Preface ........................................................ ix
Introduction ................................................. xiii
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 ............................. l41
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 TYee 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
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