Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics
Features
Summary
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.
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-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
Editorial Reviews
"The book can be viewed as a practical guide that introduces a number of techniques for constructing exact solutions of various nonlinear PDEs in Rn for arbitrary dimensions n › 1."
– Valerity A. Yumaguzhin, in Zentralblatt Math, 2009
"I find that the writing style is enjoyable to read and is easy to follow . . . This book would be a very useful resource for anyone introducing graduate students to nonlinear phenomena. It is useful also for the experienced researcher in applied nonlinear PDEs to have such a collection of interesting but simply expressed examples for guidance."
– Philip Broadbridge, in Mathematical Reviews, 2007j
