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

Abstract

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 INDEX