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

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

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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.show more

Product details

  • Hardback | 528 pages
  • 162.6 x 233.7 x 33mm | 861.84g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 63 black & white illustrations
  • 1584886633
  • 9781584886631
  • 2,411,466

Table of contents

INTRODUCTION: NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND EXACT SOLUTIONS Exact Solutions: History, Classical Symmetry Methods, Extensions Examples: Classic Fundamental Solutions belong to Invariant Subspaces Models, Targets, and Prerequisites LINEAR INVARIANT SUBSPACES IN QUASILINEAR EQUATIONS: BASIC EXAMPLES AND MODELS History: First Eexamples of Solutions on Invariant Subspaces Basic Ideas: Invariant Subspaces and Generalized Separation of Variables More Examples: Polynomial Subspaces Examples: Trigonometric Subspaces Examples: Exponential Subspaces Remarks and Comments on the Literature INVARIANT SUBSPACES AND MODULES: MATHEMATICS IN ONE DIMENSION Main Theorem on Invariant Subspaces The Optimal Estimate on Dimension of Invariant Subspaces First-Order Operators with Subspaces of Maximal Dimension Second-Order Operators with Subspaces of Maximal Dimension First- and Second-Order Quadratic Operators with Subspaces of Lower Dimensions Operators Preserving Polynomial Subspaces Extensions to ?/?t-Dependent Operators Summary: Basic Types of Equations and Solutions Remarks and Comments on the Literature Open Problems PARABOLIC EQUATIONS IN ONE DIMENSION: THIN FILM, KURAMOTO-SIVASHINSKY, AND MAGMA MODELS Thin Film Models and Polynomial Subspaces Applications to Extinction, Blow-Up, Free-Boundary Problems, and Interface Equations Exact Solutions with Zero Contact Angle Extinction Behavior for Sixth-Order Thin Film Equations Quadratic Models: Trigonometric and Exponential Subspaces 2mth-Order Thin Film Operators and Equations Oscillatory, Changing Sign Behavior in the Cauchy Problem Invariant Subspaces in Kuramoto-Sivashinsky-Type Models Quasilinear Pseudo-Parabolic Models: The Magma Equation Remarks and Comments on the Literature Open Problems ODD-ORDER ONE-DIMENSIONAL EQUATIONS: KORTEWEG-DE VRIES, COMPACTON, NONLINEAR DISPERSION, AND HARRY DYM MODELS Blow-Up and Localization for KdV-Type Equations Compactons and Shocks Waves in Higher-Order Quadratic Nonlinear Dispersion Models Higher-Order PDEs: Interface Equations and Oscillatory Solutions Compactons and Interfaces for Singular mKdV-Type Equations On Compactons in IRN for Nonlinear Dispersion Equations "Tautological" Equations and Peakons Subspaces, Singularities, and Oscillatory Solutions for Harry Dym-Type Equations Remarks and Comments on the Literature Open Problems QUASILINEAR WAVE AND BOUSSINESQ MODELS IN ONE DIMENSION: SYSTEMS OF NONLINEAR EQUATIONS Blow-Up in Nonlinear Wave Equations on Invariant Subspaces Breathers in Quasilinear Wave Equations and Blow-Up Models Quenching and Interface Phenomena, Compactons Invariant Subspaces in Systems of Nonlinear Evolution Equations Remarks and Comments on the Literature Open Problems APPLICATIONS TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN IRN Second-Order Operators and Some Higher-Order Extensions Extended Invariant Subspaces for Second-Order Operators On the Remarkable Operator in IR2 On Second-Order p-Laplacian Operators Invariant Subspaces for Operators of Monge-Ampere Type Higher-Order Thin Film Operators Moving Compact Structures in Nonlinear Dispersion Equations From Invariant Polynomial Subspaces in IR N to Invariant Trigonometric Subspaces in IR N -1 Remarks and Comments on the Literature Open Problems PARTIALLY INVARIANT SUBSPACES, INVARIANT SETS, AND GENERALIZED SEPARATION OF VARIABLES Partial Invariance for Polynomial Operators Quadratic Kuramoto-Sivashinsky Equations Method of Generalized Separation of Variables Generalized Separation and Partially Invariant Modules Evolutionary Invariant Sets for Higher-Order Equations A Separation Technique for the Porous Medium Equation in IRN Remarks and Comments on the Literature Open Problems SIGN-INVARIANTS FOR SECOND-ORDER PARABOLIC EQUATIONS AND EXACT SOLUTIONS Quasilinear Models, Definitions, and First Examples Sign-Invariants of the Form ut - ?(u) Stationary Sign-Invariants of the Form H (r, u, ur) Sign-Invariants of the Form ut - m(u)(ux)2 - M(u) General First-Order Hamilton-Jacobi Sign-Invariants Remarks and Comments on the Literature INVARIANT SUBSPACES FOR DISCRETE OPERATORS, MOVING MESH METHODS, AND LATTICES Backward Problem of Invariant Subspaces for Discrete Operators On the Forward Problem of Invariant Subspaces Invariant Subspaces for Finite-Difference Operators Invariant Properties of Moving Mesh Operators and Applications Applications to Anharmonic Lattices Remarks and Comments on the Literature Open Problems REFERENCES LIST OF FREQUENTLY USED ABBREVIATIONS INDEXshow more