Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics / Edition 1

by Victor A. Galaktionov, Sergey R. Svirshchevskii

Overview

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.

Product Details

ISBN-13:	9780367389970
Publisher:	Taylor & Francis
Publication date:	09/19/2019
Series:	Chapman & Hall/CRC Applied Mathematics and Nonlinear Science , #10
Pages:	528
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Galaktionov, Victor A.; Svirshchevskii, Sergey R.

Table of Contents

Introduction: Nonlinear Partial Differentia] Equations and Exact Solutions xi

Exact solutions: history, classical symmetry methods, extensions xi

Examples: classic fundamental solutions belong to invariant subspaces xvii

Models, targets, prerequisites xxii

Acknowledgements xxx

1 Linear Invariant Subspaces in Quasilinear Equations: Basic Examples and Models 1

1.1 History, first examples of solutions on invariant subspaces 1

1.2 Basic ideas: invariant subspaces and generalized separation of variables 16

1.3 More examples: polynomial subspaces 20

1.4 Examples: trigonometric subspaces 30

1.5 Examples: exponential subspaces 37

Remarks and comments on the literature 46

2 Invariant Subspaces and Modules: Mathematics in One Dimension 49

2.1 Main Theorem on invariant subspaces 49

2.2 The optimal estimate on dimension of invariant subspaces 54

2.3 First-order operators with subspaces of maximal dimension 57

2.4 Second-order operators with subspaces of maximal dimension 61

2.5 First and second-order quadratic operators with subspaces of lower dimensions 67

2.6 Operators preserving polynomial subspaces 72

2.7 Extensions to $$$-dependent operators 85

2.8 SUMMARY: Basic types of equations and solutions 92

Remarks and comments on the literature 96

Open problems 96

3 Parabolic Equations in One Dimension: Thin Film, Kuramoto-Sivashinsky, and Magma Models 97

3.1 Thin film models and solutions on polynomial subspaces 97

3.2 Applications to extinction, blow-up, free-boundary problems, and interface equations 106

3.3 Exact solutions with zero contact angle 120

3.4 Extinction behavior for sixth-order thin film equations 126

3.5 Quadratic models: trigonometric and exponential subspaces 128

3.6 2mth-order thin film operators and equations 134

3.7 Oscillatory, changing sign behavior in the Cauchy problem 139

3.8 Invariant subspaces in Kuramoto-Sivashinsky type models 148

3.9 Quasilinear pseudo-parabolic models: the magma equation 156

Remarks and comments on the literature 160

Open problems 162

4 Odd-Order One-Dimensional Equations: Korteweg-de Vries, Compacton, Nonlinear Dispersion, and Harry Dym Models 163

4.1 Blow-up and localization for KdV-type equations 163

4.2 Compactons and shocks waves in higher-order quadratic nonlinear dispersion models 165

4.3 Higher-order PDEs: interface equations and oscillatory solutions 183

4.4 Compactons and interfaces for singular mKdV-type equations 197

4.5 On compactons in IR^N for nonlinear dispersion equations 204

4.6 "Tautological" equations and peakons 210

4.7 Subspaces, singularities, and oscillatory solutions of Harry Dymtype equations 220

Remarks and comments on the literature 226

Open problems 234

5 Quasilinear Wave and Boussinesq Models in One Dimension. Systems of Nonlinear Equations 235

5.1 Blow-up in nonlinear wave equations on invariant subspaces 235

5.2 Breathers in quasilinear wave equations and blow-up models 241

5.3 Quenching and interface phenomena, compactons 252

5.4 Invariant subspaces in systems of nonlinear evolution equations 260

Remarks and comments on the literature 271

Open problems 274

6 Applications to Nonlinear Partial Differential Equations in R^N 275

6.1 Second-order operators and some higher-order extensions 275

6.2 Extended invariant subspaces for second-order operators 286

6.3 On the remarkable operator in IR² 293

6.4 On second-order p-Laplacian operators 300

6.5 Invariant subspaces for operators of Monge-Ampere type 304

6.6 Higher-order thin film operators 315

6.7 Moving compact structures in nonlinear dispersion equations 326

6.8 From invariant polynomial subspaces in IR^N to invariant trigonometric subspaces in IR^N-1 327

Remarks and comments on the literature 331

Open problems 336

7 Partially Invariant Subspaces, Invariant Sets, and Generalized Separation of Variables 337

7.1 Partial invariance for polynomial operators 337

7.2 Quadratic Kuramoto-Sivashinsky equations 344

7.3 Method of generalized separation of variables 346

7.4 Generalized separation and partially invariant modules 349

7.5 Evolutionary invariant sets for higher-order equations 362

7.6 A separation technique for the porous medium equation in IR^N 373

Remarks and comments on the literature 383

Open problems 384

8 Sign-Invariants for Second-Order Parabolic Equations and Exact Solutions 385

8.1 Quasilinear models, definitions, and first examples 386

8.2 Sign-invariants of the form u_t - Ψ (u) 389

8.3 Stationary sign-invariants of the form H(r, u, ur) 394

8.4 Sign-invariants of the form u_t - m(u)(u_x)² - M(u) 401

8.5 General first-order Hamilton-Jacobi sign-invariants 410

Remarks and comments on the Literature 423

9 Invariant Subspaces for Discrete Operators, Moving Mesh Methods, and Lattices 429

9.1 Backward problem of invariant subspaces for discrete operators 429

9.2 On the forward problem of invariant subspaces 433

9.3 Invariant subspaces for finite-difference operators 437

9.4 Invariant properties of moving mesh operators and applications 448

9.5 Applications to anharmonic lattices 460

Remarks and comments on the literature 466

Open problems 466

References 467

List of Frequently Used Abbreviations 493

Index 494