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

Victor A. Galaktionov and Sergey R. Svirshchevskii
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2007
Number of Pages: 
498
Format: 
Hardcover
Series: 
Chapman & Hall/CRC Applied Mathematics & Nonlinear Science 10
Price: 
89.95
ISBN: 
1584886633
Category: 
Monograph
We do not plan to review this book.

 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-Ampère 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

Dummy View - NOT TO BE DELETED