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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DOI：
10.1201/9781420011623
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发表时间：
2006-11
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影响因子：
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通讯作者：
V. Galaktionov;S. Svirshchevskii
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文献类型：
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作者：
V. Galaktionov;S. Svirshchevskii
简介：非线性偏微分方程和精确解精确解：历史，经典对称方法，扩展例子：经典基本解属于不变子空间模型，目标，和准线性方程中的Preparable线性不变子空间：基本例子和模型历史：First Eexamples of Solutions on INVARIANT Subspaces基本思想：不变子空间和广义变量分离更多例子：多项式子空间例子：三角子空间例子：指数子空间对文献INVARIANT SUBSPACES AND MODULES的评论和评论：一维数学关于不变子空间的主要定理不变子空间维数的最优估计具有最大维数子空间的一阶算子具有最大维数一阶和二阶子空间的二阶算子具有低维子空间的阶二次算子保多项式子空间的扩张t依赖运算符总结：方程和解的基本类型文献的备注和评论开放问题一维抛物方程：薄膜、KURAMOTO-SIVASHINSKY和岩浆模型薄膜模型和多项式子空间在消光、爆破、自由边界问题和界面方程中的应用六阶薄膜方程零接触角消光行为的精确解二次模型：三角和指数子空间2 m阶薄膜算子和方程柯西问题中的振荡、变号行为Kuramoto-Sivashinsky型模型中的不变子空间准线性伪抛物模型：岩浆方程对文献公开问题的评论和评论奇数阶一维方程：KdV型方程的爆破和局部化高阶二次非线性色散模型高阶偏微分方程：奇异mKdV型方程的界面方程与振动解紧子与界面关于IRN中非线性色散方程的紧子“重言式”方程和峰子空间，奇异性和Harry Dym型方程的振动解对文献公开问题的评论和评论一维拟线性波和Boussinesq模型：非线性方程组不变子空间上非线性波动方程的爆破准线性波动方程的呼吸子和爆破模型淬火和界面现象，非线性发展方程组中的紧子不变子空间对文献公开问题的评论在非线性偏微分方程中的应用IRN二阶算子及其高阶扩张二阶算子的扩展不变子空间关于IR 2中的显着算子关于二阶p-非线性色散方程中移动紧致结构的Monge-Ampere型高阶薄膜算子的Laplacian算子的不变子空间从IR N中的不变多项式子空间到IR N-1中的不变三角子空间对文献公开问题部分不变子空间，不变集，多项式算子的广义分离变量与部分不变二次Kuramoto-Sivashinsky方程的广义分离变量方法广义分离与部分不变模高阶演化不变集二阶抛物型方程的符号不变量和精确解准线性模型，定义和第一个例子ut -？形式的符号不变量(u)H型平稳符号不变量（r，u，ur）ut-m（u）（ux）2 - M（u）形式的符号不变量广义一阶Hamilton-Jacobi符号不变量对文献“离散算子的不变子空间”，“移动网格方法”，“离散算子不变子空间的正问题关于有限维不变子空间的正问题差分算子的不变性动网格算子及其应用非调和格上的应用对文献公开问题的评论
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