(3.148)

t0  2x1 D 0;

(3.149)

k t1

(3.150)

2rxk

C

ck .r2 C 2x1 / D 0; 2

(3.146)

D 0;

ek .r3 C 2x1 / D 0;

3

1

c1 q C e1 q C b1 .r C

2x1 /

(3.151)

D 0;

(3.152)

.2c2  c1 /q2 C e2 q3 C b2 .r1 C 2x1 / D 0; 2

3

1

c3 q C .2e3  e1 /q C b3 .r C

2x1 /

D 0;

.j  1 / 1 q j C 2 0 qxj D 0; 1 rxx



1 rt1

1

2

(3.154) (3.155)

3

C 2b1 p C c1 p C e1 p D 0;

(3.156)

2 rxx  2 rt2 C 2a2 x1 C b2 p1 C 2c2 p2 C e2 p3 D 0;

(3.157)

3 rxx

(3.158)



3 rt3

C

2a1 x1

(3.153)

C

2a3 x1

1

2

3

C b3 p C c3 p C 2e3 p D 0;

j

j  j qt C .aj  a1 /q j C bj p j C qxx

pkxx  k pkt C ak pk C

1 j j 1 qr 0

1 k k 1 qp 0

D 0;

D 0;

(3.159) (3.160)

where k D 1; 2; 3; j D 2; 3. Now one should solve this system w.r.t. the functions  0 ;  1 ; rk ; qk ; hk and pk taking into account the fact that the form of these functions depends essentially on the system parameters. First of all, one observes that  2 2  3 2 q C q ¤ 0;

(3.161)

otherwise the system coincides with that for searching Lie symmetry operators. Moreover, Eqs. (3.147) and (3.148) show us that two different subcases should be

106

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

examined: (i) q2 ¤ 0; q3 D 0; (ii) q2 ¤ 0; q3 ¤ 0. Formally speaking, there is a third subcase (iii) q2 D 0; q3 ¤ 0I however, it is equivalent to (i) up to the discrete transformations v ! w; a2 ! a3 ; b2 ! b3 ; c2 ! e3 ; e2 ! c3 ; w ! v; a3 ! a2 ; b3 ! b2 ; c3 ! e2 ; e3 ! c2 :

(3.162)

Let us consider subcase (i). Since c21 C d12 ¤ 0 (see (3.124)), Eq. (3.146) gives p D 0; while Eqs. (3.149)–(3.151) lead to the equations 1

1 xx D rxk D t1 D 0 .k D 1; 2; 3/:

(3.163)

Hence, Eqs. (3.147) and (3.152)–(3.153) produce the coefficient restrictions b1 D b2 D b; c1 D c2 D c; e1 D e2 D e: Moreover, (3.159) with j D 3 is reduced to the algebraic condition b3 p3 D 0 ) p3 D 0; b3 ¤ 0 (otherwise b3 D 0 and simultaneously (3.154) leads to c3 D 0; but this contradicts restriction (3.124)). Having b3 ¤ 0 and assuming c3 ¤ 0 (examination of the special subcase c3 D 0 using (3.162) leads to case 5 of Table 3.3 only), we obtain q2x D 0 differentiating Eq. (3.153) w.r.t. x and taking into account (3.163). Hence, Eq. (3.155) with j D 2 immediately gives  1 D 0 and Eq. (3.151) leads to r2 D 0. Now (3.157) is reduced to the algebraic condition cp2 D 0I hence, Eq. (3.146) produce p1 D p2 D 0. In particular, this means that Eq. (3.160) vanish. Finally, we find q2 D  bc33 r1 ; b D bc33 c 2 from (3.156), (3.163) and (3.159) with j D 2, from (3.152) and (3.154), r1 D a11 a 2 3 and r D ˇ D const from (3.158). Thus, the system of DEs (3.146)–(3.160) is completely solved in subcase (i). Substituting the functions and parameter values obtained into (3.123), (3.128) and (3.142) we conclude that the DLVS   1 ut D uxx C u a1 C bc33 cu C cv C ew ;   2 vt D vxx C v a2 C bc33 cu C cv C ew ;

(3.164)

3 wt D wxx C w .a3 C b3 u C c3 v C e3 w/ admits the Q-conditional symmetry of the first type Q D @t C

  a1  a2 b3 u @u  @v C ˇw@w : 1  2 c3

(3.165)

Because of Eq. (3.151) there are coefficient restrictions e D e3 D 0 or ˇ D 0. However, in the case e D e3 D 0; the operator X D ˇw@w is the Lie symmetry of (3.164). So, taking into account Theorem 2.1 we may set ˇ D 0 into (3.165) without

3.3 The Three-Component Diffusive Lotka–Volterra System

107

loss of generality, i.e., Q D @t C

  a1  a2 b3 u @u  @v : 1  2 c3

(3.166)

Now one realizes that the system and the first operator listed in case 1 of Table 3.3 are obtained from (3.164) and (3.166) by the transformation b3 u ! u, c3 v ! v and renaming c ! c3 b. The second operator listed in case 1 of Table 3.3 and the operators from case 2 of Table 3.3 were obtained at the third step of the proof algorithm, i.e., criterion (3.129) with M12 and M13 was subsequently applied to (3.164). Thus, subcase (i) is completely examined and cases 1, 2 and 5 of Table 3.3 were obtained. Subcase (ii) has been investigated in a quite similar way so that cases 3, 4 and 6 of Table 3.3 were derived. The examination of case (1) is now complete. It turns out that case (2) does not produce any new system and symmetry, while the last three cases of Table 3.3 were obtained by examination of case (3). In case (4) any DLVS (3.123) admits only Q-conditional symmetries of the first type, which coincide with Lie symmetries. The proof is now complete. t u Remark 3.1 We point out that the inequalities listed in the third column of Table 3.3 guarantee that the relevant operators from the fourth column are not equivalent to any Lie symmetry operators listed in Table 3.2. Remark 3.2 If a system from Table 3.3 contains as a particular case (up to local transformations of the form (3.127)) a simpler system listed in another case, then in order to get an exhaustive list of symmetries, one should consider the case involving more symmetries. For example, if the coefficients of the system from case 2 satisfy additionally the restrictions from case 3, then all the operators listed in case 3 are conditional symmetries of the system in question. We conclude that the three-component DLVS, depending on the coefficient restrictions, possesses a wide range of Q-conditional symmetries of the first type in contrast to the two-component system. It is worth stressing that there are cases when DLVS (3.123) admits the sets consisting of 5, 6 and even 7 different symmetries and each of these symmetries can be applied for finding exact solutions. From the point of view of real-world applications, the systems arising in cases 1–4 of Table 3.3 are most interesting because their nonzero coefficients do not affect the biological sense of these systems. One of these systems is examined in the next subsection.

3.3.4 Exact Solutions and Their Interpretation There are a very few papers devoted to finding exact solutions of the threecomponent DLVS [6, 8, 20]. Travelling wave solutions modelling the competition

108

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

of three species were constructed in [6]. The solutions were derived under essential restrictions on the coefficients of the system in question. Here we present an example. System (3.123) with the restrictions 1 D 2 D 3 ; a1 D a2 D a3 D a and b1 D c2 D e3 D 1 (other parameters are negative and satisfy cumbersome restrictions, which are derived in [6]) possesses the travelling wave solution: a .1 C tanh.x  ˛t// ; 2 a v D .1  tanh.x  ˛t//2 ; 4  4  1  tanh2 .x  ˛t/ ; wD 1 C e1 uD

(3.167)

1 ae1 where the wave speed ˛ D 4aC20e . One may note that (3.167) is a 2.1Ce1 / generalization of the travelling wave (3.92) on the three-component system case. Here we apply the symmetries obtained in order to find exact solutions for the biologically motivated DLVS (3.123). One notes that the DLVS in case 4 of Table 3.3 is equivalent (up to the substitution u ! bu; v ! cv; w ! ew from the set (3.127)) to the system

1 ut D uxx C u.a1  bu  cv  ew/; 2 vt D vxx C v.a2  bu  cv  ew/;

(3.168)

3 wt D wxx C w.a3  bu  cv  ew/; where the coefficients ak ; b; c and e are known positive constants. As stated above, system (3.168) is used for modelling the competition between three species in population dynamics. Let as assume that the coefficients ak ; k .k D 1; 2; 3/ satisfy the restrictions listed in case 4 of Table 3.3, so that the system admits the symmetry operators Q4i .i D 1 : : : 6/. Substituting u ! bu; v ! cv; w ! ew into the Qconditional symmetry operator Q41 one obtains Q41

    a1  a2 b 1 1 @v  @w : ! Q D @t C u @u  @v C ˛b u 1  2 c c e

(3.169)

Applying the standard procedure for reducing the given PDE system to an ODE system via the known symmetry operator (3.169), we easily find the ansatz bu D '1 .x/eıt ;  ˛  1 '1 .x/eıt ; cv D '2 .x/ C ı ˛ a1  a2 6D 0; ew D '3 .x/  '1 .x/eıt ; ı D ı 1  2

(3.170)

3.3 The Three-Component Diffusive Lotka–Volterra System

109

where '1 .x/, '2 .x/ and '3 .x/ are new unknown functions. Substituting ansatz (3.170) into (3.168) and taking into account the restriction .2  3 /a1  .1  3 /a2 C .1  2 /a3 D 0 (see case 4 of Table 3.3), one obtains the reduced system of ODEs   2 a1 '100 C '1 1a21   '  ' 2 3 D 0; 2 '200 C '2 .a2  '2  '3 / D 0;

(3.171)

'300 C '3 .a3  '2  '3 / D 0: Now exact solutions of the three-component competition system (3.168) can be easily derived by inserting solutions of this system into ansatz (3.170). Because system (3.171) is three-component system of nonlinear second-order ODEs, its general solution is unknown in an explicit form. Let us assume that the triplet .'10 .x/; '20 .x/; '30 .x// is a particular solution of (3.171) and the functions 'k0 are nonnegative and bounded on the space interval I. Then one observes that the 0 exact  (3.170) with 'k D 'k .k D 1; 2; 3/ tends to the steady-state solution  1 solution 0 1 0 0; c '2 ; e '3 of DLVS (3.168) with ı < 0 provided t ! C1. In the general   case, the solution 0; 1c '20 ; 1e '30 produces a curve in the phase space .u; v; w/, which lies in the plane .0; v; w/. Assuming that the competition of three populations take place at the space interval I, we may conclude that solution (3.170) with 'k D 'k0 .k D 1; 2; 3/ describes such competition between them that species u dies while species v and w coexist. In particular, the interesting phenomena, a limit cycle, may occur if the concentrations v D 1c '20 .x/ and w D 1e '30 .x/ create a closed curve. Let us consider an example in the simplest case when '20 .x/ and '30 .x/ are constants. It is easy to verify that the constant solution '2 D v0 ; '3 D a2  v0 of the second and third equations of (3.171) with a2 D a3 , produces the following exact solution of the three-component competition system (3.168) with a1 ¤ a2 D a3 : 1 '1 .x/eıt ; b  v0 1 ˛ vD C  1 '1 .x/eıt ; c c ı ˛ a2  v0  '1 .x/eıt ; wD e eı uD

(3.172)

where '1 .x/ is a solution of the linear ODE '100  2 ı '1 D 0:

(3.173)

It should be stressed that solution (3.172) is not obtainable by any Lie symmetry because system (3.168) is invariant only under the principal algebra (3.126), so that only travelling wave solutions can be constructed. The general solution of ODE (3.173) depends essentially on the sign of ı. The case ı > 0 leads to the unbounded

110

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

(in time) solutions (see formulae (3.172)) and it is unlikely that they can describe any competition between three populations. Obviously, Eq. (3.173) with ı < 0 possesses the general solution p  p  ı2 x C C2 sin ı2 x ; (3.174) '.x/ D C1 cos where C1 and C2 are arbitrary constants. Setting, for example, C1 D 0 and C2 D 1 in (3.174) and substituting '.x/ into (3.172), we arrive at the exact solution p  1 sin ı2 x eıt ; b  p  v0 1 ˛ vD C  1 sin ı2 x eıt ; c c ı p  ˛ a2  v0 wD  sin ı2 x eıt e eı uD

(3.175)

2 of system (3.168) with a1 ¤ a2 D a3 ; ı D a11 a 2 . Let us provide a biological interpretation of the solution obtained. Assuming that the  between three populations takes place at the space interval

competition p I D 0; = ı2 , we observe that the components of solution (3.175) satisfy the boundary conditions

a2  v0 v0 ; wD I c e p a2  v0 v0 x D = ı2 W u D 0; v D ; w D : c e

x D 0 W u D 0; v D

These conditions predict that the concentrations of all the species u; v and w are constant on boundaries (for example, there is an artificial p regulation of the population densities in a vicinity of points x D 0 and x D = ı2 ). We also note 0 if t ! C1. that this exact solution tends to the steady-state point 0; vc0 ; a2 v e All the components of the solution (3.175) are bounded and nonnegative for arbitrary given t > 0 and x 2 I provided the coefficient restrictions 0  v0  a2  ˛ı ; if ˛  ı; 1

˛ ı

 v0  a2  ˛ı ; if ı < ˛  0;

1

˛ ı

 v0  a2 ; if ˛ > 0

hold. Thus, solution (3.175) can describe the following scenarios of the competition between three species: (i) species v and w eventually coexist while species u dies provided 0 < v0 < a2 I

3.3 The Three-Component Diffusive Lotka–Volterra System

111

(ii) species v eventually dominates while species u and w die provided v0 D a2 I (iii) species w eventually dominates while species u and v die provided v0 D 0: Examples of scenarios (i) and (ii) are presented in Figs. 3.2 and 3.3, respectively.

Fig. 3.2 The components u; v (yellow) and w (green) of exact solution (3.175) of system (3.168) with 1 D 2; 2 D 3 D 1; a1 D 1:5; a2 D a3 D 2:5; b D 0:3; c D 0:5; e D 0:15; ˛ D 0:5; v0 D 1:8; ı D 1

Fig. 3.3 The components u; v (yellow) and w (green) of exact solution (3.175) of system (3.168) with 1 D 2; 2 D 3 D 1; a1 D 1:5; a2 D a3 D 2:5; b D 0:3; c D 0:5; e D 0:2; ˛ D 1:5; v0 D 2:5; ı D 1

112

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

3.4 A Hunter-Gatherer–Farmer Population Model Here we briefly examine the three-component model introduced in [2] in order to describe the spread of an initially localized population of farmers into a region occupied by hunter-gathers. Under some assumptions clearly indicated in [2], the spread and interaction between farmers and hunter-gatherers can be modelled as a reaction-diffusion process. The nonlinear reaction-diffusion system has the form Ft D df Fxx C rf F .1  . F C C/=K/ ; Ct D dc Cxx C rc C .1  . F C C/=K/ C e. F C C/H; Ht D dh Hxx C rh H .1  H=L/  e. F C C/H;

(3.176)

where F.t; x/; C.t; x/ and H.t; x/ are densities of the three populations of initial farmers, converted farmers and hunter-gatherers, respectively. Parameters df ; dc and dh are the positive diffusion constants; rf ; rc and rh are the intrinsic growth rates of initial farmers converted farmers, and hunter-gatherers, respectively; K and L are the carrying capacities of farmers and hunter-gatherers; e is the conversion rate of hunter-gatherers to initial and converted farmers. The parameters rc ; rh and e are assumed to be nonnegative, while all others are positive. We note that df D dc D dh in [2]. In our opinion, it is very unlikely that the three populations of initial farmers, converted farmers and hunter-gatherers have the same diffusivity in space, hence their diffusivities are assumed to be arbitrary. The nonlinear RDS (3.176) can be simplified using the following re-scaling of the variables s df df F D Ku; C D Kv; H D Lw; t ! t; x ! x rf rf and introducing new notations 1 D

1 1 1 df rc edf L df rh edf K ; 2 D ; 3 D ; a 1 D ; a2 D ; a3 D ; a4 D : df dc dh dc rf dc rf dh rf dh rf

In fact, the above scale transformations simplify (3.176) to the system 1 ut D uxx C u.1  u  v/; 2 vt D vxx C a1 v.1  u  v/ C a2 .u C v/w; 3 wt D wxx C a3 w.1  w/  a4 .u C v/w:

(3.177)

Notably, system (3.177) with a2 D a4 D 0 is a particular case of the threecomponent DLVS (3.123), so that we additionally assume ja2 j C ja4 j 6D 0. System (3.177) contains seven parameters. It turns out that Lie and Q-conditional symmetries of the system essentially depend on their values. A complete description

3.4 A Hunter-Gatherer–Farmer Population Model

113

of these symmetries lies beyond the scope of this monograph. Here we present only an interesting result. Because system (3.177) belongs to the general class of three-component RDSs (3.125), we can use the system of DEs (3.130)–(3.141) in order to find Q-conditional symmetries of the first type. As a result, the following statement can be formulated. Theorem 3.9 The nonlinear RDS 1 ut D uxx C u.1  u  v/; 3 v.1  u  v/ C .u C v/w; 2 vt D vxx C 21  3 3 wt D wxx  .u C v/w

(3.178)

admits Q-conditional symmetry of the first type u Q D @t C 1  3

  1  2 @u  @v C @w ; 1 ¤ 3 : 1  3

(3.179)

The proof of the theorem is rather simple because one needs only to check that the system of DEs (3.130)–(3.141) with coefficients from operator (3.179) is identically fulfilled provided the functions C1 ; C2 and C3 are taken from system (3.178). In the case 1 D 2 D , RDS (3.178) and operator (3.179) are reduced to DLVS ut D uxx C u.1  v/; vt D vxx C v.1  v C w/; 3 wt D wxx  vw u and the operator @t C 1  @u by the substitution u C v ! v. It can be noted that 3 the above operator is a linear combination of the Lie symmetry operators @t and u@u of the system obtained (see case 2 of Table 3.2). In the case 1 6D 2 , operator (3.179) is a pure Q-conditional symmetry of the nonlinear RDS (3.178). It turns out that a nontrivial exact solution can be derived using operator (3.179). In fact, the corresponding ansatz is t

u D '1 .x/e 1 3 ; t

v D '2 .x/  '1 .x/e 1 3 ; w D '3 .x/ C

t 1  2 '1 .x/e 1 3 ; 1  3

(3.180)

114

3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

where '1 .x/, '2 .x/ and '3 .x/ are new unknown functions. Substituting ansatz (3.180) into (3.178), one obtains the reduced system of ODEs   3 D 0; '100  '1 1 C ' 2 3   2 3 3 D 0; '200 C '2 21   ' C ' 2 3 3 1 3

(3.181)

'300  '2 '3 D 0: Of course, the nonlinear system (3.181) cannot be integrated in a straightforward way, however, its particular solution can be derived by setting '3 D 0. As a result, we obtain 0s 1 0s 1 1 1 '1 D C1 cos @ xA C C2 sin @ xA ; '2 D 1; '3 D 0; 3  1 3  1 (3.182) where C1 and C2 are arbitrary constants. We should additionally assume that 1 < 3 (otherwise the solution is complex). Setting, for example, C1 D 0 and C2 D 1 in (3.182) and using ansatz (3.180), we arrive at the exact solution 0s u D sin @

1 t 1 xA e 1 3 ; 3  1

0s v D 1  sin @

1 t 1 xA e 1 3 ; 3  1 0s

wD

1  2 sin @ 1  3

(3.183)

1 t 1 xA e 1 3 3  1

of system (3.178) with 1 < 3 . All the components of solution (3.183) with 1 3), the problem is still open and cannot be solved by a simple generalization of the results obtained for .m  1/-component system. Of course, some particular results can be derived.

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3 Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

For example, we have checked that the straightforward generalization of the system and the operators listed in case 2 of Table 3.3 can be done as follows: m-component system i uit D uixx C ui .ai C u1 C    C um /; i D 1; 2; : : : ; m admits m.m  1/ operators of the form Qij D @t C

ai  aj i u .@ui  @uj / ; i 6D j D 1; 2; : : : ; m; i  j

provided .ai  aj /.i  j / 6D 0. It should be noted that Q-conditional symmetry operators were found under the restriction  0 6D 0 (see formula (3.128)), hence, the case  0 D 0 (this is a socalled no-go case) should be analysed separately. We discuss this issue extensively in Chap. 4 and in the recent paper [9]. An essential part of this chapter is devoted to the construction of exact solutions of DLVSs. We present examples of travelling fronts for two- and three-component systems because finding such solutions is important from the applicability point of view (see Theorem 3.5 and its interpretation). However, we concentrate mostly on finding exact solutions with more complicated structures than travelling fronts. To the best of our knowledge, this is a relatively new topic in the case of DLVSs. All the Q-conditional symmetry operators found for the two-component DLVS were used to construct non-Lie ansätze and to reduce the system (with correctly specified coefficients) to the corresponding systems of ODEs. As result, a wide range of new exact solutions in explicit form was found. These solutions possess complicated structures and they differ from the travelling fronts obtained in [7, 21, 31]. Moreover, a realistic interpretation for two competing species was provided for some exact solutions. We have also applied the Q-conditional symmetry operator obtained for the threecomponent DLVS in order to find exact solutions, which describe the competition between three species in population dynamics. An analysis of the relevant reduced ODE system in order to provide a biological interpretation was performed. In particular, it was shown that even the simplest particular solution of the ODE system leads to the exact solution (3.175), describing different scenarios of the competition between three species. Work on the application of other symmetries for finding new exact solutions is in progress. Finally, it is worth noting that a few nonlinear BVPs were exactly solved using the solutions obtained (see Theorems 3.5–3.6 and Sect. 3.3.4). It turns out that the definitions of Lie and conditional symmetry for PDEs can be extended on BVPs (it is a highly nontrivial step). Essential progress in this direction was made very recently (see [11–13] and the papers cited therein). In particular, the relevant algorithm for finding Lie and conditional symmetry of BVPs was worked out and applied successfully for solving some nonlinear problems arising in physics. Work is in

References

117

progress on application of the algorithm for solving some biologically motivated BVPs, especially describing tumour growth.

References 1. Ablowitz, M., Zeppetella, A.: Explicit solutions of Fisher’s equation for a special wave speed. Bull. Math. Biol. 41, 835–840 (1979) 2. Aoki, K., Shida, M., Shigesada, N.: Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers. Theor. Popul. Biol. 50, 1–17 (1996) 3. Arrigo, D.J., Hill, J.M., Broadbridge, P.: Nonclassical symmetries reductions of the linear diffusion equation with a nonlinear source. IMA J. Appl. Math. 52, 1–24 (1994) 4. Beteman, H.: Higher Transcendental Functions. McGraw-Hill, New York (1955) 5. Britton, N.F.: Essential Mathematical Biology. Springer, Berlin (2003) 6. Chen, C.-C., Hung, L.-C.: Exact travelling wave solutions of three-species competitiondiffusion systems. Discrete Contin. Dyn. Syst. B 17, 2653–2669 (2012) 7. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of the diffusive Lotka–Volterra system. Math. Comput. Modelling. 54, 1238–1251 (2011) 8. Cherniha, R., Davydovych, V.: Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system. J. Phys. A Math. Theor. 46, 185204 (14 pp) (2013) 9. Cherniha, R., Davydovych, V.: Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case. Appl. Math. Comput. 268, 23–34 (2015) 10. Cherniha, R., Dutka, V.: A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions. Ukr. Math. J. 56, 1665–1675 (2004) 11. Cherniha, R., King, J.R.: Lie and conditional symmetries of a class of nonlinear (1+2)dimensional boundary value problems. Symmetry 7, 1410–1435 (2015) 12. Cherniha, R., Kovalenko, S.: Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type. J. Phys. A Math. Theor. 44, 485202 (25 pp) (2011) 13. Cherniha, R. Kovalenko, S.: Lie symmetries of nonlinear boundary value problems. Commun. Nonlinear Sci. Numer. Simulat. 17, 71–84 (2012) 14. Clarkson, P.A., Mansfield, E.L.: Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica D 70, 250–288 (1994) 15. Conway, E.D., Smoller, J.A.: Diffusion and the predator-prey interaction. SIAM J. Appl. Math. 33, 673–686 (1977) 16. Gudkov, V.V.: Exact solutions of the type of propagating waves for certain evolution equations. Dokl. Ros. Akad. Nauk 353, 439–441 (1997) 17. Guo, J.S., Wang, Y., Wu, C.H., Wu, C.C.: The minimal speed of traveling wave solutions for a diffusive three species competition system. Taiwan. J. Math. 19, 1805–1829 (2015) 18. Hastings, A.: Global stability in Lotka–Volterra systems with diffusion. J. Math. Biol. 6, 163– 168 (1978) 19. Hou, X., Leung, A.W.: Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics. Nonlinear Anal. Real World Appl. 9, 2196–2213 (2008) 20. Hung, L.-C.: Traveling wave solutions of competitive–cooperative Lotka–Volterra systems of three species. Nonlinear Anal. Real World Appl. 12, 3691–3700 (2011) 21. Hung, L.-C.: Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species. Jpn. J. Indust. Appl. Math. 29, 237–251 (2012) 22. Jorné, J., Carmi, S.: Lyapunov stability of the diffusive Lotka–Volterra equations. Math. Biosci. 37, 51–61 (1977) 23. Leung, A.W., Hou, X., Feng, W.: Traveling wave solutions for Lotka–Volterra system revisited. Discrete Contin. Dyn. Syst. B 15, 171–196 (2011) 24. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, Baltimore (1925)

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25. Lou, Y., Ni, W.-M.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131, 79–131 (1996) 26. Martinez, S.: The effect of diffusion for the multispecies Lotka–Volterra competition model. Nonlinear Anal. Real World Appl. 4, 409–436 (2003) 27. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) 28. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003) 29. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives, 2nd edn. Springer, Berlin (2001) 30. Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press Company, Boca Raton (2003) 31. Rodrigo, M., Mimura, M.: Exact solutions of a competition-diffusion system. Hiroshima Math. J. 30, 257–270 (2000) 32. Rothe, F.: Convergence to the equilibrium state in the Volterra–Lotka diffusion equations. J. Math. Biol. 3, 319–324 (1976) 33. Serov, M.I.: Conditional invariance and exact solutions of the nonlinear equation. Ukr. Math. J. 42, 1216–1222 (1990) 34. Volterra, V.: Variazionie fluttuazioni del numero d‘individui in specie animali conviventi. Mem. Acad. Lincei. 2, 31–113 (1926)

Chapter 4

Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems

Abstract Two classes of two-component nonlinear reaction-diffusion systems are studied in order to find Q-conditional symmetries of the first type (a special subset of nonclassical symmetries), to construct exact solutions, and to show their applicability. The first class involves systems with constant coefficient of diffusivity, while the second contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the reaction-diffusion systems belonging to the above classes and admitting Q-conditional symmetries of the first type. The reaction-diffusion systems obtained allow one to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravity-driven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented.

4.1 Determining Equations The main object of investigation in this chapter is the following class of twocomponent reaction-diffusion systems (RDSs) uxx D d1 .u/ut C C1 .u; v/; vxx D d2 .v/vt C C2 .u; v/;

(4.1)

where dk and Ck .k D 1; 2/ are arbitrary smooth functions of their arguments. Each system with the variable functions d1 and/or d2 is transformed to the standard RDS of the form (2.1) using the Kirchhoff substitution (2.12). Discussion concerning possible real-world applications of such systems is presented in Chap. 2. We want to find all possible Q-conditional symmetries of the first type of the form Q D  0 .t; x; u; v/@t C  1 .t; x; u; v/@x C 1 .t; x; u; v/@u C 2 .t; x; u; v/@v ;

 0 6D 0;

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6_4

(4.2) 119

120

4 Q-conditional symmetries of the first type. . .

where the functions  0 ;  1 ; 1 and 2 should be found, to apply the symmetry obtained for the construction of exact solutions and to provide possible interpretation of the solutions obtained. Since the class of RDSs (4.1) contains four arbitrary functions, dk and Ck .k D 1; 2/, the problem of symmetry classification arises. This means that we need to describe all possible Q-conditional symmetries that can be admitted by each system from this class depending on the form of these four functions. In the case of Lie symmetries, this problem is often called the group classification problem (see [52] for details). This terminology is not applicable to Q-conditional symmetries because, generally speaking, they do not generate any group. So, we introduce new terminology the ‘conditional symmetry classification problem’ (CSCP) for this purpose. In the case of a class of partial differential equations (PDEs), the Lie symmetry (group) classification problem was formulated by Ovsiannikov [52] using notions of the equivalence group Eeq , the principal group of invariance and the classical Lie algorithm. During the last few decades this problem has been further studied and a more efficient algorithm worked out (see [21–23, 25] and the references cited therein), which involves the notion of form-preserving (admissible) transformations [31, 44]. Here this algorithm is adopted for solving the CSCP. We say that the CSCP is completely solved for the given PDE class and the given kind (type) of conditional symmetry if it has been shown that (i) the set of conditional symmetry operators for each PDE (system of PDEs) from the list obtained is maximal (i.e., contains all possible symmetries of the given type); (ii) all PDEs (systems) from the list are inequivalent with respect to (w.r.t.) formpreserving transformations, which are explicitly (or implicitly) presented; (iii) any other PDE (system) from the class admitting a conditional symmetry operator (of the given type) is reduced to one from the list by a form-preserving transformation. It should be noted that a complete description of form-preserving transformations is another nontrivial problem. However, in order to solve a group classification problem, the construction of a subset (not the total set!) of these transformations is enough in many cases. Let us apply the algorithm for solving the CSCP for the RDS class (4.1) in the case of Q-conditional symmetries of the first type. In order to realize the first step (i), Definition 2.2 should be applied to the RDS class (4.1) and operator (4.2). Formally speaking, we should construct systems of determining equations (DEs) using two different manifolds M1 (see Definition 2.2). However, the class of RDSs (4.1) is invariant under discrete transformations u ! v; v ! u. Thus, we can use only the manifold, say, M1 D fS1 D 0; S2 D 0; Q.u/ D 0g. Having an exhaustive list of the conditional symmetry operators and the relevant RDSs, one may simply extend the list using these discrete transformations.

4.1 Determining Equations

121

Here we present the result under the restriction  0 ¤ 0 (the case  0 D 0 is discussed in the final section). Thus, applying Definition 2.2, the following system of DEs that corresponds to the manifold M1 D fS1 D 0; S2 D 0; Q.u/ D 0g is derived x0 D u0 D v0 D u1 D v1 D 0;

(4.3)

1v D 1uu D 2uu D 2vv D 2uv D 0;    1 2u d2  d1 C 2 0 2xu D 0;   0 1 1 t    0 t1  2 1 x1 d1   1 1 du1  2 0 1xu C  0 xx D 0;  1  0 2 2 2 2x  t d C  dv D 0;

(4.4)

1 D 0; t1 d2 C 22xv  xx

(4.8)

  . / 1 Cu1 C 2 Cv1 C 2x1  1u C1 C  0 du1   1 1 C 1t C 2x1  0  t0  0 d1  1xx D 0;   1 Cu2 C 2 Cv2  2u C1 C 2x1  2v C2   1 1 C 2t C  0 2u d2   0 2u d1  2xx D 0:

(4.5) (4.6) (4.7)

2 1

(4.9)

(4.10)

We remind the reader that we are looking for purely conditional symmetry operators, i.e., we want to exclude all such operators, which are equivalent to those presented in [20–22]. Having this in mind, we present also the system of DEs for search Lie symmetry operators, x0 D u0 D v0 D u1 D v1 D 0; 1v D 2u D 1uu D 2vv D 0;  1  2x  t0 d1 C 1 du1 D 0;  1  2x  t0 d2 C 2 dv2 D 0;

(4.11)

t1 d1 C 21xu   1 xx D 0; 1 D 0; t1 d2 C 22xv  xx   1t d1  1xx C 1 Cu1 C 2 Cv1 C 2x1  1u C1 D 0;   2t d2  2xx C 1 Cu2 C 2 Cv2 C 2x1  2v C2 D 0;

which can be easily derived using paper [22] and the Kirchhoff substitution (2.12). One notes that systems of DEs (4.3)–(4.10) and (4.11) coincide if the restrictions 2u D 0;



 2x1  t0 d1 C 1 du1 D 0

(4.12)

122

4 Q-conditional symmetries of the first type. . .

hold. Thus, we take into account only such solutions of (4.3)–(4.10), which do not satisfy at least one of the equations from (4.12). Moreover, since Q-conditional symmetry of the first type is automatically nonclassical symmetry (see Chap. 2), we should also check restrictions (4.12) for coefficients of the operator obtained by multiplying them on arbitrary smooth functions. Otherwise the Q-conditional symmetry obtained can be equivalent to a Lie symmetry. It can be noted that the functions  0 ;  1 and k .k D 1; 2/ can be partly defined independently on the functions dk and Ck .k D 1; 2/. In fact, Eqs. (4.3) and (4.4) can be easily integrated:  0 D  0 .t/;  1 D  1 .t; x/;  D r .t; x/u C p1 .t; x/; 2 D q.t; x/u C r2 .t; x/v C p2 .t; x/; 1

1

(4.13)

where  0 .t/;  1 .t; x/; q.t; x/; rk .t; x/ and pk .t; x/ (k D 1; 2) are to-be-determined functions. Thus, one may claim that the coefficients  0 and  1 of the operator Q in question do not depend on u and v while 1 and 2 are linear with w.r.t. u and v. The further implementation of the algorithm for solving the CSCP depends essentially on the diffusivities dk .k D 1; 2/ because solutions of the remaining Eqs. (4.5)–(4.10) have different structures for constant and variable diffusivities.

4.2 Reaction-Diffusion Systems with Constant Diffusivities This section is devoted to investigation of the two-component RDSs of the form ut D d1 uxx C F.u; v/; vt D d2 vxx C G.u; v/;

(4.14)

where d1 and d2 are diffusivities assumed to be positive constants. It can be noted that the diffusion coefficient d1 in system (4.14) can be omitted without loss of generality because the simple substitution t ! t=d1 ; F ! d1 C1 ; G ! d2 C2 reduces the system to the form uxx D ut C C1 .u; v/; vxx D dvt C C2 .u; v/; where d D

d1 d2 .

Thus, we consider system (4.15) in what follows.

(4.15)

4.2 Reaction-Diffusion Systems with Constant Diffusivities

123

4.2.1 Q-Conditional Symmetry of the First Type The system of DEs corresponding to the system (4.15) takes the form x0 D u0 D v0 D u1 D v1 D 0;

(4.16)

1v D 1uu D 2uu D 2vv D 2uv D 0;

(4.17)

2 0 2xu

C .d 

1/ 1 2u

D 0;

(4.18)

21xu C t1 D 0;

(4.19)

22xv

(4.20)

C

dt1

D 0;

(4.21) 2x1  t0 D 0;  1  1 1 1 2 1 1 1 1 (4.22)  Cu C  Cv C 2x  u C D xx  t ;   1 1 Cu2 C 2 Cv2 C 2x1  2v C2 D 2u C1 C .1  d/  0 2u C 2xx  d2t : (4.23) The DE system for the search for Lie symmetry operators is x0 D u0 D v0 D u1 D v1 D 0; 1v D 2u D 1uu D 2vv D 0; 21xu C t1 D 0; 22xv C dt1 D 0;

(4.24)

2x1  t0 D 0;   1 Cu1 C 2 Cv1 C 2x1  1u C1 D 1xx  1t ;   1 Cu2 C 2 Cv2 C 2x1  2v C2 D 2xx  d2t : Comparing DEs (4.16)–(4.23) with (4.24) one concludes that 2u ¤ 0 is the necessary and sufficient condition, which guarantees that we find purely conditional symmetry operators, i.e., exclude all such operators, which are equivalent to Lie symmetry operators derived in [20, 21]. Now we need to solve the nonlinear system (4.16)–(4.23). First of all, we substitute (4.13) into (4.18)–(4.23) and obtain the equivalent system of PDEs: 2 0 qx C  1 .d  1/q D 0; 2rx1

C

t1

(4.25)

D 0;

(4.26)

2rx2 C dt1 D 0;

(4.27)

2x1

(4.28)



t0

D 0;

124

4 Q-conditional symmetries of the first type. . .



     r1 u C p1 Cu1 C qu C r2 v C p2 Cv1 C 2x1  r1 C1  1  D rxx  rt1 u C p1xx  p1t ;  1      r u C p1 Cu2 C qu C r2 v C p2 Cv2 C 2x1  r2 C2   2 1 1 D qC1 C r uCp q.1  d/ C rxx  drt2 v C .qxx  dqt /u C p2xx  dp2t ; 0

(4.29)

(4.30)

to find the functions  0 .t/;  1 .t; x/; q.t; x/ ¤ 0; rk .t; x/ and pk .t; x/. In other words, all possible Q-conditional symmetries of the first type are constructed provided the   general solution of system (4.25)–(4.30) is known for each pair C1 ; C2 . Theorem 4.1 ([17]) The nonlinear RDS (4.15) with d ¤ 1 is invariant under Qconditional symmetry of the first type (4.2) if and only if (iff) it and the corresponding symmetry have the forms listed in Table 4.1. Any other RDS admitting this kind of Q-conditional operator is reduced to one of those from Table 4.1 by the transformations t ! C1 t C C2 ; x ! C3 x C C4 ; u ! C5 eC6 t u C C7 t C C8 ; v ! C9 eC10 t v C C11 t2 C C12 t C C13

(4.31)

with correctly specified constants Cl .l D 1; : : : ; 13/ and/or the discrete transformation u ! v; v ! u: Simultaneously the relevant symmetry operator is reduced (a Lie symmetry operator of the form .h1 .t; x/v C h0 .t; x//@v should be used in some cases) to that from Table 4.1. Proof In order to prove the theorem, one needs to solve the nonlinear PDE system (4.25)–(4.30) with the restriction q.t; x/ ¤ 0. We remind the reader that C1 and C2 should be treated as unknown functions. As follows from the preliminary analysis (see Eqs. (4.29) and (4.30) involving the functions C1 and C2 ), we should examine six cases separately: (1) r1 D r2 D p1 D 0; (2) r1 D r2 D 0; p1 ¤ 0; (3) r1 D p1 D 0; r2 ¤ 0; (4) r1 D 0; p1 ¤ 0; r2 ¤ 0; (5) r2 D 0; r1 ¤ 0; (6) r1 ¤ 0; r2 ¤ 0:

4.2 Reaction-Diffusion Systems with Constant Diffusivities

125

Solving system (4.25)–(4.30) in each case one obtains the list of Q-conditional symmetries of the first type together with the correctly-specified functions C1 and C2 . Note that the symmetry operators have different structures depending on the case. Let us consider case (1) in detail. Equations (4.29) and (4.30) take the form

.qu C p

2

/Cv2

.qu C p2 /Cv1 C 2x1 C1 D 0; C 2x1 C2 D qC1 C .qxx  dqt /u C p2xx  dp2t :

(4.32)

Differentiating the first equation of (4.32) w.r.t. x, one arrives at the equation .qx u C p2x / Cv1 D 0; which leads to the requirement Cv1 D 0. In fact, if qx ¤ 0, then immediately Cv1 D 0. If qx D 0, then Eq. (4.25) produces  1 D 0, hence, Cv1 D 0. Thus, the first equation of system (4.32) takes the form x1 C1 D 0 and two subcases x1 ¤ 0 and x1 D 0 should be examined. The general solution of (4.32) with x1 ¤ 0 is  C D 0; C D exp  1

2

 2x1 qxx  dqt p2xx  dp2t v g.u/ C u C ; qu C p2 2x1 2x1

(4.33)

where g.u/ is an arbitrary (at the moment) function. Because the function C2 does not depend on t and x, Eq. (4.33) with g.u/ ¤ 0 immediately produces the restrictions q D ˛1 x1 ; p2 D ˛2 x1 ; where ˛1 and ˛2 are arbitrary constants. 1 1 Differentiating Eq. (4.28) w.r.t. x, one obtains xx D 0. So, qx  ˛1 xx D 0, however, 1 this contradicts the assumption x ¤ 0. The remaining possibility g.u/ D 0 leads to the linear RDS (4.15) only. Now we examine the subcase x1 D 0, i.e.,  1 D 1 D const. The general solution of (4.32) takes the form C1 D f .u/; C2 D

qf .u/ C .qxx  dqt /u C p2xx  dp2t v C g.u/; qu C p2

(4.34)

where f .u/ and g.u/ are arbitrary (at the moment) functions. If f .u/ is an arbitrary function, then we obtain p2 D ˇq (ˇ D const). Hence C2 D

f .u/ v C ˛v C g.u/; uCˇ

t where ˛ D qxx dq . Having this, we use renaming q overdetermined system

qxx dqt q

D ˛;

2qx C 1 .d  1/q D 0:

f .u/ uCˇ

! f .u/ and solve the

126

4 Q-conditional symmetries of the first type. . .

Thus, the system of DEs (4.25)–(4.30) is completely solved (under the restrictions listed above!) and we obtain the conditional symmetry operator  Q D @t C 1 @x C 2 exp

 2 .1  d/2  4˛ 1 .1  d/ xC 1 t .u C ˇ/@v ; 2 4d

where 1 and 2 6D 0 are arbitrary constants, of the system uxx D ut C .u C ˇ/f .u/; vxx D dvt C f .u/v C ˛v C g.u/: Finally, using the simple transformation u ! u  ˇ;

1 .1  d/ ! 1 2

(4.35)

and renaming the functions f and g, one sees that it is exactly case 6 of Table 4.1. To complete the examination of case (1) we look for the correctly specified function f .u/, which satisfies (4.34) without the restriction p2 D ˇq. Indeed, if one finds the differential consequences of the second order (see equation for C2 ), 2 2 namely Cvx D 0; Cvt D 0, then the following algebraic equations for the function f .u/ are obtained:      2 qt p  qp2t f D .qxx  dqt / u C p2xx  dp2t qt u C p2t       .qxx  dqt /t u C p2xx  dp2t t qu C p2 ;  2     qx p  qp2x f D .qxx  dqt / u C p2xx  dp2t qx u C p2x       .qxx  dqt /x u C p2xx  dp2t x qu C p2 : Now one realizes that the function f .u/ D ˛1 C ˛2 u C ˛3 u2 (˛1 ; ˛2 and ˛3 should be determined) provided p2 ¤ ˇq. Substituting this expression into (4.34) and using the standard routine, one arrives at case 8 of Table 4.1 if ˛3 ¤ 0 and case 9 if ˛3 D 0. Let us consider case (6), which is the most complicated. In order to solve the linear PDEs (4.29)–(4.30) w.r.t. the unknown functions C1 .u; v/ and C2 .u; v/ using the standard technique, one obtains two essentially different expressions for invariant variable !, namely: (i) if r1 D r2 D r, then !D

  p1 rv C qu C p2 q ln u C I  ru C p1 r r

4.2 Reaction-Diffusion Systems with Constant Diffusivities

127

(ii) if r1 ¤ r2 , then  r21      r q p1 r1 p2  qp1 p1 v 1 u C C : ! D uC 1 r r  r2 r1 r1 r2

(4.36)

So, integrating Eq. (4.29) we arrive at C1 .u; v/:  C1 D u C

p1 r1

p1 p1 C xxr1 t

1 21x1 r

Z 

uC

f .!/ C p1 r1

1 r 1 rxx t r1

 21x1 2 r

Z

 u uC

!

p1 r1

 21x1 2 r

du (4.37)

du ;

where f .!/ is an arbitrary smooth function. In order to find C1 .u; v/ in an explicit form using (4.37), the following subcases should be examined: x1 D 0; 2x1 D r1 ; x1 .2x1  r1 / ¤ 0:

(4.38)

This means that examination of subcases (i) and (ii) depends essentially on the fixed restriction from (4.38). Here we consider in detail the most general subcase (ii) with the restriction x1 .2x1  r1 / ¤ 0. From (4.37), we immediately obtain 2x1  1 1 r1 1 p r1  r1 p1  p1t p1 .rxx  rt1 / ; C1 D u C 1 f .!/ C xx 1 t u C xx1  r 2x 2x  r1 2x1 .2x1  r1 /

(4.39)

where ! is given by (4.36). The further analysis consists of solving the remaining Eqs. (4.25)–(4.28) and (4.30) for the function C1 .u; v/ of the form (4.39). Because the function C1 .u; v/ depends only on the variables u and v, the second-order 1 1 differential consequences Cvt D 0 and Cvx D 0 lead to much simpler expressions. As a result, it was derived that the only restriction t1 rx1 rx2 ¤ 0 may lead to conditional symmetry operator(s), otherwise Lie symmetries are obtained. Now we realize that (4.39) cannot involve an arbitrary function f , otherwise, 1 1 taking into account (4.28), rx1 D ) xx D rx1 D 0 ( is an arbitrary nonzero 1 constant) and then the contradiction rx D 0 is obtained. In order to specify f , the second-order differential consequence of (4.39) w.r.t. v and x was calculated and it was proved that the only possibility is f D 0 (otherwise the right-hand side of (4.39) depends on t and/or x). Thus, we arrive at the linear function C1 D ˛1 u C ˛2 :

(4.40)

128

4 Q-conditional symmetries of the first type. . .

Here ˛1 and ˛2 are arbitrary constants, which satisfy the equalities 1  rt1 rxx D ˛1 ; 2x1

p1xx  p1t p1 .r1  r1 /  1 xx 1 t 1 D ˛2 : 1 1 2x  r 2x .2x  r /

(4.41)

Having C1 .u; v/, the function C2 .u; v/ is easily calculated using (4.30): 1 x  r2 2  r2 dr2 r1 p1 C D u C r1 g.!/ C xx2 1 t v C 2 1 Cr1 1 r2 .qxx  dqt x   x  1 1q r2 dr2 1 q u C p2xx  dp2t C ˛2 q C .1d/p C ˛1  xx2 1 t C .1d/r 0 1 r 2 0  2  x x    1 r2 dr2 r2 dr2 p1 qxx  dqt C ˛1  xx2 1 t C .1d/r q ;  xx2 1 t p2  2 1 Cr 1 r 2 0 

2

x

x

(4.42)

x

where g.!/ is an arbitrary smooth function. Hereafter we assume g.!/ ¤ 0 (otherwise (4.42) produces a linear C2 , so that a linear RDS will be derived). Let us assume g.!/ ¤ 0 is arbitrary. This means the right-hand side of (4.36) 2 r2 2 1 does not depend on t and x, hence we derive r1 x D ˇ and rr1 D (ˇ and are 2 1

nonzero constants). In particular, these equalities produce r1x D  ˇ leading to the restriction rx1 D 0; but t1 rx1 rx2 ¤ 0. Thus, the only possibility is to specify g in a clever way. In order to do this the 2 2 second-order differential consequences Cvx D 0 and Cvt D 0 were calculated using (4.42). Solving the equations obtained, we find g.!/ D g1 6D 0, where g1 is an arbitrary constant. We may set g1 D 1 without loss of generality because of the transformation v ! g1 v. So, formula (4.42) can be rewritten in the form ˇ  p1 C2 D u C 1 C ˛3 v C ˛4 u C ˛5 ; r

(4.43)

where the constants ˇ and ˛k (k D 3; 4; 5) should satisfy the equalities r2 2x1 r1

D ˇ;

2 dr 2 rxx t 2x1

1 2x1 Cr1 r2 1 2x1 r2



(4.44)

D ˛3 ;   qxx  dqt C ˛1  ˛3 C

p2xx  dp2t C ˛2 q C

.1d/p1 q 0

  .1d/r1 q D ˛4 ; 0    ˛3 p2  ˛4 p1 D ˛5 :

(4.45) (4.46) (4.47)

Using Eqs. (4.26), (4.27) and (4.44), one easily shows that ˇ D d 6D 1I 0. 1 Moreover, one should set pr1 D ˛0 D const in (4.43) because the function C2 depends only on u and v (the special case d D 2 should be separately examined, however no conditional symmetry was derived). So, we use the transformation u ! u  ˛0 . This means that we can set ˛0 D p1 D 0 without loss of generality,

4.2 Reaction-Diffusion Systems with Constant Diffusivities

129

simultaneously ˛2 also vanishes (see (4.41)). Moreover, ˛5 also vanishes because of the transformation ( v  ˛˛53 ; if ˛3 ¤ 0; v! (4.48) v  ˛d5 t; if ˛3 D 0: Thus, formulae (4.40) and (4.43) contain only the parameter d and three arbitrary constants ˛k (k D 1; 3; 4). Using the first equation in (4.41) and Eqs. (4.26), (4.27), (4.44), (4.45), one derives the linear PDE   d.d  5/tx1 D 4 ˛3  ˛1 d2 x1 ;

(4.49)

which can be easily integrated provided d ¤ 5. Having,  1 one readily finds the functions  0 .t/; q.t; x/; r1 .t; x/ and r2 .t; x/ from Eqs. (4.25)–(4.28). However, the functions obtained do not satisfy (4.41), (4.44)–(4.46) for any ˛k . Happily, there is the special case d D 5;

˛3 D ˛1 d2

when (4.49) vanishes. Integrating the linear equations (4.25)–(4.28), (4.41) and (4.44)–(4.47) under the above restrictions, we have found the functions  0 .t/;  1 .t; x/; q.t; x/; r1 .t; x/; r2 .t; x/ and p2 .t; x/, leading to the Q-conditional symmetry   2 C  exp  xt

  2 2 u@u Q D t2 @t C tx@x  x C2t C ˛ t 1 4    2   4˛1 t u  5x 4C2t C 5˛1 t2 v C p2 .t; x/ @v ;

(4.50)

where  ¤ 0 is an arbitrary constant, while the function p2 .t; x/ is the solution of the linear equation p2xx D 5p2t C 25˛1 p2 . Simultaneously, the restriction ˛4 D 0 was obtained, hence the corresponding RDS has the form uxx D ut C ˛1 u; vxx D 5vt C u5 C 25˛1 v:

(4.51)

Since the operator p2 .t; x/@v is a Lie symmetry of (4.51) (this follows from case 12 of Table 3 [14]), we can apply Theorem 2.1 and set p2 D 0 in (4.50) without loss of generality. Finally, applying the transformation u ! e˛1 t u; v ! e5˛1 t v

(4.52)

130

4 Q-conditional symmetries of the first type. . .

system (4.51) and operator (4.50) are reduced to the forms presented in case 26 of Table 4.1. Thus, examination of case (6) is complete. Cases (2)–(5) were treated in the similar way and the results are listed in Table 4.1. It should be noted that several transformations ((4.35), (4.48) and (4.52) are examples) have been used to reduce the number of cases and simplify structures of the relevant RDSs. These transformations can be united and presented in the form (4.31). The systems listed in Table 4.1 are inequivalent w.r.t. the set of transformations (4.31), hence step (ii) of the CSCP is realized. The proof is now complete. t u Table 4.1 Q-conditional symmetry operators of RDS (4.15) with d ¤ 1 1

C1 .u; v/ uf .!/

2

uf .!/

3

uf .!/

4

f .!/

5

f .!/

6

uf .u/

7

f .u/

8

˛1 C ˛2 u C u2

9

˛1 C ˛2 u

10 ˛1 C ˛2 u C ˛4 ln.u C v/ 11 ˛2 u C v 12 ˛2 u 13 ˛1 u ln u

C2 .u; v/ uk g.!/ C u . f .!/ C ˛.1  d// ! D uk .v  u/ u.g.!/ C ˛.1  d/ ln u Cf .!/ ln u/; ! D u exp. vu / g.!/ C u. f .!/ C ˛.1  d// ! D u exp.u  v/ eu g.!/  f .!/  ˛.1  d/ ! D eu .u C v/ uf .!/ C g.!/ C .1  d/u ! D u2  2v vf .u/ C g.u/ C ˛v .u C v/ .g.u/ C ˛ ln.u C v//  f .u/ g.u/ C uv

g.u/ C ˛3 v

Q and the coefficient restrictions @t C ˛u@u C ˛ ..1  k/u C kv/ @v ; ˛ ¤ 0; k ¤ 1 @t C ˛u@u C ˛.u C v/@v ; ˛ ¤ 0 @t C ˛u@u C ˛.u C 1/@v ; ˛ ¤ 0 @t C ˛@u C ˛.u C v  1/@v ; ˛ ¤ 0 @t C @u C u@v @t C

21 @ C 1d x 

qu@v ; 2 ¤ 0;  2 ˛ q D 2 exp 1 x C 1 d t   @t C  exp  ˛d t .u C v/@v ;  ¤ 0

21 @t C 1d @x C .qu C p2 /@v ; 1 x q D e '1.t/; '1 ¤ 0  p2 D e1 x .21 C ˛2 /'1  d'P1 21 @t C 1d @x C .qu C p2 /@v ;   2 C˛ ˛ q D 2 exp 1 x C 1 d2 3 t ;

p2xx D dp2t C ˛3 p2  ˛1 q; 2 ¤ 0 1 ˛2 ˛3 v C .˛3  ˛2 /u  ˛4 ln.u C v/ @t C ˛4˛.1d/ .u C v/@v     ˛1 1 ˛2 C .x/ exp ˛4˛.1d/ t  1d  .@u  @v /; ˛1 ˛2 ˛4 ¤ 0 2 ˛3 v C.1Cd/ v2u C˛1 u ln uC˛4 u @t C '2 .t/.u@u C v@v /  'P2 .t/u@v ; ˛2 'P2 .t/ ¤ 0 ˛3 v C ˛4 u C uk @t C 2 u@u C .'3 .t/u C 2 kv/@v ; ˛4 2 '3 ¤ 0; k ¤ 1 1

˛3 v C ˛1 v ln u C ˛2 u d

2

@t C 1d1 @x C 2 e˛1 t u@u C.qu C d2 e˛1 t v/@v ; ˛1 2 3 ¤ 0; qD   3 exp 1 x C

21 ˛3 tC d1  e˛1 t d ˛1 d 2

(continued)

4.2 Reaction-Diffusion Systems with Constant Diffusivities

131

Table 4.1 (continued) C1 .u; v/ 14 ˛1 u ln u

C2 .u; v/ ˛3 v C ˛1 v ln u C ˛4 u

Q and the coefficient restrictions ˛1 t @t C u@u  2 e  C '4 .t/u C . d2 e˛1 t C 3 /v @v ; ˛1 2 '4 ¤ 0

15 ˛1 u ln u

˛3 v C ˛1 v ln u C ˛4 u 1 C ˛2 u d ˛3 v C ˛1 dv ln u +˛1 .1  d/u ln u  ˛3 u ˛3 v C ln u

@t C 2 e˛1 t u@u C .'5 .t/uC d2 e˛1 t v/@v ; ˛1 ˛2 ˛4 2 '5 ¤ 0

16 ˛1 u ln u 17 0

@t C 2 e˛1 t u@u C .˛1 u C .2 e˛1 t  ˛1 /v/@v ; ˛1 2 ¤ 0 21 @t C 1d @x C 2 u@u C .qu C p2 /@v ; 2 2 pxx D dpt C ˛3 p2 C 2 ; 2 3 ¤ 0; q D 3 exp 1 x C

18 ˛2 u

˛3 v C ln u C ˛4 u

19 ˛2 u 20 ˛1 C ˛2 u C u2 21 ˛1 C ˛2 u

˛3 v C u ln u uv C ˛3

22 0

˛3 v C eu

21 ˛3 C2 .1d/ t d

@t C 2 u@u C .'6 .t/u C p2 /@v ; p2xx D dp2t C ˛3 p2 C 2 ; ˛4 2 '6 ¤ 0 @t C 2 u@u C .'7 .t/u C 2 v/@v ; 2 '7 ¤ 0 @t C  .v C '8 .t/.u C ˛2 /  d'P8 .t// @v ;'8 ¤ 0 21 @t C 1d @x C p1 @u C .qu C p2 /@v ; q D '9 .t/ exp.1 x/; 1 '9 ¤ 0; p2xx D dp2t C ˛3 p2 C .d   1/qp  ˛1 q; 1 2 1 p D 2 .1 C ˛2  ˛3 /'9  d'P9 exp.1 x/ 21 @t C 1d @x C 2 @u C.qu C 2 v C p2 /@v ; p2xx D dp2t C ˛3 p2  2 .1   d/q;

˛3 v C u2

q D 3 exp 1 x C

23 ˛1

˛3 v C ˛4 u C e

24 0

˛1 .u C v/ ln.u C v/ C ˛2

25 0

u2

26 0

u5

u

21 ˛3 t d

; 3 ¤ 0

@t C 2 @u C .'10 .t/u C 2 v C p2 /@v ;˛4 2 '10 ¤ 0; p2xx D dp2t C ˛3 p2  .2 .1  d/ C ˛1 /q C ˛4 2 ˛2 @t C d1 .@u  @v / C  exp. ˛d1 t/.u C v/@v ; ˛1 ˛2  ¤ 0  2 2 p exp  x @u 2t@t C x@x C 2tx 5 4t 4 t   1 x2 2 p C exp. 4t /u C 2v C p @v ; t  2 2 2 pxx D 3p2t C 2tx exp  x2t ; d D 3 4t4 t2 @t C tx@x   2 2  x C2t u@u C  exp  xt u  4 d D 5;  ¤ 0

5x2 C2t v 4



@v ;

In Table 4.1, the functions .x/, '1 .t/, '2 .t/, '4 .t/, '5 .t/, '8 .t/ and '9 .t/ are the general solutions of the linear ordinary differential equations (ODEs) 00

 

˛1 ˛2 C ˛2 ˛4 .1  d/

 D 0;

  4 C ˛2 21 C ˛1 '1 D 0; d 'R 1  221 C ˛2 'P 1 C 1 d

132

4 Q-conditional symmetries of the first type. . .

d 'R2  .˛2  ˛3 C .1  d/'2 / 'P2  ˛1 '2 D 0;     1  d ˛1 t e D 0; d'P4 C ˛3 C 2 .d  1/e˛1 t '4  ˛4 3 C 2 d   1  d ˛1 t d'P5 C ˛3 C 2 .d  1/e˛1 t '5  ˛4 2 D 0; e d ˛1 ˛3 D 0; d 'R8  ˛2 'P8 C '8 C d d and     d 'R9  21 .1 C d/ C ˛2 .1  d/  ˛3 'P9 C 41  ˛3 21  ˛2 .˛2  ˛3 / '9 D 0; respectively. The functions ( '3 .t/ D





˛2 ˛3 C2 .1d/ t d ˛4 2 .1k/ t C 3 ; if ˛2  d

3 exp

(

'10 .t/ D

C

˛4 2 .1k/ ; ˛2 ˛3 C2 .1d/

if ˛2 ¤ ˛3  2 .1  d/;

D ˛3  2 .1  d/I   3 exp  ˛d3 t C ˛˛4 3 2 ; if ˛3 ¤ 0; ˛4 2 d t; if ˛3 D 0:

Finally, the function '6 .t/ D '3 .t/ at k D 0; while '7 .t/ D '3 .t/ at k D 0 and ˛4 D 1. Hereafter the upper dot index denotes differentiation w.r.t. the variable t. Theorem 4.1 presents all possible RDSs with constant diffusivities admitting Q-conditional symmetries of the first type, which are inequivalent up to any transformation of the form (4.31). A natural question is: Can we claim that the 26 systems listed in Table 4.1 are inequivalent up to arbitrary local (point) transformations (not only of the form (4.31)!)? It turns out that the answer is positive. The rigorous proof of this statement is presented in [17] (see Sect. 3) and is based on finding a set of form-preserving transformations. As a result, it was shown that Table 4.1 cannot be shortened. Thus, one may claim that the CSCP for the RDS class (4.15) is completely solved, i.e., the classification obtained satisfies items (i), (ii) and (iii).

4.2.2 Reductions, Exact Solutions and Their Interpretation Table 4.1 presents a wide range of RDSs with the constant diffusivities, which admit Q-conditional symmetry of the first type. The most general RDSs occur in cases 1– 7 because they contain the systems involving arbitrary smooth functions f and g. Depending on the form of f and g, one may extract RDSs arising in applications, construct exact solutions and provide some interpretation for them. Here we present a few examples.

4.2 Reaction-Diffusion Systems with Constant Diffusivities

133

Let us construct exact solutions of the nonlinear RDS listed in case 1 of Table 4.1, when the system and the corresponding symmetry operator have the form ut D uxx  uf .!/; ! D uk .v  u/; dvt D vxx  uk g.!/  u. f .!/ C ˛.1  d//

(4.53)

Q D @t C ˛u@u C ˛ ..1  k/u C kv/ @v :

(4.54)

and

It is well known that using any Q-conditional symmetry, one reduces the given system of PDEs to a system of ODEs via the same procedure as for classical Lie symmetries. Since each Q-conditional symmetry of the first type is automatically one of the second type, i.e., the standard Q-conditional symmetry, we apply this procedure for finding exact solutions. The system Q .u/ D 0;

Q .v/ D 0

(4.55)

for the operator (4.54) takes the form ut D ˛u; vt D ˛.1  k/u C ˛kv and its general solution produces the ansatz u D '.x/e˛t ; v D .x/ek˛t C '.x/e˛t ;

(4.56)

where '.x/ and .x/ are new unknown functions. Substituting ansatz (4.56) into (4.53), one obtains the reduced system of ODEs ' 00 D ' .˛ C f .!// ; 00 D ' k g.!/ C ˛kd ; ! D

' k ;

(4.57)

(hereafter ' 00 D 'xx ; 00 D xx ). Because system (4.57) is nonlinear (excepting, of course, some special cases) it can be integrated only for the correctly specified functions f and g. We specify f and g in a such way that the RDS in question will be still nonlinear (otherwise the result will be rather trivial) but (4.57) will be integrable. Thus, setting 1

f .!/ D ! k  ˛; g.!/ D ˇ!

134

4 Q-conditional symmetries of the first type. . .

(ˇ and 6D 0 are arbitrary constants), RDS (4.53) takes the form 1

ut D uxx  .v  u/ k C ˛u; 1

dvt D vxx  .v  u/ k  ˇv C .ˇ C ˛d/u;

(4.58)

while the corresponding reduced system is 1

' 00 D k ; 00 D .ˇ C ˛kd/ : The general solution of (4.59) can be easily constructed:  Z Z 1 k '.x/ D

.x/ dx dx C c3 x C c4 ;

(4.59)

(4.60)

.x/ D c1 exp.x/ C c2 exp.x/; if 2 D ˇ C ˛kd > 0; .x/ D c1 cos.x/ C c2 sin.x/; if 2 D .ˇ C ˛kd/ > 0;

(4.61)

.x/ D c1 x C c2 ; if ˇ C ˛kd D 0: Thus, substituting (4.60) and (4.61) into (4.56), the four-parameter family of solutions for the nonlinear RDS (4.58) is constructed. Hereafter we highlight the solutions satisfying the zero Neumann boundary conditions, which widely arise in biologically motivated problems. Hence, setting c3 D ˇ D 0; k D 13 ; 2 D  ˛d D c1 cos.x/ in (4.60) and (4.61), one 3 > 0; obtains the periodic solution  c31  cos2 .x/ C 6 cos.x/e˛t C c4 e˛t ; u D ˛d ˛t v D c1 cos.x/e 3 C u

(4.62)

of the nonlinear RDS ut D uxx  .v  u/3 C ˛u; dvt D vxx  .v  u/3 C ˛du:

(4.63)

Obviously, the exact solution (4.62) is bounded and nonnegative in time and space provided ˛ < 0 and c4 > 0 is sufficiently large. It can also be noted that this solution satisfies the zero Neumann boundary conditions ux jxD0 D 0; vx jxD0 D 0; ux jxDj  D 0; vx jxDj  D 0; where j 2 N. Thus, formulae (4.62) present the solution of n bounded nonnegative  o

the nonlinear RDS (4.63) in the domain ˝ D .t; x/ 2 .0; C1/  0; j  , which satisfies the zero Neumann boundary conditions.

4.2 Reaction-Diffusion Systems with Constant Diffusivities

135

Let us set f .!/ D .a1 C b!/; g.!/ D .˛.1  d/  a1 /! (hereafter ˛; a1 and b are arbitrary nonzero constants) in (4.53), hence, it takes the form ut D uxx C a1 u  bu2k C bvu1k ; dvt D vxx  a2 v  bu2k C bvu1k ;

(4.64)

where a2 D ˛.1  d/  a1 . The corresponding reduced system of ODEs is ' 00 C b ' 1k C .a1  ˛/' D 0; 00 D .˛kd C a2 / :

(4.65)

Although we have not constructed the general solution of system (4.65), its particular solution was found by setting D ı; ı 6D 0. In this case, the firstorder ODE r 2bı 2k a1 0 ' ; k¤2 (4.66) ' D ˙ .˛  a1 /' 2 C C c1 ; ˛ D 2k d.k  1/ C 1 for the function ' is obtained (the value k D 2 is special and leads to the equation p ' 0 D ˙ .˛  a1 /' 2 C 2bı ln ' C c1 ). If c1 ¤ 0, then the general solution of (4.66) can be expressed via hypergeometric functions. In order to avoid cumbersome formulae, here we present the solution of ODE (4.66) with c1 D 0; k ¤ 0:  p   1k   k a1 ˛ .x ˙ c / C 1 ; if a1 > ˛; '.x/ D ˇ tan2 2 2  p     1k k ˛a1 '.x/ D ˇ tanh2 .x ˙ c2 /  1 ; if a1 < ˛; 2

(4.67)

  ˇ D .a1 ˛/.2k/ , which seems to be the most interesting. Note that an arbitrary 2bı constant can be removed by the trivial substitution x ˙ c2 ! x. Now we rewrite system (4.64) setting v ! v in order to obtain a biologically motivated model. So the system takes form   ut D uxx C u a1  bu1k  bvu1k ;   dvt D vxx C v a2 C bu1k C bu2k ;

(4.68)

where all coefficients (except k) should be positive. System (4.68) can be treated as a prey–predator model in population dynamics. In fact, the species u is prey and described by the first equation. Its population decreases proportionally to the predator density v. The natural birth–death rule for the prey is u.a1  bu1k / and

136

4 Q-conditional symmetries of the first type. . .

can be treated as a generalization of the standard logistic rule u.a1  bu/ (see, e.g., [48]). Similar arguments are also valid for the second equation, which describes the predator density v. Notably this system with k D 0 is the classical prey– predator model [48] with the additional term bu2 . Such models usually involve the zero Neumann boundary conditions (zero-flux on the boundaries), which reflect the natural assumption that both species cannot be widespread over the globe and they occupy a bounded domain. Using ansatz (4.56) with v ! v, D ı and (4.67) with a1 > ˛ we construct the exact solution    p   1k k a1 ˛ u D ˇ tan2 x C 1 e˛t ; 2 (4.69)  p     1k ˛kt 2 k a1 ˛ ˛t v D ıe  ˇ tan x C1 e 2 of (4.68). It turns out that solution can describe the interaction between prey and predator on the space interval Œ0; l (here l D kp2 j ; j 2 N) provided a1 ˛ 1 0 0; k > 0 hold. Moreover, solution (4.102) is nonnegative and satisfies the standard zero-flux conditions on the correctly specified intervals. For instance, solution (4.102) with C2 D 0 satisfies the boundary conditions Ux jxD0 D 0; Vx jxD0 D 0; Ux jxDj h D 0; Vx jxDj h D 0

 on the space interval 0; j h ; j 2 N;ˇ and its components are positive provided o nˇ ˇ ˇ .lC1/ 11 12 < 0; C4 > max ˇ .kC1/22 C1 ˇ ; jC1 j : Thus, we have established that the 11 exact solutions obtained can satisfy the typical requirements occurring in physically and biologically motivated problems. For example, the exact solution of the model describing the gravity-driven flow of thin films of the size L D j h is presented in Fig. 4.2.

4.4 Concluding Remarks In this chapter, RDSs with constant and variable diffusivities were studied in order to find Q-conditional symmetries of the first type, to construct exact solutions, and to show applicability of the solutions obtained. The main theoretical results are presented in the form of Theorems 4.1 and 4.2. These theorems contain exhaustive lists (up to the given sets of point transformations) of the nonlinear RDSs of the forms (4.15) and (4.1) admitting Q-conditional symmetries of the first type. Thus, we have shown that the CSCP for RDSs with constant and variable diffusivities can be completely solved provided we restrict ourselves to search Q-conditional symmetries of the first type. Because such symmetries are only a subset of the

148

4 Q-conditional symmetries of the first type. . .

standard Q-conditional symmetries (nonclassical symmetries), the CSCP is still an open task in the general case. It should be noted that Q-conditional symmetry operators were found under restriction  0 6D 0 (see formula (4.2)), hence, the case  0 D 0 (this is a so-called no-go case) should be analysed separately. It is well known that the problem of finding Q-conditional symmetry operators with  0 D 0 for a single evolution PDE is equivalent to that of the construction of the general solution for the given PDE (see [56, 65] for details). Thus, this problem cannot be completely solved for any nonintegrable evolution equation. Of course, particular solutions of the equation in question may help one to solve this problem partly, for example, an interesting approach was presented in [50] (see [33] for its recent application). It turns out that the situation is different for systems of evolution PDEs if one is seeking for Q-conditional symmetry operators of the first type and this was demonstrated in the recent paper [18]. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of RDSs of the form (4.1) admitting such symmetry is derived in the no-go case. The symmetries are applied to find exact solutions. As a result, multiparameter families of exact solutions for nonlinear RDSs with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established [18]. It is worth pointing out that the structure of the operator coefficients in (4.2) can be generalized by involving the derivatives of the functions u and v. As a result, the relevant definitions and the algorithm for finding generalized conditional symmetries of the first type for RDSs can be worked out. We make no attempt to explore such matters here in detail. However, we predict that the system of DEs obtained for finding such generalized conditional symmetries will be much more complicated than (4.3)–(4.10). Of course, some particular solutions in the case of a fixed RDS can be obtained (similarly to those constructed very recently in [62]) but we do not expect that any complete result (like that presented in Theorems 4.1 and 4.2) will be derived. Tables 4.1 and 4.2 present a wide range of RDSs with constant and variable diffusivities, which admit Q-conditional symmetry operators. Many of them involve the functions d 1 , d2 , f and g, which can be arbitrary smooth functions of the relevant arguments. This gives a variety of possibilities for extracting specific systems, which occur in real-world models. In this chapter, a few examples are presented, which show how the symmetry operators derived can be applied for searching for exact solutions. Moreover, exact solutions with attractive properties were found for some RDSs (see systems (4.68) and (4.101)) and their possible biological and physical interpretations were presented. Work is in progress to construct exact solutions for other RDSs from Tables 4.1 and 4.2, which are used in real-world applications. It is worth noting that there are many papers [1, 2, 4, 5, 10, 14, 19–21, 32, 34, 37, 53, 57, 61] devoted to finding exact solutions of nonlinear RDSs of the form (4.1) because they are governing equations for relevant models describing various biological, ecological and chemical processes. The handbook [55, Chap. 17] summarizes many exact solutions presented in the papers cited above. It can be noted that a large majority of the solutions presented therein were constructed by

4.4 Concluding Remarks

149

application (explicitly or implicitly) of Lie symmetry of the systems in question. Here and in Chap. 3, several families of exact solutions are derived using Qconditional symmetry and it is shown that they cannot be constructed by application of Lie symmetry, i.e., they are non-Lie solutions. In conclusion, we present the following observation. All symmetry-based methods for solving nonlinear PDEs have a clear connection to the method of differential constraints, which has been formulated in [58, 64] (actually the main ideas of this method were published much earlier, see, e.g., the historical review in [30]). In fact, a common property that underlies all the symmetry-based methods can be described as follows: in order to find exact solutions one solves a nonlinear PDE (system of PDEs) together with the differential constraint(s) generated by a symmetry operator. The corresponding symmetry can be of different types: Lie symmetry, Q-conditional symmetry, generalized conditional symmetry, etc. Because the overdetermined system consisting of the given PDE and the constraint generated by the symmetry is compatible, one can find its solutions in a much simpler way. This means that the main problem of the method of differential constraints, how to define suitable constraint(s) for the given PDE in a such way that the overdetermined system obtained will be compatible, is automatically solved. There are several techniques, which propose to use the correctly specified differential constraints in order to find exact solutions for some classes of PDEs without knowledge of any symmetries. Let us restrict ourselves (just for simplicity) to a second-order two-dimensional equation of the form L .t; x; u; ut ; ux ; utt ; utx ; uxx / D 0:

(4.103)

If PDE (4.103) can be reduced to one with quadratic nonlinearities, then the method of additional generating conditions [11–13, 22, 49] and the method of determining equations [38, 39], which were independently worked out in the 1990s, are applicable for constructing exact solutions. Very similar techniques were also used in later papers (see, e.g., [9, 35]). According to the method of additional generating conditions, PDE (4.103) should be examined together with the additional generating conditions (i.e., specified differential constrains) in the form of the linear mth-order homogeneous ODE ˛1 .t; x/

du dm1 u dm u C    C ˛m1 .t; x/ m1 C m D 0; dx dx dx

(4.104)

where ˛1 .t; x/; : : : ; ˛m1 .t; x/ are arbitrary smooth functions and the variable t is considered as a parameter. As a result, the given PDE is reduced to a system of ODEs with the structure determined by PDE (4.103) and condition (4.104) (see [12, 13] for details). Having any solution of the obtained system of ODEs, we immediately derive an exact solution of PDE (4.103). Several approaches based on substitutions of the special form, which are usually called ansatz, should be mentioned. Ansatz reduces the given PDE to a simpler equation (e.g., ODE) or a system of simpler equations, which can be integrated

150

4 Q-conditional symmetries of the first type. . .

(at least partly). In order to construct the relevant ansatz, either some physical (biological, chemical, etc.) motivation or an ad hoc approach is used. The most typical is the plane wave ansatz u D .!/; ! D x  vt; v 2 R;

(4.105)

which reduces Eq. (4.103) to an ODE provided the function L does not depend explicitly on t and x. Although solving the nonlinear ODE obtained can also be a nontrivial problem, the solution (at least partial) can usually be found by using classical methods or handbooks like [36, 54]. Notably, there exist some recent techniques (e.g., the tanh- and exp-function methods) allowing to construct such solutions of the nonlinear ODE obtained, which lead to travelling waves (fronts) of the given PDE (see, e.g., [46, 47, 63]), however their wide applicability is often questionable [45]. The second classical ansatz follows from the Fourier method (the method of separation of variables) and reads as u D T.t/X.x/;

(4.106)

where T.t/ and X.x/ are to-be-determined functions. Depending on the form of L, PDE (4.103) can be reduced to a system of two ODEs. Both ansätze have been well known since the nineteenth century and are related to the Lie symmetry of the equation in question. The first one reflects invariance under time and space translations, while the second is related to scale invariance of the given equation. Notably, J. Boussinesq was the first to construct an exact solution of the nonlinear physically motivated problem using ansatz (4.106) [8]. Several new ansätze have been suggested during recent decades in order to construct exact solutions for nonlinear PDEs, especially evolution equations. Many of them can be united and written in the form of the ansatz with separated variables A.u/ D '0 .t/g0 .x/ C    C 'm1 .t/gm1 .x/:

(4.107)

Of course, (4.107) is a straightforward generalization of ansatz (4.106). Here some functions are known, and others should be found. Typically, the functions '0 ; : : : ; 'm1 .t/ are unknown in the case of evolution equations, the functions g0 .x/; : : : ; gm1 .x/ are specified by either physically motivated reasons or an ad hoc approach, while A.u/ usually reflects nonlinearities in the given PDE. As a result, an mdimensional system of ODEs should be obtained for finding '0 ; : : : ; 'm1 .t/ by using ansatz (4.107). For example, ansatz (4.107) with the power-law functions gk D xk , i.e., A.U/ D '0 .t/ C '1 .t/x C    C 'm1 .t/xm1 ;

(4.108)

was extensively used for constructing exact solutions of reaction-diffusion equations with power and exponential nonlinearities in the 1990s [28, 29, 40, 41, 43, 59].

References

151

It should be mentioned that ansatz (4.108) was also applied in the earlier papers [51, 60]), while later, in the 2000s, it was shown that the ansatz is applicable for reduction of some nonlinear PDEs when the functions gk D xk contain rational (and even irrational) exponents k (see, e.g., examples in [24, 30]). It is worth noting that a generalization of ansatz (4.108) for solving multi-dimensional PDEs was suggested in [42] (see also examples in [30]). Several cases of ansatz (4.107) with the functions gk 6D xk were used for construction of new solutions to nonlinear evolution PDEs (see, e.g., [13, 15, 29, 59]). Papers [6, 51] were the first in this direction. In the 1990s, the method of linear invariant subspaces was developed [29, 59] and was extensively applied for constructing exact solutions of a wide range of nonlinear PDEs (almost all the results are summarized in monograph [30]). One may note that the method is reduced to finding exact solutions in the form (4.107) provided the functions g0 .x/; : : : ; gm1 .x/ generate the basis of an invariant subspace for a given PDE. However, it is worth noting that there are exact solutions, which cannot be derived by the method of invariant subspaces (see an example in [13] on page 8196). Thus, there are suitable differential constraints, which are not related to invariant subspaces. Finally, the direct method [27], which is also called the Clarkson–Kruskal method, should be mentioned. The basic idea of the method is to reduce PDE (4.103) to ODEs (a system of ODE and PDE) using the ansatz u D '.t; x; !/;

(4.109)

where the function !.z/ is a solution of some ODE, while the variable z.t; x/ should be additionally defined (e.g., it can be an invariant variable corresponding to a Lie symmetry). The function ' is usually assumed to be linear with respect to !.z/. It turns out that this ansatz leads to new exact solutions for some PDEs arising in real-world applications [26, 27]. It was shown later that a consistency criterion can be formulated when the direct method and the nonclassical method [7] lead to the same solutions [3]. On the other hand, there are examples of the exact solutions found by the nonclassical method, which are not obtainable by the direct method [3], hence the nonclassical method seems to be a more general tool for finding exact solutions of nonlinear PDEs.

References 1. Archilla, J., Romero, J., Romero, F., Palmero, F.: Lie symmetries and multiple solutions in ˛  ! reaction-diffusion systems. J. Phys. A Math. Gen. 30, 185–194 (1997) 2. Archilla, J., Romero, J., Romero, F., Palmero, F.: Spiral wave solutions ineaction-diffusion equations with symmetries. Analysis through specific models. J. Phys. A Math. Gen. 30, 4259– 4271 (1997) 3. Arrigo, D.J., Broadbridge, P., Hill, J.M.: Nonclassical symmetry solutions and the methods of Bluman–Cole and Clarkson–Kruskal. J. Math. Phys. 34, 4692–4703 (1993)

152

4 Q-conditional symmetries of the first type. . .

4. Barannyk, T.: Symmetry and exact solutions for systems of nonlinear reaction-diffusion equations. Proc. Inst. Math. Nat. Acad. Sci. Ukraine 43, 80–85 (2002) 5. Barannyk, T.A., Nikitin, A.G.: Solitary wave solutions for heat equations. Proc. Inst. Math. Nat. Acad. Sci. Ukraine 50, 34–39 (2004) 6. Bertsch, M., Kersner, R., Peletier, L.A.: Positivity versus localization in degenerate diffusion equations. Nonliniar Anal. TMA 9, 987–1008 (1985) 7. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, 1025–1042 (1969) 8. Boussinesq, J.: Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur débit de sources (in French). J. Math. Pures Appl. 10, 5–78 (1904) 9. Carini, M., Fusco, D., Manganaro, N.: Wave-like solutions for a class of parabolic models. Nonlinear Dyn. 32, 211–222 (2003) 10. Cherniha, R.M.: On exact solutions of a nonlinear diffusion-type system. In: Symmetry Analysis and Exact Solutions of Equations of Mathematical Physics. Inst. Math., Acad. Sci. of USSR, Kyiv, pp. 49–53 (1988) 11. Cherniha, R.: Symmetry and exact solutions of heat-and-mass transfer equations in Tokamak plasma (in Ukrainian). Proc. Acad. Sci. Ukraine 4, 17–21 (1995) 12. Cherniha, R.: A constructive method for construction of new exact solutions of nonlinear evolution equations. Rep. Math. Phys. 38, 301–312 (1996) 13. Cherniha, R.: New non-Lie ansätze and exact solutions of nonlinear reaction-diffusionconvection equations. J. Phys. A Math. Gen. 31, 8179–8198 (1998) 14. Cherniha, R.: Lie symmetries of nonlinear two-dimensional reaction-diffusion systems. Rept. Math. Phys. 46, 63–76 (2000) 15. Cherniha, R.: Nonlinear Galilei-invariant PDEs with infinite-dimensional Lie symmetry. J. Math. Anal. Appl. 253, 126–141 (2001) 16. Cherniha, R., Davydovych, V.: Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities. Commun. Nonlinear Sci. Numer. Simul. 17, 3177–3188 (2012) 17. Cherniha, R., Davydovych, V.: Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations. In: Algebra, Geometry and Mathematical Physics, vol. 85, pp. 533–553. Springer, Berlin (2014) 18. Cherniha, R., Davydovych, V.: Nonlinear reaction-diffusion systems with a non-constant diffusivity: conditional symmetries in no-go case. Appl. Math. Comput. 268, 23–34 (2015) 19. Cherniha, R., Dutka, V.: A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions. Ukr. Math. J. 56, 1665–1675 (2004) 20. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. J. Phys. A Math. Gen. 33, 267–282, 7839–7841 (2000) 21. Cherniha, R., King, J.R.: Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II. J. Phys. A Math. Gen. 36, 405–425 (2003) 22. Cherniha, R., King, J.R.: Nonlinear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions. J. Math. Anal. Appl. 308, 11–35 (2005) 23. Cherniha, R.M., Serov, M.I.: Symmetries, ansätze and exact solutions of nonlinear secondorder evolution equations with convection term II. Euro. J. Appl. Math. 17, 597–605 (2006) 24. Cherniha, R., Waniewski, J.: Exact solutions of a mathematical model for fluid transport in peritoneal dialysis. Ukr. Math. J. 57, 1316–1324 (2005) 25. Cherniha, R., Serov, M., Rassokha, I.: Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations. J. Math. Anal. Appl. 342, 1363–1379 (2008) 26. Clarkson, P.A.: New similarity solutions for the modified Boussinesq equation. J. Phys. A Math. Gen. 22, 2355 (1989) 27. Clarkson, P.A., Kruskal, M.D.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989) 28. Fushchych, W.I., Zhdanov, R.Z.: Antireduction and exact solutions of nonlinear heat equations. J. Nonlinear Math. Phys. 1 60–64 (1994)

References

153

29. Galaktionov, V.A.: Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. Roy. Soc. Edinb. Sect. A 125 225–246 (1995) 30. Galaktionov, V.A., Svirshchevskii, S.R.: Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. Chapman & Hall/CRC, Boca Raton (2007) 31. Gazeau, J.P., Winternitz, P.: Symmetries of variable coefficient Korteweg-de Vries equations. J. Math. Phys. 33, 4087–4102 (1992) 32. Gudkov, V.V.: Exact solutions of the type of propagating waves for certain evolution equations. Dokl. Ros. Akad. Nauk 353, 439–441 (1997) 33. Hashemi, M.S., Nucci, M.C.: Nonclassical symmetries for a class of reaction-diffusion equations: the method of heir-equations. J. Nonlinear Math. Phys. 20, 44–60 (2013) 34. Hung, L.-C.: Exact traveling wave solutions for diffusive Lotka–Volterra systems of two competing species. Jpn. J. Indust. Appl. Math. 29, 237–251 (2012) 35. Ji, L.: The method of linear determining equations to evolution system and application for reaction-diffusion system with power diffusivities. Symmetry 8, 157 (2016) 36. Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. I: Gewöhnliche (in German). Stuttgart (1977) 37. Kaptsov, O.V.: Construction of exact solutions of systems of diffusion equations (in Russian). Math. Model. 7, 107–115 (1995) 38. Kaptsov, O.V.: Determining equations and differential constraints. J. Nonlinear Math. Phys. 2, 283–291 (1995) 39. Kaptsov, O.V., Verevkin. V.: Differential constraints and exact solutions of nonlinear diffusion equations. J. Phys. A Math. Gen. 36, 1401 (2003) 40. King, J.R.: Exact results for the nonlinear diffusion equations @u=@t D .@=@x/.u4=3 @u=@x/ and @u=@t D .@=@x/.u2=3 @u=@x/. J. Phys. A Math. Gen. 24, 5721–5745 (1991) 41. King, J.R.: Some non-self-similar solutions to a nonlinear diffusion equation. J. Phys. A Math. Gen. 25, 4861–4868 (1992) 42. King, J.R.: Exact multidimensional solutions to some nonlinear diffusion equations. Quart. J. Mech. Appl. Math. 46, 419–436 (1993) 43. King, J.R.: Exact polynomial solutions to some nonlinear diffusion equations. Physica D 64, 35–65 (1993) 44. Kingston, J.G., Sophocleous, C.: On form-preserving point transformations of partial differential equations. J. Phys. A 31, 1597–1619 (1998) 45. Kudryashov, N.A.: Comment on: “A novel approach for solving the Fisher equation using Exp-function method” [Phys. Lett. A 372 3836 (2008)] [MR2418599]. Phys. Lett. A 373, 1196–1197 (2009) 46. Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its applications. Phys. Scr. 82, 065003 (2010) 47. Malfliet, W.: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. Comp. Appl. Math. 164, 529–541 (2004) 48. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989) 49. Myroniuk, L.P., Cherniha, R.M.: Reduction and solutions of a class of nonlinear reactiondiffusion systems with the power nonliniarities (in Ukrainian). Proc. Inst. Math. NAS Ukraine 3, 217–224 (2006) 50. Nucci, M.C.: Iterations of the non-classical symmetries method and conditional Lie-Bäcklund symmetries. J. Phys. A Math. Gen. 29, 8117–8122 (1996) 51. Oron, A. Rosenau, P.: Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A 118, 172–176 (1986) 52. Ovsiannikov, L.V.: The Group Analysis of Differential Equations. Academic, New York (1980) 53. Polyanin, A.D.: Exact solutions of nonlinear sets of equations of the theory of heat and mass trannsfer in reactive media and mathematical bilogy. Theor. Found. Chem. Eng. 38, 622–635 (2004) 54. Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, London (2003)

154

4 Q-conditional symmetries of the first type. . .

55. Polyanin, A.D., Zaitsev, V.F.: Handbook of Nonlinear Partial Differential Equations, 2nd edn. CRC Press, Boca Raton (2012) 56. Popovych, R.O.: On a class of Q-conditional symmetries and solutions of evolution equations (in Ukrainian). In: Symmetry and Analytic Methods in Mathematical Physics. Proc. Inst. Math.of Ukraine, Kyiv, vol. 19, pp. 194–199 (1998) 57. Rodrigo, M., Mimura, M.: Exact solutions of a competition-diffusion system. Hiroshima Math. J. 30, 257–270 (2000) 58. Sidorov, A.F., Shapeev, V.P., Yanenko, N.N.: Method of Differential Relations and its Application to Gas Dynamics (in Russian). Nauka, Novosibirsk (1984) 59. Svirshchevskii, S.: Invariant linear spaces and exact solutions of nonlinear evolution equations. J. Nonlinear Math. Phys. 3, 164–79 (1996) 60. Titov, S.S.: Solutions of non-linear partial differential equations in the form of polynomials with respect to one of variables (in Russian). Chislennye Metody Mekhaniki Sploshnoy Sredy 8, 144–149 (1977) 61. Vyaz’mina, E.A., Polyanin, A.D.: New classes of exact solutions to general nonlinear diffusion-kinetic equations. Theor. Found. Chem. Eng. 40, 595–603 (2006) 62. Wang, J., Ji, L.: Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems. J. Math. Anal. Appl. 427, 1101–1118 (2015) 63. Wazwaz, A.M.: The extended tanh method for the Zakharo–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear Sci. Numer. Simulat. 13, 1039–1047 (2008) 64. Yanenko, N.N.: Compatibility theory and methods of integration of systems of nonlinear partial differential equations (in Russian). In: Proceedings of Fourth All-Union Mathematical Conference, pp. 613–621. Nauka, Leningrad (1964) 65. Zhdanov, R.Z., Lahno, V.I.: Conditional symmetry of a porous medium equation. Phys. D 122, 178–186 (1998)

Appendix A

List of Reaction-Diffusion Systems and Exact Solutions

In Chaps. 2–4, a great variety of Q-conditional symmetries was derived for twoand three-component reaction-diffusion systems. Some symmetries were applied in order to construct exact solutions. The solutions obtained are summarized in Table A.1.

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6

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d1 .u/ ut D uxx  11 u  12 v C ˛u d1 .u/ v ˛ vt D vxx  21 u  22 v C ˛1 v ˛C1

ut D uxx  .v  u/3 C ˛u dvt D vxx  .v  u/3 C ˛du   ut D uxx C u a1  bu1k  bvu1k dvt D vxx C v a2 C bu1k C bu2k

@t C

1 ut D uxx C u.1  u  v/ 3 2 vt D vxx C 21  v.1  u  v/ C uw C vw 3 3 wt D wxx  uw  vw 1 ut D uxx  .v  u/ k C ˛u 1 dvt D vxx  .v  u/ k  ˇv C .ˇ C ˛d/u u 1 3

a1 a2 1 2 b c

1 2 @ 1 3 w



  @v C˛b u 1c @v 

 @u  @v C

 u @u 

3

2

 u C 13 v @v

  et @t C ˛u @u C ˇv @v

@t C ˛u@u  ˛ ..1  k/u  kv/ @v

@t C ˛u@u C ˛

@t C ˛u@u C ˛ ..1  k/u C kv/ @v

@t C

1 ut D uxx C u.a1  bu  cv  ew/ 2 vt D vxx C v.a2  bu  cv  ew/ 3 wt D wxx C w.a3  bu  cv  ew/

2 vt D vxx C v.a2  bu  cv/

1 ut D uxx C u.a1  bu  cv/ 1 e

@w



  .1  2 /@t C.a1 cv C a2 bu  a1 a2 /  1b @u C 1c @v

.1  2 /@t .a1 v C a2 u C a1 a2 /.@u  @v /

1 ut D uxx C u.a1 C u C v/

2 vt D vxx C v.a2 C u C v/

Q-conditional symmetry

Reaction-diffusion system

Table A.1

(4.96)

(4.69), Fig. 4.1

(4.62)

(4.56), (4.60), (4.61)

(3.183)

(3.172), (3.175), Fig. 3.2, Fig. 3.3

(3.122), Fig. 3.1

(3.108), (3.109), (3.117)–(3.120)

Exact solutions

156 A List of reaction-diffusion systems. . .

Index

ansatz, 23, 26, 79, 93, 108, 113, 133, 143, 150

basic operator, 37, 99 Belousov–Zhabotinskii system, 47, 72, 73, 92 boundary condition, 2, 22, 91, 97, 110, 147 boundary value problem (BVP), 91, 97, 116 Burgers equation, 18, 20, 41

competition of populations, 61, 79, 109 competition of species, 46, 92, 97, 108 conditional symmetry, 4, 40, 47, 48, 56, 72, 105, 120

differential consequence, 8, 15, 56, 81, 126, 127, 139, 141 diffusion equation, 34, 70, 80 diffusive Lotka–Volterra system (DLVS), 46, 47, 78–80, 82, 88, 91, 94, 98, 100, 102, 103, 107 Dirichlet condition, 38, 97

equivalence transformation, 137, 139, 141, 143 evolution equation, 8, 9, 48, 50, 148, 150 exact solution, 1, 3, 21–24, 29, 30, 36, 37, 39, 48, 60, 79, 80, 90, 91, 94, 96, 107, 109, 115, 120, 132, 134, 136, 143, 147, 148

fast diffusion, 35, 39, 41 Fisher equation, 2, 3, 21–23, 84, 90

Fitzhugh–Nagumo equation (FNE), 24, 32, 41 form-preserving transformation, 120, 132, 143

general solution, 12, 18, 28, 29, 34, 55, 62, 83, 93, 95, 110, 125, 131, 134, 139, 141, 142 generalized conditional symmetry, 4, 41, 48, 72 generalized Fitzhugh–Nagumo equation (FNE), 26, 28, 31–33 generalized Kolmogorov–Petrovskii–Piskunov equation (KPPE), 26, 28, 32

heat equation, 4, 13, 41, 129 Huxley equation, 16, 41

independent variable, 9, 48, 53, 79 infinitesimal operator, 49 invariance criterion, 5, 9, 14, 49, 50, 99 inverse function, 13, 52, 53

Kirchhoff substitution, 13, 52, 61, 119, 121 Kolmogorov–Petrovskii–Piskunov equation (KPPE), 3, 26, 41

Lie algebra, 6, 79, 80, 99, 100 Lie method, 23, 24 Lie symmetry, 4, 6, 16, 23, 48, 50, 79, 80, 97, 99, 100, 122, 129, 150, 151 logistic model, 1, 2

© Springer International Publishing AG 2017 R. Cherniha, V. Davydovych, Nonlinear Reaction-Diffusion Systems, Lecture Notes in Mathematics 2196, DOI 10.1007/978-3-319-65467-6

157

158 manifold, 5, 6, 9, 14, 15, 48–53, 56, 101, 120, 121 Murray equation, 23, 35, 39, 41 mutualism, 46, 78

Neumann condition, 91, 134, 136 non-Lie solution, 30, 36, 37, 149 nonclassical symmetry, 4, 13, 48–50, 72

overdetermined system, 19, 27, 55, 90, 104, 125, 149

plane wave solution, 3, 27, 30, 32, 79, 89, 92, 97 population, 1–3, 26, 35, 46, 61, 78, 91, 92, 110, 135 porous-Fisher equation, 35 porous-Murray equation, 35, 38 potential symmetry, 41 power-law diffusivity, 3, 39, 61 predator–prey interaction, 46, 77, 78 predator–prey model, 135, 136 principal algebra, 79, 99, 109

Index reaction-diffusion equation (RDE), 3, 11, 12, 24, 34, 73 reaction-diffusion system (RDS), 45, 46, 52, 56, 60, 61, 63–65, 68, 80, 99, 119, 122, 124, 130, 137 reaction-diffusion-convection equation (RDCE), 15–17, 19–21, 33, 72 reduction, 6, 97, 99, 151

slow diffusion, 35 smooth function, 5, 8, 13, 14, 46, 48, 57, 68, 87, 99, 119, 127, 128, 139, 149 steady-state point, 22, 24, 31, 32, 110, 147 steady-state solution, 22, 109 system of determining equations, 4, 14, 15, 48, 52, 55, 57, 62, 81, 85, 86, 101, 120, 121, 123

thin film, 146, 147 travelling front, 3, 21–25, 32, 39, 91, 116, 150 travelling wave solution, 21, 31, 107, 109 trivial algebra, 23, 26, 79, 99

weak symmetry, 41, 72 Weierstrass function, 34, 95 Q-conditional symmetry, 4–6, 9, 14, 15, 24, 48–51, 61, 79, 82, 92, 101, 119, 132, 148

Zeldovich equation, 16, 24

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