Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics: 1st Edition (Hardback) book cover

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

1st Edition

By Victor A. Galaktionov, Sergey R. Svirshchevskii

Chapman and Hall/CRC

528 pages | 63 B/W Illus.

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Description

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.

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

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

About the Series

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics
SCI041000
SCIENCE / Mechanics / General