Valentin F. Zaitsev. Chapman and Hall/CRC ...... 33, pp. 305â309, 1976. Danilov, V. G. and Subochev, P. Yu., Wave solutions of semilinear parabolic equations,.
HANDBOOK OF
NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION UPDATED, REVISED AND EXTENDED Andrei D. Polyanin Valentin F. Zaitsev Chapman and Hall/CRC Press (January 2012) ISBN-10: 1584883553 ISBN-13: 978-1584883555
PREFACE TO THE SECOND EDITION The Handbook of Nonlinear Partial Differential Equations, a unique reference for scientists and engineers, contains over 3,000 nonlinear partial differential equations with solutions, as well as exact, symbolic, and numerical methods for solving nonlinear equations. First-, second-, third-, fourth-, and higher-order nonlinear equations and systems of equations are considered. Equations of parabolic, hyperbolic, elliptic, mixed, and general types are discussed. A large number of new exact solutions to nonlinear equations are described. In total, the handbook contains several times more nonlinear PDEs and exact solutions than any other book currently available. In selecting the material, the authors gave the highest priority to the following five major types of equations: • Equations that arise in various applications (heat and mass transfer theory, wave theory, nonlinear mechanics, hydrodynamics, gas dynamics, plasticity theory, nonlinear acoustics, combustion theory, nonlinear optics, theoretical physics, differential geometry, control theory, chemical engineering sciences, biology, and others). • Equations of general form that depend on arbitrary functions; exact solutions of such equations are of principal value for testing numerical and approximate methods. • Equations for which the general solution or solutions of quite general form, with arbitrary functions, could be obtained. • Equations that involve many free parameters. • Equations whose solutions are reduced to solving linear partial differential equations or linear integral equations. The second edition has been substantially updated, revised, and expanded. More than 1,500 new equations with exact solutions, as well some methods and many examples, have been added. The new edition has been increased by a total of over 1,000 pages. New to the second edition: • • • • •
First-order nonlinear partial differential equations with solutions. Some second-, third-, fourth-, and higher-order nonlinear equations with solutions. Parabolic, hyperbolic, elliptic, and other systems of equations with solutions. Some exact methods and transformations. Symbolic and numerical methods for solving nonlinear PDEs with Maple, Mathematica, and MATLAB. • Many new examples and tables included for illustrative purposes. • A large list of references consisting of over 1,300 sources. • Extensive table of contents and detailed index. Note that the first part of the book can be used as a database of test problems for numerical and approximate methods for solving nonlinear partial differential equations. We would like to express our deep gratitude to Alexei Zhurov and Alexander Aksenov for fruitful discussions and valuable remarks. We are very thankful to Inna Shingareva and Carlos Liz´arraga-Celaya who wrote three chapters (39–41) of the book at our request. i
ii
P REFACE
The authors hope that the handbook will prove helpful for a wide audience of researchers, university and college teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, chemistry, biology, economics, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev September 2011
PREFACE TO THE FIRST EDITION Nonlinear partial differential equations are encountered in various fields of mathematics, physics, chemistry, and biology, and numerous applications. Exact (closed-form) solutions of differential equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. Exact solutions of nonlinear equations graphically demonstrate and allow unraveling the mechanisms of many complex nonlinear phenomena such as spatial localization of transfer processes, multiplicity or absence of steady states under various conditions, existence of peaking regimes, and many others. Furthermore, simple solutions are often used in teaching many courses as specific examples illustrating basic tenets of a theory that admit mathematical formulation. Even those special exact solutions that do not have a clear physical meaning can be used as “test problems” to verify the consistency and estimate errors of various numerical, asymptotic, and approximate analytical methods. Exact solutions can serve as a basis for perfecting and testing computer algebra software packages for solving differential equations. It is significant that many equations of physics, chemistry, and biology contain empirical parameters or empirical functions. Exact solutions allow researchers to design and run experiments, by creating appropriate natural conditions, to determine these parameters or functions. This book contains more than 1,600 nonlinear mathematical physics equations and nonlinear partial differential equations and their solutions. A large number of new exact solutions to nonlinear equations are described. Equations of parabolic, hyperbolic, elliptic, mixed, and general types are discussed. Second-, third-, fourth-, and higher-order nonlinear equations are considered. The book presents exact solutions to equations of heat and mass transfer, wave theory, nonlinear mechanics, hydrodynamics, gas dynamics, plasticity theory, nonlinear acoustics, combustion theory, nonlinear optics, theoretical physics, differential geometry, control theory, chemical engineering sciences, biology, and other fields. Special attention is paid to general-form equations that depend on arbitrary functions; exact solutions of such equations are of principal value for testing numerical and approximate methods. Almost all other equations contain one or more arbitrary parameters (in fact, this book deals with whole families of partial differential equations), which can be fixed by the reader at will. In total, the handbook contains significantly more nonlinear PDEs and exact solutions than any other book currently available.
P REFACE
iii
The supplement of the book presents exact analytical methods for solving nonlinear mathematical physics equations. When selecting the material, the authors have given a pronounced preference to practical aspects of the matter; that is, to methods that allow effectively “constructing” exact solutions. Apart from the classical methods, the book also describes wide-range methods that have been greatly developed over the last decade (the nonclassical and direct methods for symmetry reductions, the differential constraints method, the method of generalized separation of variables, and others). For the reader’s better understanding of the methods, numerous examples of solving specific differential equations and systems of differential equations are given throughout the book. For the convenience of a wide audience with different mathematical backgrounds, the authors tried to do their best, wherever possible, to avoid special terminology. Therefore, some of the methods are outlined in a schematic and somewhat simplified manner, with necessary references made to books where these methods are considered in more detail. Many sections were written so that they could be read independently from each other. This allows the reader to quickly get to the heart of the matter. The handbook consists of chapters, sections, and subsections. Equations and formulas are numbered separately in each subsection. The equations within subsections are arranged in increasing order of complexity. The extensive table of contents provides rapid access to the desired equations. Separate parts of the book may be used by lecturers of universities and colleges for practical courses and lectures on nonlinear mathematical physics equations for graduate and postgraduate students. Furthermore, the books may be used as a database of test problems for numerical and approximate methods for solving nonlinear partial differential equations. We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. The authors hope that this book will be helpful for a wide range of scientists, university teachers, engineers, and students engaged in the fields of mathematics, physics, mechanics, control, chemistry, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev September 2003
CONTENTS Preface to the new edition
xxvii
Preface to the first edition
xxviii
Authors
xxxi
Some notations and remarks
xxxiii
Part I Exact Solutions of Nonlinear Partial Differential Equations 1. First-Order Quasilinear Equations 1.1. Equations with Two Independent Variables Containing Arbitrary Parameters . 1.1.1. Coefficients of Equations Contain Power-Law Functions . . . . . . . . . . . . 1.1.2. Coefficients of Equations Contain Exponential Functions . . . . . . . . . . . . 1.1.3. Coefficients of Equations Contain Hyperbolic Functions . . . . . . . . . . . . . 1.1.4. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . 1.1.5. Coefficients of Equations Contain Trigonometric Functions . . . . . . . . . . 1.2. Equations with Two Independent Variables Containing Arbitrary Functions . . 1.2.1. Equations Contain Arbitrary Functions of One Variable . . . . . . . . . . . . . 1.2.2. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . 1.3. Other Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Equations with Three Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Equations with Arbitrary Number of Independent Variables . . . . . . . . . . 2. First-Order Equations with Two Independent Variables Quadratic in Derivatives 2.1. Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂w 2.1.1. Equations of the Form ∂w ∂x ∂y = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . ∂w ∂w 2.1.2. Equations of the Form f (x, y, w) ∂w ∂x ∂y + g(x, y, w) ∂x = h(x, y, w) . . ∂w ∂w 2.1.3. Equations of the Form f (x, y, w) ∂w ∂x ∂y + g(x, y, w) ∂x + h(x, y, w) ∂w ∂y = s(x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ∂w )2 2.1.4. Equations of the Form ∂w = g(x, y, w) . . . . . . . . . . . ∂x + f (x, y, w) ∂y ( ∂w )2 ∂w 2.1.5. Equations of the Form ∂x + f (x, y, w) ∂x + g(x, y, w) ∂w ∂y = h(x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( )2 ( )2 2.1.6. Equations of the Form f (x, y, w) ∂w +g(x, y, w) ∂w = h(x, y, w) ∂x ∂y ( ∂w )2 ∂w ∂w 2.1.7. Equations of the Form f (x, y) ∂x + g(x, y) ∂x ∂y = h(x, y, w) . . . . 2.1.8. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Equations Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂w 2.2.1. Equations of the Form ∂w ∂x ∂y = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . ( ∂w )2 2.2.2. Equations of the Form f (x, y) ∂w = h(x, y, w) . . . . ∂x + g(x, y, w) ∂y ( ∂w )2 ∂w 2.2.3. Equations of the Form ∂x + f (x, y, w) ∂x + g(x, y, w) ∂w ∂y = h(x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( )2 ( )2 +g(x, y, w) ∂w = h(x, y, w) 2.2.4. Equations of the Form f (x, y, w) ∂w ∂x ∂y v
1 3 3 3 11 14 16 17 19 19 30 35 35 39 43 43 43 45 47 51 59 63 69 73 77 77 79 86 89
vi
C ONTENTS
( )2 ∂w + f (x, y, w) ∂w 2.2.5. Equations of the Form ∂w ∂x ∂x ∂y = g(x, y, w) . . . . . . . . 2.2.6. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92 95
3. First-Order Nonlinear Equations with Two Independent Variables of General Form 3.1. Nonlinear Equations Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . 3.1.1. Equations Contain the Fourth Powers of Derivatives . . . . . . . . . . . . . . . . 3.1.2. Equations Contain Derivatives in Radicands . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Equations Contain Arbitrary Powers of Derivatives . . . . . . . . . . . . . . . . . 3.1.4. More Complicated Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Equations Containing Arbitrary Functions of Independent Variables . . . . . . . . . 3.2.1. Equations Contain One Arbitrary Power of Derivative . . . . . . . . . . . . . . . 3.2.2. Equations Contain Two or Three Arbitrary Powers of Derivatives . . . . . 3.3. Equations Containing Arbitrary Functions of Derivatives . . . . . . . . . . . . . . . . . . 3.3.1. Equations Contain Arbitrary Functions of One Variable . . . . . . . . . . . . . 3.3.2. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . 3.3.3. Equations Contain Arbitrary Functions of Three Variables . . . . . . . . . . . 3.3.4. Equations Contain Arbitrary Functions of Four Variables . . . . . . . . . . . .
99 99 99 101 102 105 107 107 111 113 113 116 120 123
4. First-Order Nonlinear Equations with Three or More Independent Variables 4.1. Nonlinear Equations with Three Variables Quadratic in Derivatives . . . . . . . . . 4.1.1. Equations Contain Squares of One or Two Derivatives . . . . . . . . . . . . . . 4.1.2. Equations Contain Squares of Three Derivatives . . . . . . . . . . . . . . . . . . . . 4.1.3. Equations Contain Products of Derivatives with Respect to Different Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Equations Contain Squares and Products of Derivatives . . . . . . . . . . . . . 4.2. Other Nonlinear Equations with Three Variables Containing Parameters . . . . . 4.2.1. Equations Cubic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Equations Contain Roots and Moduli of Derivatives . . . . . . . . . . . . . . . . . 4.2.3. Equations Contain Arbitrary Powers of Derivatives . . . . . . . . . . . . . . . . . 4.3. Nonlinear Equations with Three Variables Containing Arbitrary Functions . . . 4.3.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Equations with Power Nonlinearity in Derivatives . . . . . . . . . . . . . . . . . . 4.3.3. Equations with Arbitrary Dependence on Derivatives . . . . . . . . . . . . . . . . 4.3.4. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Nonlinear Equations with Four Independent Variables . . . . . . . . . . . . . . . . . . . . . 4.4.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Equations Contain Power-Law Functions of Derivatives . . . . . . . . . . . . . 4.5. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Equations with Power-Law Nonlinearity in Derivatives . . . . . . . . . . . . . . 4.6. Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Equations with Power-Law Nonlinearity in Derivatives . . . . . . . . . . . . . . 4.6.3. Equations Contain Arbitrary Functions of Two Variables . . . . . . . . . . . . 4.6.4. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125 125 125 130 131 133 134 134 136 136 139 139 146 148 150 154 154 156 158 158 161 162 162 167 168 169
C ONTENTS
5. Second-Order Parabolic Equations with One Space Variable
vii 175
5.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 ∂2w 2 5.1.1. Equations of the Form ∂w ∂t = a ∂x2 + bw + cw . . . . . . . . . . . . . . . . . . . . . 175 ∂w ∂2w 2 3 ........ ∂t = a ∂x2 + b0 + b1 w + b2 w + b3 w 2 ∂ w k m n ............. 5.1.3. Equations of the Form ∂w ∂t = a ∂x2 + bw + cw + sw 2 ∂ w 5.1.4. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . ∂2w k ∂w 5.1.5. Equations of the Form ∂w ∂t = a ∂x2 + bw ∂x + f (w) . . . . . . . . . . . . . . . . ∂w ∂2w 5.1.6. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) ∂x + g(x, t, w) . . . . . . . ( ) 2 ∂ w ∂w 2 5.1.7. Equations of the Form ∂w + f (x, t, w) . . . . . . . . . . . . ∂t = a ∂x2 + b ∂x ( ) ∂w ∂2w 5.1.8. Equations of the Form ∂t = a ∂x2 + f x, t, w, ∂w ∂x . . . . . . . . . . . . . . . . . ( ) k ∂ 2 w + f x, t, w, ∂w 5.1.9. Equations of the Form ∂w = aw .............. 2 ∂t ∂x ∂x( ) ∂ ∂w ∂w m 5.1.10. Equations of the Form ∂t = a ∂x w ∂x . . . . . . . . . . . . . . . . . . . . . . . ( m ∂w ) ∂ k 5.1.11. Equations of the Form ∂w = a ∂t ∂x w ∂x + bw . . . . . . . . . . . . . . . . . 5.1.12. Equations of the Form ∂w ∂t = ( m ∂w ) ∂ a ∂x w ∂x + bw + c1 wk1 + c2 wk2 + c3 wk3 . . . . . . . . . . . . . . . . . . . . . [ ] ∂ (w) ∂w + g(w) . . . . . . . . . . . . . . . . 5.1.13. Equations of the Form ∂w ∂t = ∂x f ∂x ( m ∂w ) ∂w ∂ 5.1.14. Equations of the Form ∂t = a ∂x w ∂x + f (x, t, w) . . . . . . . . . . . . [ ] ∂w ∂w ∂ 5.1.15. Equations of the Form ∂w ∂t = ∂x [f (w) ∂x ] + g(w) ∂x . . . .). . . . . . . . . ( ∂w ∂w ∂ 5.1.16. Equations of the Form ∂t = ∂x f (w) ∂x + g x, t, w, ∂w ∂x . . . . . . . .
5.1.2. Equations of the Form
176 182 185 186 190 193 195 197 210 219
228 233 236 237 242 5.1.17. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.2. Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂2w 5.2.1. Equations of the Form ∂w b1 eλw + b2 e2λw . . . . . . . . . . . . ∂t = a ∂x2( + b0 + ) ∂ 5.2.2. Equations of the Form ∂w eλw ∂w ∂t = a ∂x ∂x ] + f (w) . . . . . . . . . . . . . . . . . . [ ∂w ∂ ∂w 5.2.3. Equations of the Form ∂t = ∂x f (w) ∂x + g(w) . . . . . . . . . . . . . . . . . . 5.2.4. Other Equations Explicitly Independent of x and t . . . . . . . . . . . . . . . . . . 5.2.5. Equations Explicitly Dependent on x and/or t . . . . . . . . . . . . . . . . . . . . . .
254 254 256 259 262 266
5.3. Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Equations Involving Hyperbolic Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Equations Involving Hyperbolic Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Equations Involving Hyperbolic Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Equations Involving Hyperbolic Cotangent . . . . . . . . . . . . . . . . . . . . . . . .
268 268 269 270 270
5.4. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 ∂2w 5.4.1. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . 271 5.4.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.5. Equations with Trigonometric Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Equations Involving Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2. Equations Involving Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3. Equations Involving Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4. Equations Involving Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5. Equations Involving Inverse Trigonometric Functions . . . . . . . . . . . . . . .
276 276 277 278 278 279
viii
C ONTENTS
5.6. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 ∂2w 5.6.1. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . . 279 ∂w ∂2w ∂w ∂t = a ∂x2 + f (x, t) ∂x + g(x, t, w) . . . . . . . . . . ∂2w ∂w 5.6.3. Equations of the Form ∂w ∂t = a ∂x2 + f (x, t, w) ∂x + g(x, t, w) . . . . . . . ( ∂w )2 ∂2w 5.6.4. Equations of the Form ∂w + f (x, t, w) . . . . . . . . . . . . ∂t = a ∂x2 + b ∂x ( ∂w )2 ∂w ∂2w 5.6.5. Equations of the Form ∂t = a ∂x2 +b ∂x +f (x, t, w) ∂w ∂x +g(x, t, w) ∂w 5.6.6. Equations of the Form ∂t = ( )2 2 a ∂∂xw2 + f (x, t, w) ∂w + g(x, t, w) ∂w ∂x ∂x + h(x, t, w) . . . . . . . . . . . . . . . . ( ) ∂w ∂2w 5.6.7. Equations of the Form ∂t = a ∂x2 + f x, t, w, ∂w ∂x . . . . . . . . . . . . . . . . . ( ) ∂2w ∂w 5.6.8. Equations of the Form ∂w ............ ∂t = f (x, t) ∂x2 + g x, t, w, ∂x 2 ∂ w ∂w 5.6.9. Equations of the Form ∂w ∂t = aw ∂x2 + f (x, t, w) ∂x + g(x, t, w) . . . . . . 5.6.10. Equations of the Form ∂w = ∂t ( ∂w )2 ∂2w (aw + b) ∂x2 + f (x, t, w) ∂x + g(x, t, w) ∂w ∂x + h(x, t, w) . . . . . . . . ∂w ∂2w m 5.6.11. Equations of the Form ∂t = aw ∂x2 + f (x, t) ∂w ∂x + g(x, t, w) . . . . . ( ∂w ) ∂w ∂ 5.6.12. Equations of the Form ∂t = a ∂x w ∂x + f (x, t) ∂w ∂x + g(x, t, w) . . ( m ∂w ) ∂w ∂ 5.6.13. Equations of the Form ∂t = a ∂x w ∂x + f (x, t) ∂w ∂x + g(x, t, w) . ( λw ∂w ) ∂w ∂ 5.6.14. Equations of the Form ∂t = a ∂x e ∂x + f (x, t, w) . . . . . . . . . . . . [ ] ( ) ∂w ∂w ∂ 5.6.15. Equations of the Form ∂w ∂t = ∂x f (w) ∂x + g x, t, w, ∂x . . . . . . . . ∂2w 5.6.16. Equations of the Form ∂w ∂t = f (x, w) ∂x2 . . . . . . . . . . . . . . . . . . . . . . . . ) ( ∂2w ∂w ....... 5.6.17. Equations of the Form ∂w ∂t = f (x, t, w) ∂x2 + g x, t, w, ∂x ) ) ( ( ∂w ∂w ∂ 2 w ∂w 5.6.18. Equations of the Form ∂t = f x, w, ∂x ∂x2 + g x, t, w, ∂x . . . . .
5.6.2. Equations of the Form
284 286 290 293 294 298 299 303 306 308 311 312 316 318 327 329
339 5.6.19. Evolution Equations Nonlinear in the Second Derivative . . . . . . . . . . . 346 5.6.20. Nonlinear Equations of the Thermal (Diffusion) Boundary Layer . . . 346
5.7. Nonlinear Schr¨odinger Equations and Related Equations . . . . . . . . . . . . . . . . . . ∂2w 5.7.1. Equations of the Form i ∂w ∂t + ∂x2 + f (|w|)w = 0 Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . .(. . . . . .). . . . . . . . . . . . . . . . . . . . . . . . . 1 ∂ n ∂w 5.7.2. Equations of the Form i ∂w ∂t + xn ∂x x ∂x + f (|w|)w = 0 Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 5.7.4. Equations with Cubic Nonlinearities Involving Arbitrary Functions . . . 5.7.5. Equations of General Form Involving Arbitrary Functions of a Single Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.6. Equations of General Form Involving Arbitrary Functions of Two Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
348
352 354 355
6. Second-Order Parabolic Equations with Two or More Space Variables
367
6.1. Equations with Two Space Variables Involving Power-Law Nonlinearities ... [ ] ∂ [ ] ∂ ∂w ∂w ∂w p 6.1.1. Equations of the Form ∂t = ∂x f (x) ∂x + ∂y g(y) ∂y + aw . . . . . ( n ∂w ) ( k ∂w ) ∂ ∂ 6.1.2. Equations of the Form ∂w ............ ∂t = a ∂x w ∂x + b ∂y w ∂y [ ] ∂ [ ] ∂w ∂ ∂w ∂w 6.1.3. Equations of the Form ∂t = ∂x f (w) ∂x + ∂y g(w) ∂y + h(w) . . . . 6.1.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367 367
348
358 362
368 374 377
C ONTENTS
ix
6.2. Equations with Two Space Variables Involving Exponential Nonlinearities . . . 384 [ ] ∂ [ ] ∂w ∂ ∂w ∂w λw 6.2.1. Equations of the Form ∂t = ∂x f (x) ∂x + ∂y g(y) ∂y + ae . . . . . 384 ( βw ∂w ) ( λw ∂w ) ∂ ∂ 6.2.2. Equations of the Form ∂w ∂t = a ∂x e ∂x + b ∂y e ∂y + f (w) . . . 385 6.3. Other Equations with Two Space Variables Involving Arbitrary Parameters . . . 388 6.3.1. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 388 6.3.2. Equations with Trigonometrical Nonlinearities . . . . . . . . . . . . . . . . . . . . . 389 6.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions . . . . . . . . . [. . . . . . . . .] . . . . . [. . . . . . . .]. . . . . . . . . . . . . ∂ ∂ ∂w ∂w 6.4.2. Equations of the Form ∂w ∂t = ∂x f (x) ∂x + ∂y g(y) ∂y + h(w) . . . . . [ ] [ ] ∂ ∂w ∂ ∂w 6.4.3. Equations of the Form ∂w ∂t = ∂x f (w) ∂x + ∂y g(w) ∂y + h(t, w) . . 6.4.4. Other Equations Linear in the Highest Derivatives . . . . . . . . . . . . . . . . . . 6.4.5. Nonlinear Diffusion Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . 6.4.6. Equations Nonlinear in the Highest Derivatives . . . . . . . . . . . . . . . . . . . . . 6.5. Equations with Three or More Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1. Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Heat Equations with Power-Law or Exponential Temperature-Dependent Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . 6.5.3. Equations of Heat and Mass Transfer in Anisotropic Media . . . . . . . . . . 6.5.4. Other Equations with Three Space Variables . . . . . . . . . . . . . . . . . . . . . . . 6.5.5. Equations with n Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
390 390 392 394 398 402 404 406 406 409 412 414 417
6.6. Nonlinear Schr¨odinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.6.1. Two-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.6.2. Three and n-Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 7. Second-Order Hyperbolic Equations with One Space Variable
433
7.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 2 ∂2w n 2n−1 . . . . . . . . . . . . 433 7.1.1. Equations of the Form ∂∂tw 2 = ∂x2 + aw + bw + cw 7.1.2. Equations of the Form 7.1.3. Equations of the Form 7.1.4. Equations of the Form
7.1.7. 7.1.8. 7.1.9.
2
= a ∂∂xw2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . 436 ) ( 2 . . . . . . . . . . . . . . . . 439 = a ∂∂xw2 + f x, t, w, ∂w ∂x ) ( ∂2w ∂w = f (x) ∂x2 + g x, t, w, ∂x . . . . . . . . . . . . . 442 2
= awn ∂∂xw2 + f (x, w) . . . . . . . . . . . . . . . . . . . ( n ∂w ) ∂ Equations of the Form = a ∂x w ∂x . . . . . . . . . . . . . . . . . . . . . . . . ( ) ∂ k .................. Equations of the Form wn ∂w = a ∂x ∂x + bw ( ) ∂ k1 k2 k3 . . Equations of the Form wn ∂w = a ∂x ∂x + b1 w + b2 w + b3 w Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.5. Equations of the Form 7.1.6.
∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2 ∂2w ∂t2
447 450 454 458 460
7.2. Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 2 ∂2w βw + ceγw . . . . . . . . . . . . . . . . . . 469 7.2.1. Equations of the Form ∂∂tw 2 = a ∂x2 + be ∂2w ∂t2 ∂2w ∂t2
2
= a ∂∂xw2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . 472 ( ) 2 7.2.3. Equations of the Form = f (x) ∂∂xw2 + g x, t, w, ∂w . . . . . . . . . . . . . 475 ∂x 7.2.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 7.2.2. Equations of the Form
x
C ONTENTS
7.3. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. Sine-Gordon Equation and Other Equations with Trigonometric Nonlinearities . . . . . . . . .2. . . . . . . . . . . . . .[ . . . . . . . .]. . . . . . . . . . . . . . . . . . . ∂w ∂ ∂w 7.3.4. Equations of the Form ∂∂tw ................. 2 + a ∂t = ∂x f (w) ∂x [ ] ∂2w ∂w ∂ ∂w 7.3.5. Equations of the Form ∂t2 + f (w) ∂t = ∂x g(w) ∂x . . . . . . . . . . . . . . 7.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ∂2w 7.4.1. Equations of the Form ∂∂tw 2 = a ∂x2 + f (x, t, w) . . . . . . . . . . . . . . . . . . . . ( ) 2 ∂2w ∂w 7.4.2. Equations of the Form ∂∂tw ................ 2 = a ∂x2 + f x, t, w, ∂x ( ) ∂2w ∂2w ∂w 7.4.3. Equations of the Form ∂t2 = f (x) ∂x2 + g x, t, w, ∂x . . . . . . . . . . . . . ( ) 2 ∂2w ∂w ............ 7.4.4. Equations of the Form ∂∂tw 2 = f (w) ∂x2 + g x, t, w, ∂x ( ) ∂2w ∂2w ∂w 7.4.5. Equations of the Form ∂t2 = f (x, w) ∂x2 + g x, t, w, ∂x . . . . . . . . . . ) ( 2 ∂2w ∂w 7.4.6. Equations of the Form ∂∂tw ........... 2 = f (t, w) ∂x2 + g x, t, w, ∂x 7.4.7. Other Equations Linear in the Highest Derivatives . . . . . . . . . . . . . . . . . . ) ( ∂2w ∂w ....................... 7.5. Equations of the Form ∂x∂y = F x, y, w, ∂w , ∂x ∂y
485 485 486 490 494 496 499 499 505 511 517 526 529 530 540
∂2w ∂x∂y
7.5.1. Equations Involving Arbitrary Parameters of the Form = f (w) . . 540 7.5.2. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 544 7.5.3. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 546 8. Second-Order Hyperbolic Equations with Two or More Space Variables 8.1. Equations with Two Space Variables Involving Power-Law Nonlinearities . . . [ ] ∂ [ ] 2 ∂w ∂w ∂ p ..... 8.1.1. Equations of the Form ∂∂tw 2 = ∂x f (x) ∂x + ∂y g(y) ∂y + aw ( ) ( ) 2 ∂ ∂ n ∂w k ∂w 8.1.2. Equations of the Form ∂∂tw ........... 2 = a ∂x w ∂x + b ∂y w ∂y [ ] [ ] ∂2w ∂w ∂ ∂w ∂ 8.1.3. Equations of the Form ∂t2 = ∂x f (w) ∂x + ∂y g(w) ∂y . . . . . . . . . . 8.1.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Equations with Two Space Variables Involving Exponential Nonlinearities . . . [ ] ∂ [ ] 2 ∂w ∂w ∂ λw . . . . 8.2.1. Equations of the Form ∂∂tw 2 = ∂x f (x) ∂x + ∂y g(y) ∂y + ae ( βw ∂w ) ( λw ∂w ) 2 ∂ ∂ 8.2.2. Equations of the Form ∂∂tw ......... 2 = a ∂x e ∂x + b ∂y e ∂y 8.2.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Nonlinear Telegraph Equations with Two Space Variables . . . . . . . . . . . . . . . . . 8.3.1. Equations Involving Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . 8.3.2. Equations Involving Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . 8.4. Equations with Two Space Variables Involving Arbitrary Functions . . . . . . . . . [ ] ∂ [ ] 2 ∂ ∂w ∂w 8.4.1. Equations of the Form ∂∂tw 2 = ∂x f (x) ∂x + ∂y g(y) ∂y + h(w) . . . . [ ] ∂ [ ] 2 ∂ ∂w ∂w 8.4.2. Equations of the Form ∂∂tw 2 = ∂x f (w) ∂x + ∂y g(w) ∂y + h(w) . . . 8.4.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Equations with Three Space Variables Involving Arbitrary Parameters . . . . . . . 2 8.5.1. Equations of the Form ∂∂tw 2 = [ ] ∂ [ ] ∂[ ] ∂ ∂w ∂w ∂w p ................. ∂x f (x) ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z + aw 8.5.2.
2 Equations of the Form ∂∂tw 2 = [ ] ∂ [ ] ∂[ ] ∂ ∂w ∂w ∂w λw ∂x f (x) ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z + ae
553 553 553 555 565 570 574 574 577 582 583 583 587 589 589 593 599 604 604
. . . . . . . . . . . . . . . . . 606
C ONTENTS
xi
2
8.5.3. Equations of the Form ∂∂tw = ( n ∂w ) ( m ∂w2) ( k ∂w ) ∂ ∂ ∂ a ∂x w ∂x + b ∂y w ∂y + c ∂z w ∂z + swp . . . . . . . . . . . . . . . . . . 608 2
= 8.5.4. Equations of the Form ∂∂tw ( λ w ∂w ) ( λ w2∂w ) ( λ w ∂w ) ∂ ∂ ∂ βw . . . . . . . . . . . . . 615 1 2 3 a ∂x e ∂x + b ∂y e ∂y + c ∂z e ∂z + se 8.6. Equations with Three or More Space Variables Involving Arbitrary Functions 624 2 8.6.1. Equations of the Form ∂∂tw 2 = [ ] [ ] ∂[ ] ∂ ∂w ∂ ∂w ∂w f (x) + f (y) + f (z) 1 2 3 ∂x ∂x ∂y ∂y ∂z ∂z + g(w) . . . . . . . . . . . . . . 624 2
8.6.2. Equations of the Form ∂∂tw 2 = [ ] ∂ [ ] ∂[ ] ∂ ∂w ∂w ∂w ∂x f1 (w) ∂x + ∂y f2 (w) ∂y + ∂z f3 (w) ∂z + g(w) . . . . . . . . . . . . . 629 8.6.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 9. Second-Order Elliptic Equations with Two Space Variables 641 9.1. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 2 2 9.1.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = aw + bwn + cw2n−1 . . . . . . . . . . . . 641 9.1.2. Equations of the Form 9.1.3. Equations of the Form 9.1.4. Equations of the Form
2 ∂2w + ∂∂yw2 = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . ∂x2 ) ( 2 ∂2w ∂w ........... + a ∂∂yw2 = F x, y, w, ∂w ∂x , ∂y ∂x2[ ] [ ] ∂ ∂w ∂ + ∂y f2 (x, y) ∂w = g(w) . . . . ∂x [f1 (x, y) ∂x ] [ ] ∂y ∂ ∂w ∂ ∂w ∂x f1 (w) ∂x + ∂y f2 (w) ∂y = g(w) . . . . . . .
9.1.5. Equations of the Form 9.1.6. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 9.2. Equations with Exponential Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 9.2.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = a + beβw + ceγw . . . . . . . . . . . . . . . 9.2.2. Equations of the Form 9.2.3. Equations of the Form
2 ∂2w + ∂∂yw2 = f (x, y, w) . . . . . . . . . . . . . . . . . . . . . ∂x2[ ] ∂ [ ] ∂w ∂ + ∂y f2 (x, y) ∂w = g(w) . . . . ∂x [f1 (x, y) ∂x ∂y ] [ ] ∂ ∂w ∂ ∂w ∂x f1 (w) ∂x + ∂y f2 (w) ∂y = g(w) . . . . . . .
9.2.4. Equations of the Form 9.2.5. Other Equations Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . 9.3. Equations Involving Other Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. Equations with Hyperbolic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Equations with Logarithmic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Equations with Trigonometric Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 9.4. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 9.4.1. Equations of the Form ∂∂xw2 + ∂∂yw2 = F (x, y, w) . . . . . . . . . . . . . . . . . . . . ( ) 2 2 ∂w 9.4.2. Equations of the Form a ∂∂xw2 + b ∂∂yw2 = F x, y, w, ∂w .......... ∂x , ∂y 9.4.3. Heat Mass Equations of the Form [ and ∂w ] Transfer [ ] ∂ ∂ ∂w f (x) + g(y) = h(w) .............................. ∂x ∂x ∂y ∂y[ ] ∂ [ ] ∂ ∂w 9.4.4. Equations of the Form ∂x f (x, y, w) ∂x + ∂y g(x, y, w) ∂w ∂y = h(x, y, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.5. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
644 645 646 648 654 662 662 664 665 668 671 675 675 677 680 682 682 690 697 699 707
10. Second-Order Elliptic Equations with Three or More Space Variables 713 10.1. Equations with Three Space Variables Involving Power-Law Nonlinearities . 713 10.1.1. Equations of] the Form [ [ ] ∂[ ] ∂ ∂w ∂ ∂w ∂w p . . . . . . . . . . . . . . 713 f (x) ∂x ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z = aw
xii
C ONTENTS
10.1.2. Equations of] the Form [ [ ] ∂[ ] ∂ ∂w ∂ ∂w ∂w f (w) ∂x ∂x + ∂y g(w) ∂y + ∂z g(w) ∂z = 0 . . . . . . . . . . . . . . . . 716 10.2. Equations with Three Space Variables Involving Exponential Nonlinearities . 722 10.2.1. Equations of] the Form [ [ ] ∂[ ] ∂ ∂w ∂ ∂w ∂w λw . . . . . . . . . . . . . 722 f (x) ∂x ∂x + ∂y g(y) ∂y + ∂z h(z) ∂z = ae 10.2.2. Equations the ( λ wof ) Form ( λ w ∂w ) ( λ w ∂w ) ∂ ∂w ∂ ∂ βw . . . . . . 725 1 2 2 a1 ∂x e ∂x + a2 ∂y e ∂y + a3 ∂y e ∂y = be 10.3. Three-Dimensional Equations Involving Arbitrary Functions . . . . . . . . . . . . . . 10.3.1. Heat Equations [ and Mass ] Transfer [ ] ∂ [of the Form ] ∂ ∂w ∂ ∂w ∂w f (x) + f (y) ∂x 1 ∂x ∂y 2 ∂y + ∂z f3 (z) ∂z = g(w) . . . . . . . . . . . 10.3.2. Heat and Mass Transfer Equations with Complicating Factors . . . . . 10.3.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Equations with n Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1. Equations of the [ ] Form ∂ [ ] ∂w ∂w ∂ f (x ) 1 1 ∂x1 ∂x1 + · · · + ∂xn fn (xn ) ∂xn = g(x1 , . . . , xn , w) . . . . . 10.4.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Second-Order Equations Involving Mixed Derivatives and Some Other Equations 11.1. Equations Linear in the Mixed Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1. Calogero Equation and Related Equations . . . . . . . . . . . . . . . . . . . . . . 11.1.2. Khokhlov–Zabolotskaya and Related Equations . . . . . . . . . . . . . . . . . 11.1.3. Equation of Unsteady Transonic Gas Flows . . . . . . . . . . . . . . . . . . . . . ) ( ∂2w ∂w ∂ 2 w ∂w ∂w ..... 11.1.4. Equations of the Form ∂w ∂y ∂x∂y − ∂x ∂y 2 = F x, y, ∂x , ∂y 11.1.5. Other Equations with Two Independent Variables . . . . . . . . . . . . . . . . 11.1.6. Other Equations with Three and More Independent Variables . . . . . 11.2. Equations Quadratic in the Highest Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 11.2.1. Equations of the Form ∂∂xw2 ∂∂yw2 = F (x, y) . . . . . . . . . . . . . . . . . . . . . . ( ∂ 2 w )2 ∂ 2 w ∂ 2 w − 2 ∂y2 = F (x, y) . . . . . . . . . 11.2.2. Monge–Amp`ere Equation ∂x∂y ) ( ∂ 2 w )2 ∂ 2 w ∂x ( 2 ∂w 11.2.3. Equations of the Form ∂x∂y − ∂x2 ∂∂yw2 = F x, y, w, ∂w .. ∂x , ∂y ( ∂ 2 w )2 ∂2w ∂2w 11.2.4. Equations of the Form ∂x∂y = f (x, y) ∂x2 ∂y2 + g(x, y) . . . . . . . 11.2.5. Other Equations with Two Independent Variables . . . . . . . . . . . . . . . . 11.2.6. Pleba´nski Heavenly Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Bellman Type Equations and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1. Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2. Equations with Power-Law Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 12. Second-Order Equations of General Form 12.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ∂w ∂ 2 w ) 12.1.1. Equations of the Form ∂w ∂t = F (w, ∂x , ∂x2 ). . . . . . . . . . . . . . . . . . . ∂w ∂ 2 w 12.1.2. Equations of the Form ∂w ∂t = F (t, w, ∂x , ∂x2 ) . . . . . . . . . . . . . . . . . . ∂w ∂ 2 w 12.1.3. Equations of the Form ∂w ∂t = F (x, w, ∂x , ∂x2 ). . . . . . . . . . . . . . . . . ∂w ∂ 2 w 12.1.4. Equations of the Form ∂w ............... 2 ∂t( = F x, t, w, ∂x , ∂x) ∂w ∂w ∂ 2 w 12.1.5. Equations of the Form F x, t, w, ∂t , ∂x , ∂x2 = 0 . . . . . . . . . . . . . 12.1.6. Equations with Three Independent Variables . . . . . . . . . . . . . . . . . . . .
730 730 734 737 739 739 742 745 745 745 749 756 760 762 770 772 772 774 787 794 798 802 805 805 808 811 811 811 820 825 830 836 838
C ONTENTS
xiii
12.2. Equations Involving Two or More Second Derivatives . . . . . . . . . . . . . . . . . . . . ( ∂w ∂ 2 w ) 2 12.2.1. Equations of the Form ∂∂tw ................... 2 = F w, ∂x , ∂x2 ( ) ∂2w ∂w ∂w ∂ 2 w 12.2.2. Equations of the Form ∂t2 = F x, t, w, ∂x , ∂t , ∂x2 . . . . . . . . . . . 12.2.3. Equations Linear in the Mixed Derivative . . . . . . . . . . . . . . . . . . . . . . . 12.2.4. Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.5. Equations with n Independent Variables . . . . . . . . . . . . . . . . . . . . . . . .
839 839 843 847
13. Third-Order Equations 13.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂3w ∂w 13.1.1. Korteweg–de Vries Equation ∂w ∂t + a ∂x3 + bw ∂x = 0 . . . . . . . . . . . . 13.1.2. Cylindrical, Spherical, and Modified Korteweg–de Vries Equations ∂3w ∂w 13.1.3. Generalized Korteweg–de Vries Equation ∂w ∂t +a ∂x3 +f (w) ∂x = 0 13.1.4. Equations Reducible to the Korteweg–de Vries Equation . . . . . . . . . ( ∂w ) ∂3w 13.1.5. Equations of the Form ∂w + a + F w, =0 .............. 3 ∂t ∂x ( ∂x ∂w ) ∂w ∂3w 13.1.6. Equations of the Form ∂t + a ∂x3 + F x, t, w, ∂x = 0 . . . . . . . . . . ( ) ∂w ∂ 2 w ∂ 3 w 13.1.7. Equations of the Form ∂w = F x, w, , ............. , 2 3 ∂t ∂x ∂x ∂x 13.2. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1. Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3. Hydrodynamic Boundary Layer Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1. Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2. Steady Boundary Layer Equations for Non-Newtonian Fluids . . . . . 13.3.3. Unsteady Boundary Layer Equations for a Newtonian Fluid . . . . . . . 13.3.4. Unsteady Boundary Layer Equations for Non-Newtonian Fluids . . . 13.3.5. Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Equations of Motion of Ideal Fluid (Euler Equations) . . . . . . . . . . . . . . . . . . . . 13.4.1. Stationary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2. Nonstationary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Other Third-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1. Equations Involving Second-Order Mixed Derivatives . . . . . . . . . . . . 13.5.2. Equations Involving Third-Order Mixed Derivatives . . . . . . . . . . . . . 13.5.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
857 857 857 866 871 875 879 882 884 896 896 900 903
14. Fourth-Order Equations 14.1. Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . . . ) ( ∂4w ∂w 14.1.1. Equations of the Form ∂w ............. ∂t = a ∂x4 + F x, t, w, ∂x 14.1.2. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Boussinesq Equation and Its Modifications . . . . . . . . . . . . . . . . . . . . . 14.2.2. Other Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . 14.2.3. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3. Equations Involving Mixed Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1. Kadomtsev–Petviashvili Equation and Related Equations . . . . . . . . . 14.3.2. Stationary Hydrodynamic Equations (Navier–Stokes Equations) . . . 14.3.3. Nonstationary Hydrodynamic Equations (Navier–Stokes Equations) 14.3.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
977 977 977 982 987 987 993 998 1000 1000 1003 1013 1028
849 853
903 911 917 930 935 938 938 942 949 949 958 973
xiv
C ONTENTS
15. Equations of Higher Orders
1031
15.1. Equations Involving the First Derivative in t and Linear in the Highest Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1. Fifth-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2. Some Equations with Sixth- to Ninth-Order . . . . . . . . . . . . . . . . . . . . . ∂nw 15.1.3. Equations of the Form ∂w ∂t = a ∂xn + f (x, t, w) . . . . . . . . . . . . . . . . . . ∂nw ∂w 15.1.4. Equations of the Form ∂w ∂t = a ∂xn + f (w) ∂x . . . . . . . . . . . . . . . . . . . n ∂ w t, w) ∂w 15.1.5. Equations of the Form ∂w g(x, t, w) . . . . ∂t = a ∂xn + f (x, ∂x + ( ) n ∂w ∂ w ∂w 15.1.6. Equations of the Form ∂t = a ∂xn + F x, t, w, ∂x . . . . . . . . . . . . . ( ) ∂w ∂nw ∂ n−1 w 15.1.7. Equations of the Form ∂w = a + F x, t, w, , . . . , .. n n−1 ∂t ∂x ∂x ∂x
1031 1031 1039 1043 1044 1047 1051
∂w ∂t
n = aw ∂∂xw n
+ f (x, t, w) ∂w ∂x
15.1.8. Equations of the Form + g(x, t, w) . . 15.1.9. Other Equations Involving Arbitrary Parameters and/or Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.10. Nonlinear Equations Involving Arbitrary Linear Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.11. Equations of the Burgers and the Korteweg–de Vries Hierarchies and Related Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1057 1059 1062 1065 1067
15.2. General Form Equations Involving the First Derivative in t . . . . . . . . . . . . . . . . ( ∂w ) ∂nw 15.2.1. Equations of the Form ∂w ∂t = F (w, ∂x , . . . , ∂xn ). . . . . . . . . . . . . . . ∂w ∂nw 15.2.2. Equations of the Form ∂w ∂t = F (t, w, ∂x , . . . , ∂xn ) . . . . . . . . . . . . . . ∂w ∂nw 15.2.3. Equations of the Form ∂w ∂t = F (x, w, ∂x , . . . , ∂xn ). . . . . . . . . . . . . ∂w ∂nw 15.2.4. Equations of the Form ∂w ........... ∂t = F x, t, w, ∂x , . . . , ∂xn
1070 1070 1077 1079 1084
15.3. Equations Involving the Second Derivative in t . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ∂nw 15.3.1. Equations of the Form ∂∂tw 2 = a ∂xn + f (x, t, w) . . . . . . . . . . . . . . . . . ) ( 2 ∂nw ∂w ............ 15.3.2. Equations of the Form ∂∂tw 2 = a ∂xn + F x, t, w, ∂x ( n−1 ) ∂2w ∂nw ∂w 15.3.3. Equations of the Form ∂t2 = a ∂xn + F x, t, w, ∂x , . . . , ∂∂xn−1w .
1088 1088
∂2w ∂t2
1089 1094
∂nw ∂xn
15.3.4. Equations of the Form + f (x, t, w) ∂w = aw ∂x + g(x, t, w) . 1098 15.3.5. Other Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . 1099 15.3.6. Equations Involving Arbitrary Differential Operators . . . . . . . . . . . . . 1101 15.4. Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 15.4.1. Equations Involving Mixed Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 1103 n ∂mw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 15.4.2. Equations Involving ∂∂xw n and ∂y m 16. Systems of Two First-Order Partial Differential Equations
1115
∂w 16.1. Systems of the Form ∂u ∂x = F (u, w), ∂t = G(u, w) . . . . . . . . . . . . . . . . . . . . . . 1115 16.1.1. Systems Involving Arbitrary Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1115 16.1.2. Systems Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1117
16.2. Other Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 16.2.1. Gas Dynamic Type Systems Linearizable with the Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 16.2.2. Other Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130
xv
C ONTENTS
17. Systems of Two Parabolic Equations
1133
∂2u ∂x2
∂2w ∂x2
17.1. Systems of the Form ∂u + F (u, w), ∂w + G(u, w) . . . . . . . . ∂t = a ∂t = b 17.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . 17.1.3. Arbitrary Functions Depend on the Product of Powers of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.4. Arbitrary Functions Depend on Sum or Difference of Squares of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.5. Arbitrary Functions Depend on the Unknowns in a Complex Way . 17.1.6. Some Systems Depending on Arbitrary Parameters . . . . . . . . . . . . . . ( n ∂u ) a ∂ 17.2. Systems of the Form ∂u = n x ∂x x ∂x + F (u, w), ( n ∂w ) ∂t ∂w b ∂ = x + G(u, w) ........................................ ∂t xn ∂x ∂x 17.2.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . 17.2.3. Arbitrary Functions Depend on the Product of Powers of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4. Arbitrary Functions Depend on Sum or Difference of Squares of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.5. Arbitrary Functions Have Different Arguments . . . . . . . . . . . . . . . . . . 17.3. Other Systems of Two Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1. Second-Order Equations Involving Real Functions of Real Variables 17.3.2. Second-Order Nonlinear Equations of Laser Systems . . . . . . . . . . . . 17.3.3. Systems Involving Third-Order Evolution Equations . . . . . . . . . . . . .
1156
18. Systems of Two Second-Order Klein–Gordon Type Hyperbolic Equations
1173
∂2u ∂t2
2 = a ∂∂xu2
∂2w ∂t2
2 = b ∂∂xw2
1133 1133 1137 1145 1146 1148 1150
1157 1159 1162 1163 1164 1165 1165 1170 1172
18.1. Systems of the Form + F (u, w), + G(u, w) . . . . . . . 18.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . 18.1.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( n ∂u ) 2 ∂ x ∂x + F (u, w), 18.2. Systems of the Form ∂∂t2u = xan ∂x ( n ∂w ) ∂2w b ∂ = xn ∂x x ∂x + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂t2 18.2.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . 18.2.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1173
19. Systems of Two Elliptic Equations 19.1. Systems of the Form ∆u = F (u, w), ∆w = G(u, w) . . . . . . . . . . . . . . . . . . . . . 19.1.1. Arbitrary Functions Depend on a Linear Combination of the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2. Arbitrary Functions Depend on the Ratio of the Unknowns . . . . . . . 19.1.3. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2. Other Systems of Two Second-Order Elliptic Equations . . . . . . . . . . . . . . . . . . 19.3. Von K´arm´an Equations (Fourth-Order Elliptic Equations) . . . . . . . . . . . . . . . . .
1185 1185
1173 1175 1177 1178 1178 1181 1183
1185 1187 1189 1191 1192
xvi
C ONTENTS
20. First-Order Hydrodynamic and Other Systems Involving Three or More Equations 20.1. Equations of Motion of Ideal Fluid (Euler Equations) . . . . . . . . . . . . . . . . . . . . 20.1.1. Euler Equations in Various Coordinate Systems . . . . . . . . . . . . . . . . . 20.1.2. Two-Dimensional Euler Equations for Incompressible Ideal Fluid (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.3. Other Solutions with Two Nonzero Components of the Fluid Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.4. Rotationally Symmetric Motions of Fluid . . . . . . . . . . . . . . . . . . . . . . . 20.1.5. Euler Equations for Barotropic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . 20.2. Adiabatic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2. One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.3. Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.4. Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3. Systems Describing Fluid Flows in the Atmosphere, Seas, and Oceans . . . . . 20.3.1. Equations of Breezes and Monsoons . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2. Equations of Atmospheric Circulation in the Equatorial Region . . . . 20.3.3. Equations of Dynamic Convection in the Sea . . . . . . . . . . . . . . . . . . . . 20.3.4. Equations of Flows in the Baroclinic Layer of the Ocean . . . . . . . . . 20.4. Chromatography Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1. Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2. Generalized Langmuir Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3. Power Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.4. Exponential Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5. Other Hydrodynamic-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1. Hydrodynamic-Type Systems of Diagonal Form . . . . . . . . . . . . . . . . . 20.5.2. Hydrodynamic-Type Systems of Nondiagonal Form . . . . . . . . . . . . . 20.6. Ideal Plasticity with the von Mises Yield Criterion . . . . . . . . . . . . . . . . . . . . . . . 20.6.1. Two-Dimensional Equations. Plane Case . . . . . . . . . . . . . . . . . . . . . . . 20.6.2. Axisymmetric Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6.3. Three-Dimensional Equations. Steady-State Case . . . . . . . . . . . . . . . . 20.6.4. Dynamic Case. Two-Dimensional Equations . . . . . . . . . . . . . . . . . . . . 21. Navier–Stokes and Related Equations 21.1. Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1. Navier–Stokes Equations in Various Coordinate Systems . . . . . . . . . 21.1.2. General Properties of the Navier–Stokes Equations . . . . . . . . . . . . . . 21.2. Solutions with One Nonzero Component of the Fluid Velocity . . . . . . . . . . . . . 21.2.1. Unidirectional Plane Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2. Unidirectional Flows in Tubes of Various Cross-Sections. External Flow Around a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3. One-Dimensional Rotation Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4. Purely Radial Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. Solutions with Two Nonzero Components of the Fluid Velocity . . . . . . . . . . . . 21.3.1. Two-Dimensional Solutions in the Rectangular Cartesian Coordinates (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2. Two-Dimensional Solutions in the Cylindrical Coordinates (Plane Flows) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1197 1197 1197 1198 1199 1200 1201 1204 1204 1205 1215 1219 1223 1223 1225 1227 1229 1231 1231 1233 1234 1235 1236 1236 1236 1238 1238 1239 1240 1244 1247 1247 1247 1249 1251 1251 1253 1257 1260 1263 1263 1270
C ONTENTS
21.4.
21.5.
21.6.
21.7.
xvii
21.3.3. Axisymmetric Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 21.3.4. Other Fluid Flows with Two-Nonzero Velocity Components . . . . . . 1282 Solutions with Three Nonzero Fluid Velocity Components Dependent on Two Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 21.4.1. Quasi-plane Flows (with the Fluid Velocity Components Independent of z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 21.4.2. Cylindrical and Conical Vortex Flows. Von K´arm´an-Type Rotationally Symmetric Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 21.4.3. Rotationally Symmetric Motions of General Form . . . . . . . . . . . . . . . 1297 Solutions with Three Nonzero Fluid Velocity Components Dependent on Three Space Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1302 21.5.1. Three-Dimensional Stagnation-Point Type Flows . . . . . . . . . . . . . . . . 1302 21.5.2. Solutions with Linear Dependence of the Velocity Components on Two Space Variables. Axial Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303 21.5.3. Solutions with Linear Dependence of the Velocity Components on Two Space Variables. General Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1317 21.5.4. Solutions with the Linear Dependence of the Velocity Components on One Space Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323 21.5.5. Other Three-Dimensional Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 Convective Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 21.6.1. Equations for Convective Fluid Motions . . . . . . . . . . . . . . . . . . . . . . . . 1329 21.6.2. Steady-State Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 21.6.3. Unsteady Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1331 Boundary Layer Equations (Prandtl Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . 1333 21.7.1. Equations and Boundary Conditions. Stream Function . . . . . . . . . . . 1333 21.7.2. Self-Similar Solutions of Some Boundary Layer Problems . . . . . . . . 1333 21.7.3. Other Solutions of the Boundary Layer Equations . . . . . . . . . . . . . . . 1336
22. Systems of General Form 22.1. Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3. Other Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4. Nonlinear Systems of Many Equations Involving the First Derivatives with Respect to t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1337 1337 1344 1347 1348
Part II Exact Methods for Nonlinear Partial Differential Equations
1353
23. Methods for Solving First-Order Quasilinear Equations 23.1. Characteristic System. General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1.1. Equations with Two Independent Variables. General Solution . . . . . 23.1.2. Quasilinear Equations with n Independent Variables . . . . . . . . . . . . . 23.2. Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . 23.2.1. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.2. Procedure of Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . 23.2.3. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1355 1355 1355 1356 1357 1357 1358 1359
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23.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . 23.3.1. Model Equation of Gas Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.2. Solution of the Cauchy Problem. Rarefaction Wave. Wave “Overturn” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.3. Shock Waves. Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.4. Utilization of Integral Relations for Determining Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.5. Conservation Laws. Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . 23.3.6. Hopf’s Formula for the Generalized Solution . . . . . . . . . . . . . . . . . . . . 23.3.7. Problem of Propagation of a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1360 1360 1360 1362 1364 1365 1366 1367
23.4. Quasilinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4.1. Quasilinear Equations in Conservative Form . . . . . . . . . . . . . . . . . . . . 23.4.2. Generalized Solution. Jump Condition and Stability Condition . . . . 23.4.3. Method for Constructing Stable Generalized Solutions . . . . . . . . . . .
1368 1368 1369 1370
24. Methods for Solving First-Order Nonlinear Equations
1373
24.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1. Complete, General, and Singular Integrals . . . . . . . . . . . . . . . . . . . . . . 24.1.2. Method of Separation of Variables. Equations of Special Form . . . . 24.1.3. Lagrange–Charpit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.4. Construction of a Complete Integral with the Aid of Two First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.5. Case where the Equation Does Not Depend on w Explicitly . . . . . . . 24.1.6. Hamilton–Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1373 1373 1374 1376 1377 1378 1379
24.2. Cauchy Problem. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . 24.2.1. Statement of the Problem. Solution Procedure . . . . . . . . . . . . . . . . . . . 24.2.2. Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.3. Cauchy Problem for the Hamilton–Jacobi Equation . . . . . . . . . . . . . . 24.2.4. Examples of Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . .
1379 1379 1380 1380 1381
24.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . . 24.3.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.2. Viscosity Solutions Based on the Use of a Parabolic Equation . . . . . 24.3.3. Viscosity Solutions Based on Test Functions and Differential Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.4. Local Structure of Generalized Viscosity Solutions . . . . . . . . . . . . . . 24.3.5. Generalized Classical Method of Characteristics . . . . . . . . . . . . . . . . . 24.3.6. Examples of Viscosity (Nonsmooth) Solutions . . . . . . . . . . . . . . . . . .
1382 1382 1382 1383 1383 1385 1386
25. Classification of Second-Order Nonlinear Equations
1389
25.1. Semilinear Equations in Two Independent Variables . . . . . . . . . . . . . . . . . . . . . 25.1.1. Types of Equations. Characteristic Equation . . . . . . . . . . . . . . . . . . . . 25.1.2. Canonical Form of Parabolic Equations (Case b2 − ac = 0) . . . . . . . . 25.1.3. Canonical Form of Hyperbolic Equations (Case b2 − ac > 0) . . . . . . 25.1.4. Canonical Form of Elliptic Equations (Case b2 − ac < 0) . . . . . . . . . .
1389 1389 1390 1390 1392
25.2. Nonlinear Equations in Two Independent Variables . . . . . . . . . . . . . . . . . . . . . . 1392 25.2.1. Nonlinear Equations of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . 1392 25.2.2. Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393
C ONTENTS
26. Transformations of Equations of Mathematical Physics 26.1. Point Transformations: Overview and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 26.1.1. General Form of Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . 26.1.2. Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1.3. Simple Nonlinear Point Transformations . . . . . . . . . . . . . . . . . . . . . . . 26.2. Hodograph Transformations (Special Point Transformations) . . . . . . . . . . . . . . 26.2.1. One PDE: One of the Independent Variables Is Taken to Be the Dependent One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2.2. One PDE: Method of Conversion to an Equivalent System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2.3. System of Two PDEs: One of the Independent Variables Is Taken to Be the Dependent One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2.4. System of Two PDEs: Both of the Independent Variables Are Taken to Be the Dependent Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3. Contact Transformations. Legendre and Euler Transformations . . . . . . . . . . . . 26.3.1. Preliminary Remarks. Contact Transformations for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.2. General Form of Contact Transformations for Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.3. Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.4. Euler Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.5. Legendre Transformation with Many Variables . . . . . . . . . . . . . . . . . . 26.4. Differential Substitutions. Von Mises Transformation . . . . . . . . . . . . . . . . . . . . 26.4.1. Differential Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4.2. Von Mises Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5. B¨acklund Transformations. RF Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.1. B¨acklund Transformations for Second-Order Equations . . . . . . . . . . 26.5.2. RF Pairs and Their Use for Constructing B¨acklund Transformations 26.6. Some Other Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.1. Crocco Transformation. Order Reduction of Hydrodynamic Type Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.2. Transformations Based on Conservation Laws . . . . . . . . . . . . . . . . . . 27. Traveling-Wave Solutions and Self-Similar Solutions 27.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2. Traveling-Wave Solutions. Invariance of Equations under Translations . . . . . 27.2.1. General Form of Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . 27.2.2. Invariance of Solutions and Equations under Translation Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.3. Functional Equation Describing Traveling-Wave Solutions . . . . . . . . 27.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.1. General Form of Self-Similar Solutions. Similarity Method . . . . . . . 27.3.2. Examples of Self-Similar Solutions to Mathematical Physics Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.3. More General Approach Based on Solving a Functional Equation. Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.4. Generalized Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xix 1395 1395 1395 1396 1396 1397 1397 1398 1402 1402 1403 1403 1405 1406 1407 1408 1409 1409 1410 1413 1413 1415 1422 1422 1425 1429 1429 1429 1429 1430 1431 1431 1431 1432 1434 1436
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28. Elementary Theory of Using Invariants for Solving Equations 28.1. Introduction. Symmetries. General Scheme of Using Invariants for Solving Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.1. Symmetries. Transformations Preserving the Form of Equations. Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.2. General Scheme of Using Invariants for Solving Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2. Algebraic Equations and Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1. Algebraic Equations with Even Powers . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2. Reciprocal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.3. Systems of Algebraic Equations Symmetric with Respect to Permutation of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3. Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.1. Transformations Preserving the Form of Equations. Invariants . . . . . 28.3.2. Order Reduction Procedure for Equations with n ≥ 2 (Reduction to Solvable Form with n = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.3. Simple Transformations. Invariant Determination Procedure . . . . . . 28.3.4. Analysis of Some Ordinary Differential Equations. Useful Remarks 28.4. Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.1. Transformations Preserving the Form of Equations. Invariants . . . . . 28.4.2. Procedure for Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . 28.4.3. Examples of Constructing Invariant Solutions to Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.4. Simple Inverse Problems (Determination of the Form of Equations from Their Properties) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5. General Conclusions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Method of Generalized Separation of Variables 29.1. Exact Solutions with Simple Separation of Variables . . . . . . . . . . . . . . . . . . . . . 29.1.1. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . . . . . 29.1.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.1. General Form of Solutions. Classes of Nonlinear Equations Considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.2. General Form of Functional Differential Equations . . . . . . . . . . . . . . 29.3. Simplified Scheme for Constructing Generalized Separable Solutions . . . . . . 29.3.1. Description of the Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions . . . . . . . . . 29.3.2. Examples of Finding Exact Solutions of Second- and Third-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4. Solution of Functional Differential Equations by Differentiation . . . . . . . . . . . 29.4.1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4.2. Examples of Constructing Exact Generalized Separable Solutions . 29.5. Solution of Functional-Differential Equations by Splitting . . . . . . . . . . . . . . . . 29.5.1. Preliminary Remarks. Description of the Method . . . . . . . . . . . . . . . . 29.5.2. Solutions of Simple Functional Equations and Their Application . .
1439 1439 1439 1440 1441 1441 1442 1444 1445 1445 1446 1446 1447 1450 1450 1450 1451 1455 1457 1459 1459 1459 1459 1462 1464 1464 1464 1465 1465 1465 1467 1467 1467 1471 1471 1472
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29.6. Titov–Galaktionov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6.1. Method Description. Linear Subspaces Invariant under a Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6.2. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6.3. Finding Linear Subspaces Invariant Under a Given Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6.4. Generalizations to Pseudo-Differential Equations . . . . . . . . . . . . . . . . 30. Method of Functional Separation of Variables
xxi 1477 1477 1479 1480 1483 1487
30.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 30.1.1. Structure of Functional Separable Solutions . . . . . . . . . . . . . . . . . . . . . 1487 30.1.2. Solution by Reduction to Equations with Quadratic (or Power) Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 30.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 30.2.1. Special Functional Separable and Generalized Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 30.2.2. General Scheme for Constructing Generalized Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489 30.3. Differentiation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492 30.3.1. Basic Ideas of the Method. Reduction to a Standard Equation . . . . . 1492 30.3.2. Examples of Constructing Functional Separable Solutions . . . . . . . . 1492 30.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4.1. Splitting Method. Reduction to a Standard Functional Equation . . . 30.4.2. Three-Argument Functional Equations of Special Form . . . . . . . . . . 30.4.3. Functional Equation f (t) + g(x) = Q(z), with z = φ(x) + ψ(t) . . . . 30.4.4. Functional Equation f (t) + g(x) + h(x)Q(z) + R(z) = 0, with z = φ(x) + ψ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4.5. Functional Equation f (t) + g(x)Q(z) + h(x)R(z) = 0, with z = φ(x) + ψ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4.6. Equation f1 (x) + f2 (y) + g1 (x)P (z) + g2 (y)Q(z) + R(z) = 0, z = φ(x) + ψ(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1501
31. Direct Method of Symmetry Reductions of Nonlinear Equations
1503
31.1. Clarkson–Kruskal Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1. Simplified Scheme. Examples of Constructing Exact Solutions . . . . 31.1.2. Description of the Method: A Special Form for Symmetry Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3. Description of the Method: the General Form for Symmetry Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1503 1503
1496 1496 1497 1498 1498 1500
1505 1506
31.2. Some Modifications and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507 31.2.1. Symmetry Reductions Based on the Generalized Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507 31.2.2. Similarity Reductions in Equations with Three or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510
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32. Classical Method of Symmetry Reductions 32.1. One-Parameter Transformations and Their Local Properties . . . . . . . . . . . . . . . 32.1.1. One-Parameter Transformations. Infinitesimal Operator . . . . . . . . . . 32.1.2. Invariant of an Infinitesimal Operator. Transformations in the Plane 32.1.3. Formulas for Derivatives. Coordinates of the First and Second Prolongations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition . . . 32.2.1. Invariance Condition. Splitting in Derivatives . . . . . . . . . . . . . . . . . . . 32.2.2. Examples of Finding Symmetries of Nonlinear Equations . . . . . . . . . 32.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1. Using Symmetries of Equations for Constructing One-Parameter Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.2. Procedure for Constructing Invariant Solutions . . . . . . . . . . . . . . . . . . 32.3.3. Examples of Constructing Invariant Solutions to Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.4. Solutions Induced by Linear Combinations of Admissible Operators 32.4. Some Generalizations. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1. One-Parameter Lie Groups of Point Transformations. Group Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.2. Group Invariants. Local Transformations of Derivatives . . . . . . . . . . 32.4.3. Invariant Condition. Splitting Procedure. Invariant Solutions . . . . . . 32.5. Symmetries of Systems of Equations of Mathematical Physics . . . . . . . . . . . . 32.5.1. Basic Relations Used in Symmetry Analysis of Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5.2. Symmetries of Equations of Steady Hydrodynamic Boundary Layer 33. Nonclassical Method of Symmetry Reductions 33.1. General Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.1. Description of the Method. Invariant Surface Condition . . . . . . . . . . 33.1.2. Scheme for Constructing Exact Solutions by the Nonclassical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2. Examples of Constructing Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.1. Newell–Whitehead Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.2. Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34. Method of Differential Constraints 34.1. Preliminary Remarks. Method of Differential Constraints for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.1. Description of the Method. First-Order Differential Constraints . . . 34.1.2. Differential Constraints of Arbitrary Order. General Consistency Method for Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.3. Some Generalizations. The Case of Several Differential Constraints 34.2. Description of the Method for Partial Differential Equations . . . . . . . . . . . . . . 34.2.1. Preliminary Remarks. A Simple Example . . . . . . . . . . . . . . . . . . . . . . 34.2.2. General Description of the Method of Differential Constraints . . . . . 34.3. First-Order Differential Constraints for PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.1. Second-Order Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.2. Second-Order Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.3. Second-Order Equations of General Form . . . . . . . . . . . . . . . . . . . . . .
1513 1513 1513 1514 1515 1516 1516 1516 1520 1520 1520 1522 1524 1526 1526 1526 1527 1527 1527 1528 1533 1533 1533 1533 1534 1534 1536 1539 1539 1539 1542 1543 1545 1545 1546 1547 1547 1551 1553
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34.4. Second-Order Differential Constraints for PDEs. Some Generalized . . . . . . . 34.4.1. Second-Order Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 34.4.2. Higher-Order Differential Constraints. Determining Equations . . . . 34.4.3. Usage of Several Differential Constraints. Systems of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5. Connection Between the Method of Differential Constraints and Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.1. Generalized/Functional Separation of Variables vs. Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.2. Direct Method of Symmetry Reductions and Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.3. Nonclassical Method of Symmetry Reductions and Differential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxiii 1553 1553 1555 1557 1561 1561 1562 1562
35. Painlev´e Test for Nonlinear Equations of Mathematical Physics 1565 35.1. Movable Singularities of Solutions of Ordinary Differential Equations . . . . . . 1565 35.1.1. Examples of Solutions Having Movable Singularities . . . . . . . . . . . . 1565 35.1.2. Classification Results for Nonlinear First- and Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565 35.1.3. Painlev´e Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566 35.1.4. Painlev´e Test for Ordinary Differential Equations . . . . . . . . . . . . . . . . 1566 35.1.5. Remarks on the Painlev´e Test. Fuchs Indices. Examples . . . . . . . . . . 1567 35.1.6. The Painlev´e Test for Systems of Ordinary Differential Equations . . 1569 35.2. Solutions of Partial Differential Equations with a Movable Pole. Method Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569 35.2.1. Simple Scheme for Studying Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1570 35.2.2. General Scheme for Analysis of Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1570 35.2.3. Basic Steps of the Painlev´e Test for Nonlinear Equations . . . . . . . . . 1570 35.2.4. Some Remarks. Truncated Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 1571 35.3. Performing the Painlev´e Test and Truncated Expansions for Studying Some Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572 35.3.1. Equations Passing the Painlev´e Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572 35.3.2. Checking Whether Nonlinear Systems of Equations of Mathematical Physics Pass the Painlev´e Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576 35.3.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev´e Test, Using Truncated Expansions . . . . . . . . . . . . . . . . . . . . . 1577 36. Methods of the Inverse Scattering Problem (Soliton Theory) 36.1. Method Based on Using Lax Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1.1. Method Description. Consistency Condition. Lax Pairs . . . . . . . . . . . 36.1.2. Examples of Lax Pairs for Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2. Method Based on a Compatibility Condition for Systems of Linear Equations 36.2.1. General Scheme. Compatibility Condition. Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2.2. Solution of the Determining Equations in the Form of Polynomials in λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1579 1579 1579 1580 1581 1581 1582
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36.3. Method Based on Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3.1. Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3.2. Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4. Solution of the Cauchy Problem by the Inverse Scattering Problem Method . 36.4.1. Preliminary Remarks. Direct and Inverse Scattering Problems . . . . . 36.4.2. Solution of the Cauchy Problem for Nonlinear Equations by the Inverse Scattering Problem Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4.3. N -Soliton Solution of the Korteweg–de Vries Equation . . . . . . . . . . 37. Conservation Laws 37.1. Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.1.1. General Form of Conservation Laws. Integrals of Motion . . . . . . . . . 37.1.2. Conservation Laws for Some Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2. Equations Admitting Variational Form. Noetherian Symmetries . . . . . . . . . . . 37.2.1. Lagrangian. Euler–Lagrange Equation. Noetherian Symmetries . . . 37.2.2. Examples of Constructing Conservation Laws Using Noetherian Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. Nonlinear Systems of Partial Differential Equations 38.1. Overdetermined Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.1.1. Overdetermined Systems of First-Order Equations in One Unknown 38.1.2. Other Overdetermined Systems of Equations in One Unknown . . . . 38.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.1. Pfaffian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.2. Condition for Integrability of the Pfaffian Equation by a Single Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2.3. Pfaffian Equations Not Satisfying the Integrability Condition . . . . . . 38.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.3.1. Traveling-Wave Solutions and Some Other Invariant Solutions . . . . 38.3.2. Systems Reducible to an Ordinary Differential Equation . . . . . . . . . . 38.3.3. Some Nonlinear Problems of Suspension Transport in Porous Media 38.3.4. Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.1. Systems of Two Quasilinear Equations. Systems in the Form of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.2. Self-Similar Continuous Solutions. Hyperbolic Systems . . . . . . . . . . 38.4.3. Simple Riemann Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.4. Linearization of Systems of Gas Dynamic Type by the Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4.5. Cauchy and Riemann Problems. Qualitative Features of Solutions . 38.4.6. Reduction of Systems to the Canonical Form. Riemann Invariants . 38.4.7. Hyperbolic n × n Systems of Conservation Laws. Exact Solutions . 38.4.8. Shock Waves. Rankine–Hugoniot Jump Conditions . . . . . . . . . . . . . . 38.4.9. Shock Waves. Evolutionary Conditions. Lax Condition . . . . . . . . . . . 38.4.10. Solutions for the Riemann Problem. Solutions Describing Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1584 1584 1585 1587 1587 1589 1591 1593 1593 1593 1593 1595 1595 1596 1599 1599 1599 1600 1600 1600 1601 1602 1602 1602 1603 1604 1606 1607 1607 1607 1609 1610 1611 1611 1614 1615 1616 1618
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38.5. Systems of Second-Order Equations of Reaction-Diffusion Type . . . . . . . . . . 1620 38.5.1. Traveling-Wave Solutions and Some Other Invariant Solutions . . . . 1620 38.5.2. Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620
Part III Symbolic and Numerical Solutions of Nonlinear PDEs with Maple, Mathematica, and MATLAB 39. Nonlinear Partial Differential Equations with Maple 39.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2. Brief Introduction to Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2.1. Maple Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3. Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.1. Constructing Analytical Solutions in Terms of Predefined Functions 39.3.2. Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . 39.3.3. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 39.3.4. Ansatz Methods (Tanh-Coth Method, Sine-Cosine Method, and Exp-Function Method) for Constructing Traveling-Wave Solutions . 39.3.5. Constructing Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3.6. Constructing Solutions along Characteristics . . . . . . . . . . . . . . . . . . . . 39.3.7. Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.4. Analytical Solutions of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.4.1. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 39.4.2. Constructing Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . 39.5. Constructing Exact Solutions Using Symbolic Computation. What Can Go Wrong? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.5.1. Constructing New Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.5.2. Removing Redundant Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 39.6. Some Errors That People Commonly Do When Constructing Exact Solutions with the Use of Symbolic Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.6.1. General Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.6.2. Examples in Which “New Solutions” Are Obtained . . . . . . . . . . . . . . 39.7. Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.7.1. Constructing Numerical Solutions in Terms of Predefined Functions 39.7.2. Constructing Finite Difference Approximations . . . . . . . . . . . . . . . . . 39.8. Analytical-Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.8.1. Analytical Derivation of Numerical Methods . . . . . . . . . . . . . . . . . . . . 39.8.2. Constructing Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.8.3. Comparison of Asymptotic and Numerical Solutions . . . . . . . . . . . . . 40. Nonlinear Partial Differential Equations with Mathematica 40.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2. Brief Introduction to Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2.1. Mathematica Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3. Analytical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3.1. Constructing Solutions Using Predefined Functions . . . . . . . . . . . . . . 40.3.2. Constructing Solutions via Transformations . . . . . . . . . . . . . . . . . . . . . 40.3.3. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 40.3.4. Ansatz Methods (Tanh-Coth Method, Sine-Cosine Method, and Exp-Function Method) for Constructing Traveling-Wave Solutions .
1623 1625 1625 1625 1627 1629 1629 1635 1637 1641 1645 1647 1651 1659 1659 1661 1662 1662 1665 1668 1668 1669 1672 1672 1677 1680 1680 1683 1684 1687 1687 1687 1689 1691 1692 1694 1697 1700
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40.3.5. Constructing Self-Similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3.6. Constructing Solutions Along Characteristics . . . . . . . . . . . . . . . . . . . 40.3.7. Constructing Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4. Analytical Solutions of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4.1. Constructing Traveling-Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . 40.4.2. Constructing Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . 40.5. Numerical Solutions and Their Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.5.1. Constructing Numerical Solutions in Terms of Predefined Functions 40.5.2. Constructing Finite-Difference Approximations . . . . . . . . . . . . . . . . . 40.6. Analytical-Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6.1. Analytical Derivation of Numerical Methods . . . . . . . . . . . . . . . . . . . . 40.6.2. Constructing Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6.3. Comparison of Asymptotic and Numerical Solutions . . . . . . . . . . . . .
1705 1706 1710 1719 1719 1721 1722 1722 1725 1728 1729 1731 1733
41. Nonlinear Partial Differential Equations with MATLAB 41.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Brief Introduction to MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1. MATLAB Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Numerical Solutions Via Predefined Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1. Scalar Nonlinear PDEs in One Space Dimension . . . . . . . . . . . . . . . . 41.3.2. Systems of Nonlinear PDEs in One Space Dimension . . . . . . . . . . . . 41.3.3. Nonlinear Elliptic PDEs in Two Space Dimensions . . . . . . . . . . . . . . 41.3.4. Systems of Nonlinear Elliptic PDEs in Two Space Dimensions . . . . 41.3.5. Nonlinear Elliptic PDEs. Geometrical Models . . . . . . . . . . . . . . . . . . 41.4. Solving Cauchy Problems. Method of Characteristics . . . . . . . . . . . . . . . . . . . . 41.5. Constructing Finite-Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . .
1735 1735 1735 1738 1741 1742 1745 1748 1752 1754 1756 1760
Part Supplements. Painlev´e Transcendents. Functional Equations
1767
42. Painlev´e Transcendents 42.1. Preliminary Remarks. Singular Points of Solutions . . . . . . . . . . . . . . . . . . . . . . 42.2. First Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Second Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4. Third Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5. Fourth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6. Fifth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7. Sixth Painlev´e Transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.8. Examples of Solutions to Nonlinear Equations in Terms of Painlev´e Transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1769 1769 1770 1771 1772 1775 1776 1777
43. Functional Equations 43.1. Method of Differentiation in a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1.1. Classes of Equations. Description of the Method . . . . . . . . . . . . . . . . 43.1.2. Examples of Solutions of Some Specific Functional Equations . . . . 43.2. Method of Differentiation in Independent Variables . . . . . . . . . . . . . . . . . . . . . . 43.2.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2.2. Examples of Solutions of Some Specific Functional Equations . . . .
1783 1783 1783 1784 1785 1785 1785
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43.3. Method of Argument Elimination by Test Functions . . . . . . . . . . . . . . . . . . . . . 43.3.1. Classes of Equations. Description of the Method . . . . . . . . . . . . . . . . 43.3.2. Examples of Solutions of Specific Functional Equations . . . . . . . . . . 43.4. Nonlinear Functional Equations Reducible to Bilinear Equations . . . . . . . . . . 43.4.1. Bilinear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4.2. Functional-Differential Equations Reducible to a Bilinear Equation 43.4.3. Nonlinear Functional Equations Containing the Complex Argument
1786 1786 1787 1789 1789 1790 1790
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