Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010222217	QA377 G345 2007	Open Access Book	Book	Searching... Unknown
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Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book focuses on the existence of new exact solutions on linear invariant subspaces for nonlinear operators and their crucial new properties.

This practical reference deals with various partial differential equations (PDEs) and models that exhibit some common nonlinear invariant features. It begins with classical as well as more recent examples of solutions on invariant subspaces. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including reaction-diffusion and gas dynamics models, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym models, quasilinear magma equations, and Green-Naghdi equations. Using exact solutions, they describe the evolution properties of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, changing sign behavior of weak solutions near interfaces for nonlinear PDEs of various types and orders.

The techniques surveyed in Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics serve as a preliminary introduction to the general theory of nonlinear evolution PDEs of different orders and types.

Author Notes

Galaktionov, Victor A.; Svirshchevskii, Sergey R.

Introduction: Nonlinear Partial Differential Equations and Exact Solutions	p. xi
Exact solutions: history, classical symmetry methods, extensions	p. xi
Examples: classic fundamental solutions belong to invariant subspaces	p. xvii
Models, targets, prerequisites	p. xxii
Acknowledgements	p. xxx
1 Linear Invariant Subspaces in Quasilinear Equations: Basic Examples and Models	p. 1
1.1 History: first examples of solutions on invariant subspaces	p. 1
1.2 Basic ideas: invariant subspaces and generalized separation of variables	p. 16
1.3 More examples: polynomial subspaces	p. 20
1.4 Examples: trigonometric subspaces	p. 30
1.5 Examples: exponential subspaces	p. 37
Remarks and comments on the literature	p. 46
2 Invariant Subspaces and Modules: Mathematics in One Dimension	p. 49
2.1 Main Theorem on invariant subspaces	p. 49
2.2 The optimal estimate on dimension of invariant subspaces	p. 54
2.3 First-order operators with subspaces of maximal dimension	p. 57
2.4 Second-order operators with subspaces of maximal dimension	p. 61
2.5 First and second-order quadratic operators with subspaces of lower dimensions	p. 67
2.6 Operators preserving polynomial subspaces	p. 72
2.7 Extensions to [Characters not reproducible]-dependent operators	p. 85
2.8 Summary: Basic types of equations and solutions	p. 92
Remarks and comments on the literature	p. 96
Open problems	p. 96
3 Parabolic Equations in One Dimension: Thin Film, Kuramoto-Sivashinsky, and Magma Models	p. 97
3.1 Thin film models and solutions on polynomial subspaces	p. 97
3.2 Applications to extinction, blow-up, free-boundary problems, and interface equations	p. 106
3.3 Exact solutions with zero contact angle	p. 120
3.4 Extinction behavior for sixth-order thin film equations	p. 126
3.5 Quadratic models: trigonometric and exponential subspaces	p. 128
3.6 2mth-order thin film operators and equations	p. 134
3.7 Oscillatory, changing sign behavior in the Cauchy problem	p. 139
3.8 Invariant subspaces in Kuramoto-Sivashinsky type models	p. 148
3.9 Quasilinear pseudo-parabolic models: the magma equation	p. 156
Remarks and comments on the literature	p. 160
Open problems	p. 162
4 Odd-Order One-Dimensional Equations: Korteweg-de Vries, Compacton, Nonlinear Dispersion, and Harry Dym Models	p. 163
4.1 Blow-up and localization for KdV-type equations	p. 163
4.2 Compactons and shocks waves in higher-order quadratic nonlinear dispersion models	p. 165
4.3 Higher-order PDEs: interface equations and oscillatory solutions	p. 183
4.4 Compactons and interfaces for singular mKdV-type equations	p. 197
4.5 On compactons in IR[supercript N] for nonlinear dispersion equations	p. 204
4.6 "Tautological" equations and peakons	p. 210
4.7 Subspaces, singularities, and oscillatory solutions of Harry Dym-type equations	p. 220
Remarks and comments on the literature	p. 226
Open problems	p. 234
5 Quasilinear Wave and Boussinesq Models in One Dimension. Systems of Nonlinear Equations	p. 235
5.1 Blow-up in nonlinear wave equations on invariant subspaces	p. 235
5.2 Breathers in quasilinear wave equations and blow-up models	p. 241
5.3 Quenching and interface phenomena, compactons	p. 252
5.4 Invariant subspaces in systems of nonlinear evolution equations	p. 260
Remarks and comments on the literature	p. 271
Open problems	p. 274
6 Applications to Nonlinear Partial Differential Equations in IR[superscript N]	p. 275
6.1 Second-order operators and some higher-order extensions	p. 275
6.2 Extended invariant subspaces for second-order operators	p. 286
6.3 On the remarkable operator in IR[superscript 2]	p. 293
6.4 On second-order p-Laplacian operators	p. 300
6.5 Invariant subspaces for operators of Monge-Ampere type	p. 304
6.6 Higher-order thin film operators	p. 315
6.7 Moving compact structures in nonlinear dispersion equations	p. 326
6.8 From invariant polynomial subspaces in IR[superscript N] to invariant trigonometric subspaces in IR[superscript N-1]	p. 327
Remarks and comments on the literature	p. 331
Open problems	p. 336
7 Partially Invariant Subspaces, Invariant Sets, and Generalized Separation of Variables	p. 337
7.1 Partial invariance for polynomial operators	p. 337
7.2 Quadratic Kuramoto-Sivashinsky equations	p. 344
7.3 Method of generalized separation of variables	p. 346
7.4 Generalized separation and partially invariant modules	p. 349
7.5 Evolutionary invariant sets for higher-order equations	p. 362
7.6 A separation technique for the porous medium equation in IR[superscript N]	p. 373
Remarks and comments on the literature	p. 383
Open problems	p. 384
8 Sign-Invariants for Second-Order Parabolic Equations and Exact Solutions	p. 385
8.1 Quasilinear models, definitions, and first examples	p. 386
8.2 Sign-invariants of the form u[subscript t] - [psi](u)	p. 389
8.3 Stationary sign-invariants of the form H(r, u, u[subscript r])	p. 394
8.4 Sign-invariants of the form u[subscript t] - m(u)(u[subscript x])[superscript 2] - M(u)	p. 401
8.5 General first-order Hamilton-Jacobi sign-invariants	p. 410
Remarks and comments on the literature	p. 423
9 Invariant Subspaces for Discrete Operators, Moving Mesh Methods, and Lattices	p. 429
9.1 Backward problem of invariant subspaces for discrete operators	p. 429
9.2 On the forward problem of invariant subspaces	p. 433
9.3 Invariant subspaces for finite-difference operators	p. 437
9.4 Invariant properties of moving mesh operators and applications	p. 448
9.5 Applications to anharmonic lattices	p. 460
Remarks and comments on the literature	p. 466
Open problems	p. 466
References	p. 467
List of Frequently Used Abbreviations	p. 493
Index	p. 494

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