Applications of Symmetry Methods to Partial Differential Equations (eBook)

Contents 6
Preface 10
Introduction 13
Chapter 1 Local Transformations and Conservation Laws 20
1.1 Introduction 20
1.2 Local Transformations 24
1.2.1 Point transformations 25
1.2.2 Contact transformations 27
1.2.3 Higher-order transformations 29
1.2.4 One-parameter higher-order transformations 29
1.2.5 Point symmetries 35
1.2.6 Contact and higher-order symmetries 39
1.2.7 Equivalence transformations and symmetry classification 40
1.2.8 Recursion operators for local symmetries 43
1.3 Conservation Laws 57
1.3.1 Local conservation laws 57
1.3.2 Equivalent conservation laws 61
1.3.3 Multipliers for conservation laws. Euler operators 62
1.3.4 The direct method for construction of conservation laws. Cauchy-Kovalevskaya form 65
1.3.5 Examples 69
1.3.6 Linearizing operators and adjoint equations 72
1.3.7 Determination of fluxes of conservation laws from multipliers 75
1.3.8 Self-adjoint PDE systems 83
1.4 Noether's Theorem 89
1.4.1 Euler-Lagrange equations 90
1.4.2 Noether's formulation of Noether's theorem 91
1.4.3 Boyer's formulation of Noether's theorem 94
1.4.4 Limitations of Noether's theorem 96
1.4.5 Examples 98
1.5 Some Connections Between Symmetries and Conservation Laws 108
1.5.1 Use of symmetries to find new conservation laws from known conservation laws 109
1.5.2 Relationships among symmetries, solutions of adjoint equations, and conservation laws 126
1.6 Discussion 136
Chapter 2 Construction of Mappings Relating Differential Equations 140
2.1 Introduction 140
2.2 Notations Mappings of Infinitesimal Generators
2.2.1 Theorems on invertible mappings 146
2.3 Mapping of a Given PDE to a Specific Target PDE 147
2.3.1 Construction of non-invertible mappings 148
2.3.2 Construction of an invertible mapping by a point transformation 152
2.4 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Symmetries 158
2.4.1 Invertible mappings of nonlinear PDE systems (with at least two dependent variables) to linear PDE systems 160
2.4.2 Invertible mappings of nonlinear PDE systems (with one dependent variable) to linear PDE systems 165
2.5 Invertible Mappings of Linear PDEs to Linear PDEs with Constant Coefficients 177
2.5.1 Examples of mapping variable coefficient linear PDEs to constant coefficient linear PDEs through invertible point transformations 182
2.5.2 Example of finding the most general mapping of a given constant coefficient linear PDE to some constant coefficient linear PDE 187
2.6 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Conservation Law Multipliers 192
2.6.1 Computational steps 196
2.6.2 Examples of linearizations of nonlinear PDEs through conservation law multipliers 198
2.7 Discussion 203
Chapter 3 Nonlocally Related PDE Systems 206
3.1 Introduction 206
3.2 Nonlocally Related Potential Systems and Subsystems in Two Dimensions 210
3.2.1 Potential systems 211
3.2.2 Nonlocally related subsystems 212
3.3 Trees of Nonlocally Related PDE Systems 218
3.3.1 Basic procedure of tree construction 219
3.3.2 A tree for a nonlinear diffusion equation 221
3.3.3 A tree for planar gas dynamics (PGD) equations 223
3.4 Nonlocal Conservation Laws 228
3.4.1 Conservation laws arising from nonlocally related systems 229
3.4.2 Nonlocal conservation laws for diffusion-convection equations 231
3.4.3 Additional conservation laws of nonlinear telegraph equations 233
3.5 Extended Tree Construction Procedure 241
3.5.1 An extended tree construction procedure 242
3.5.2 An extended tree for a nonlinear diffusion equation 244
3.5.3 An extended tree for a nonlinear wave equation 247
3.5.4 An extended tree for the planar gas dynamics equations 251
3.6 Discussion 261
Chapter 4 Applications of Nonlocally Related PDE Systems 264
4.1 Introduction 264
4.2 Nonlocal Symmetries 267
4.2.1 Nonlocal symmetries of a nonlinear diffusion equation 270
4.2.2 Nonlocal symmetries of a nonlinear wave equation 275
4.2.4 Nonlocal symmetries of nonlinear telegraph equations with power law nonlinearities 290
4.2.5 Nonlocal symmetries of the planar gas dynamics equations 295
4.3 Construction of Non-invertible Mappings Relating PDEs 302
4.3.1 Non-invertible mappings of nonlinear PDE systems to linear PDE systems 303
4.3.2 Non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficients 309
4.4 Discussion 313
Chapter 5 Further Applications of Symmetry Methods: Miscellaneous Extensions 316
5.1 Introduction 316
5.2 Applications of Symmetry Methods to the Construction of Solutions of PDEs 320
5.2.1 The classical method 321
5.2.2 The nonclassical method 325
5.2.3 Invariant solutions arising from nonlocal symmetries that are local symmetries of nonlocally related systems 333
5.2.4 Further extensions of symmetry methods for construction of solutions of PDEs connected with nonlocally related systems 339
5.3 Nonlocally Related PDE Systems in Three or More Dimensions 352
5.3.1 Divergence-type conservation laws and resulting potential systems 353
5.3.2 Nonlocally related subsystems 355
5.3.3 Tree construction, nonlocal conservation laws, and nonlocal symmetries 356
5.3.4 Lower-degree conservation laws and related potential systems 360
5.3.5 Examples of applications of nonlocally related systems in higher dimensions 362
5.3.6 Symmetries and exact solutions of the three-dimensional MHD equilibrium equations 369
5.4 Symbolic Software 376
5.4.1 An example of symbolic computation of point symmetries 376
5.4.2 An example of point symmetry classiffication 378
5.4.3 An example of symbolic computation of conservation laws 382
5.5 Discussion 383
References 387
Theorem, Corollary and Lemma Index 400
Author Index 402
Subject Index 406