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1 Local Transformations and Conservation Laws 1

1.1 Introduction 1

1.2 Local Transformations 5

1.2.1 Point transformations 6

1.2.2 Contact transformations 8

1.2.3 Higher-order transformations 10

1.2.4 One-parameter higher-order transformations 10

1.2.5 Point symmetries 16

1.2.6 Contact and higher-order symmetries 20

1.2.7 Equivalence transformations and symmetry classification 21

1.2.8 Recursion operators for local symmetries 24

1.3 Conservation Laws 38

1.3.1 Local conservation laws 38

1.3.2 Equivalent conservation laws 42

1.3.3 Multipliers for conservation laws.Euler operators 43

1.3.4 The direct method for construction of conservation laws.Cauchy-Kovalevskaya form 46

1.3.5 Examples 50

1.3.6 Linearizing operators and adjoint equations 53

1.3.7 Determination of fluxes of conservation laws from multipliers 56

1.3.8 Self-adjoint PDE systems 64

1.4 Noether's Theorem 70

1.4.1 Euler-Lagrange equations 71

1.4.2 Noether's formulation of Noether's theorem 72

1.4.3 Boyer's formulation of Noether's theorem 75

1.4.4 Limitations of Noether's theorem 77

1.4.5 Examples 79

1.5 Some Connections Between Symmetries and Conservation Laws 89

1.5.1 Use of symmetries to find new conservation laws from known conservation laws 90

1.5.2 Relationships among symmetries,solutions of adjoint equations,and conservation laws 107

1.6 Discussion 117

2 Construction of Mappings Relating Differential Equations 121

2.1 Introduction 121

2.2 Notations;Mappings of Infinitesimal Generators 123

2.2.1 Theorems on invertible mappings 127

2.3 Mapping of a Given PDE to a Specific Target PDE 128

2.3.1 Construction of non-invertible mappings 129

2.3.2 Construction of an invertible mapping by a point transformation 133

2.4 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Symmetries 139

2.4.1 Invertible mappings of nonlinear PDE systems (with at least two dependent variables)to linear PDE systems 141

2.4.2 Invertible mappings of nonlinear PDE systems (with one dependent variable) to linear PDE systems 146

2.5 Invertible Mappings of Linear PDEs to Linear PDEs with Constant Coefficients 158

2.5.1 Examples of mapping variable coefficient linear PDEs to constant coefficient linear PDEs through invertible point transformations 163

2.5.2 Example of finding the most general mapping of a given constant coefficient linear PDE to some constant coefficient linear PDE 168

2.6 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Conservation Law Multipliers 173

2.6.1 Computational steps 177

2.6.2 Examples of linearizations of nonlinear PDEs through conservation law multipliers 179

2.7 Discussion 184

3 Nonlocally Related PDE Systems 187

3.1 Introduction 187

3.2 Nonlocally Related Potential Systems and Subsystems in Two Dimensions 191

3.2.1 Potential systems 192

3.2.2 Nonlocally related subsystems 193

3.3 Trees of Nonlocally Related PDE Systems 199

3.3.1 Basic procedure of tree construction 200

3.3.2 A tree for a nonlinear diffusion equation 202

3.3.3 A tree for planar gas dynamics(PGD)equations 204

3.4 Nonlocal Conservation Laws 209

3.4.1 Conservation laws arising from nonlocally related systems 210

3.4.2 Nonlocal conservation laws for diffusion-convection equations 212

3.4.3 Additional conservation laws of nonlinear telegraph equations 214

3.5 Extended Tree Construction Procedure 222

3.5.1 An extended tree construction procedure 223

3.5.2 An extended tree for a nonlinear diffusion equation 225

3.5.3 An extended tree for a nonlinear wave equation 228

3.5.4 An extended tree for the planar gas dynamics equations 232

3.6 Discussion 242

4 Applications of Nonlocally Related PDE Systems 245

4.1 Introduction 245

4.2 Nonlocal Symmetries 248

4.2.1 Nonlocal symmetries of a nonlinear diffusion equation 251

4.2.2 Nonlocal symmetries of a nonlinear wave equation 256

4.2.3 Classification of nonlocal symmetries of nonlinear telegraph equations arising from point symmetries of potential systems 270

4.2.4 Nonlocal symmetries of nonlinear telegraph equations with power law nonlinearities 271

4.2.5 Nonlocal symmetries of the planar gas dynamics equations 276

4.3 Construction of Non-invertible Mappings Relating PDEs 283

4.3.1 Non-invertible mappings of nonlinear PDE systems to linear PDE systems 284

4.3.2 Non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficients 290

4.4 Discussion 294

5 Further Applications of Symmetry Methods:Miscellaneous Extensions 297

5.1 Introduction 297

5.2 Applications of Symmetry Methods to the Construction of Solutions of PDEs 301

5.2.1 The classical method 302

5.2.2 The nonclassical method 306

5.2.3 Invariant solutions arising from nonlocal symmetries that are local symmetries of nonlocally related systems 314

5.2.4 Further extensions of symmetry methods for construction of solutions of PDEs connected with nonlocally related systems 320

5.3 Nonlocally Related PDE Systems in Three or More Dimensions 333

5.3.1 Divergence-type conservation laws and resulting potential systems 334

5.3.2 Nonlocally related subsystems 336

5.3.3 Tree construction,nonlocal conservation laws,and nonlocal symmetries 337

5.3.4 Lower-degree conservation laws and related potential systems 341

5.3.5 Examples of applications of nonlocally related systems in higher dimensions 343

5.3.6 Symmetries and exact solutions of the three-dimensional MHD equilibrium equations 350

5.4 Symbolic Software 357

5.4.1 An example of symbolic computation of point symmetries 357

5.4.2 An example of point symmetry classification 359

5.4.3 An example of symbolic computation of conservation laws 363

5.5 Discussion 364

References 369

Theorem,Corollary and Lemma Index 383

Author Index 385

Subject Index 389