Handbook of Nonlinear Partial Differential Equations, Second Edition

EXACT SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
First-Order Quasilinear Equations
Equations with Two Independent Variables Containing Arbitrary Parameters
Equations with Two Independent Variables Containing Arbitrary Functions
Other Quasilinear Equations





First-Order Equations with Two Independent Variables Quadratic in Derivatives
Equations Containing Arbitrary Parameters
Equations Containing Arbitrary Functions





First-Order Nonlinear Equations with Two Independent Variables of General Form
Nonlinear Equations Containing Arbitrary Parameters
Equations Containing Arbitrary Functions of Independent Variables
Equations Containing Arbitrary Functions of Derivatives





First-Order Nonlinear Equations with Three or More Independent Variables
Nonlinear Equations with Three Variables Quadratic in Derivatives
Other Nonlinear Equations with Three Variables Containing Parameters
Nonlinear Equations with Three Variables Containing Arbitrary Functions
Nonlinear Equations with Four Independent Variables
Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters
Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions





Second-Order Parabolic Equations with One Space Variable
Equations with Power Law Nonlinearities
Equations with Exponential Nonlinearities
Equations with Hyperbolic Nonlinearities
Equations with Logarithmic Nonlinearities
Equations with Trigonometric Nonlinearities
Equations Involving Arbitrary Functions
Nonlinear Schrödinger Equations and Related Equations





Second-Order Parabolic Equations with Two or More Space Variables
Equations with Two Space Variables Involving Power Law Nonlinearities
Equations with Two Space Variables Involving Exponential Nonlinearities
Other Equations with Two Space Variables Involving Arbitrary Parameters
Equations Involving Arbitrary Functions
Equations with Three or More Space Variables
Nonlinear Schrödinger Equations





Second-Order Hyperbolic Equations with One Space Variable
Equations with Power Law Nonlinearities
Equations with Exponential Nonlinearities
Other Equations Involving Arbitrary Parameters
Equations Involving Arbitrary Functions
Equations of the Form





Second-Order Hyperbolic Equations with Two or More Space Variables
Equations with Two Space Variables Involving Power Law Nonlinearities
Equations with Two Space Variables Involving Exponential Nonlinearities
Nonlinear Telegraph Equations with Two Space Variables
Equations with Two Space Variables Involving Arbitrary Functions
Equations with Three Space Variables Involving Arbitrary Parameters
Equations with Three or More Space Variables Involving Arbitrary Functions





Second-Order Elliptic Equations with Two Space Variables
Equations with Power Law Nonlinearities
Equations with Exponential Nonlinearities
Equations Involving Other Nonlinearities
Equations Involving Arbitrary Functions





Second-Order Elliptic Equations with Three or More Space Variables
Equations with Three Space Variables Involving Power Law Nonlinearities
Equations with Three Space Variables Involving Exponential Nonlinearities
Three-Dimensional Equations Involving Arbitrary Functions
Equations with n Independent Variables





Second-Order Equations Involving Mixed Derivatives and Some Other Equations
Equations Linear in the Mixed Derivative
Equations Quadratic in the Highest Derivatives
Bellman-Type Equations and Related Equations





Second-Order Equations of General Form
Equations Involving the First Derivative in t
Equations Involving Two or More Second Derivatives





Third-Order Equations
Equations Involving the First Derivative in t
Equations Involving the Second Derivative in t
Hydrodynamic Boundary Layer Equations
Equations of Motion of Ideal Fluid (Euler Equations)
Other Third-Order Nonlinear Equations





Fourth-Order Equations
Equations Involving the First Derivative in t
Equations Involving the Second Derivative in t
Equations Involving Mixed Derivatives





Equations of Higher Orders
Equations Involving the First Derivative in t and Linear in the Highest Derivative
General Form Equations Involving the First Derivative in t
Equations Involving the Second Derivative in t
Other Equations





Systems of Two First-Order Partial Differential Equations
Systems of the Form
Other Systems of Two Equations





Systems of Two Parabolic Equations
Systems of the Form
Other Systems of Two Parabolic Equations


Systems of Two Second-Order Klein–Gordon Type Hyperbolic Equations
Systems of the Form





Systems of Two Elliptic Equations
Systems of the Form
Other Systems of Two Second-Order Elliptic Equations
Von Kármán Equations (Fourth-Order Elliptic Equations)





First-Order Hydrodynamic and Other Systems Involving Three or More Equations
Equations of Motion of Ideal Fluid (Euler Equations)
Adiabatic Gas Flow
Systems Describing Fluid Flows in the Atmosphere, Seas, and Oceans
Chromatography Equations
Other Hydrodynamic-Type Systems
Ideal Plasticity with the von Mises Yield Criterion





Navier–Stokes and Related Equations
Navier–Stokes Equations
Solutions with One Nonzero Component of the Fluid Velocity
Solutions with Two Nonzero Components of the Fluid Velocity
Solutions with Three Nonzero Fluid Velocity Components Dependent on Two Space Variables
Solutions with Three Nonzero Fluid Velocity Components Dependent on Three Space Variables
Convective Fluid Motions
Boundary Layer Equations (Prandtl Equations)





Systems of General Form
Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to t
Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to t
Other Nonlinear Systems of Two Equations
Nonlinear Systems of Many Equations Involving the First Derivatives with Respect to t





EXACT METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Methods for Solving First-Order Quasilinear Equations
Characteristic System. General Solution
Cauchy Problem. Existence and Uniqueness Theorem
Qualitative Features and Discontinuous Solutions of Quasilinear Equations
Quasilinear Equations of General Form


Methods for Solving First-Order Nonlinear Equations
Solution Methods
Cauchy Problem. Existence and Uniqueness Theorem
Generalized Viscosity Solutions and Their Applications





Classification of Second-Order Nonlinear Equations
Semilinear Equations in Two Independent Variables
Nonlinear Equations in Two Independent Variables





Transformations of Equations of Mathematical Physics
Point Transformations: Overview and Examples
Hodograph Transformations (Special Point Transformations)
Contact Transformations. Legendre and Euler Transformations
Differential Substitutions. Von Mises Transformation
Bäcklund Transformations. RF Pairs
Some Other Transformations





Traveling-Wave Solutions and Self-Similar Solutions
Preliminary Remarks
Traveling-Wave Solutions. Invariance of Equations under Translations
Self-Similar Solutions. Invariance of Equations under Scaling Transformations





Elementary Theory of Using Invariants for Solving Equations
Introduction. Symmetries. General Scheme of Using Invariants for Solving Mathematical Equations
Algebraic Equations and Systems of Equations
Ordinary Differential Equations
Partial Differential Equations
General Conclusions and Remarks





Method of Generalized Separation of Variables
Exact Solutions with Simple Separation of Variables
Structure of Generalized Separable Solutions
Simplified Scheme for Constructing Generalized Separable Solutions
Solution of Functional Differential Equations by Differentiation
Solution of Functional Differential Equations by Splitting
Titov–Galaktionov Method





Method of Functional Separation of Variables
Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities
Special Functional Separable Solutions. Generalized Traveling-Wave Solutions
Differentiation Method
Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications





Direct Method of Symmetry Reductions of Nonlinear Equations
Clarkson–Kruskal Direct Method
Some Modifications and Generalizations


Classical Method of Symmetry Reductions
One-Parameter Transformations and Their Local Properties
Symmetries of Nonlinear Second-Order Equations. Invariance Condition
Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions
Some Generalizations. Higher-Order Equations
Symmetries of Systems of Equations of Mathematical Physics





Nonclassical Method of Symmetry Reductions
General Description of the Method
Examples of Constructing Exact Solutions





Method of Differential Constraints
Preliminary Remarks. Method of Differential Constraints for Ordinary Differential Equations
Description of the Method for Partial Differential Equations
First-Order Differential Constraints for PDEs
Second-Order Differential Constraints for PDEs. Some Generalized
Connection between the Method of Differential Constraints and Other Methods





Painlevé Test for Nonlinear Equations of Mathematical Physics
Movable Singularities of Solutions of Ordinary Differential Equations
Solutions of Partial Differential Equations with a Movable Pole. Method Description
Performing the Painlevé Test and Truncated Expansions for Studying Some Nonlinear Equations





Methods of the Inverse Scattering Problem (Soliton Theory)
Method Based on Using Lax Pairs
Method Based on a Compatibility Condition for Systems of Linear Equations
Method Based on Linear Integral Equations
Solution of the Cauchy Problem by the Inverse Scattering Problem Method





Conservation Laws
Basic Definitions and Examples
Equations Admitting Variational Form. Noetherian Symmetries


Nonlinear Systems of Partial Differential Equations
Overdetermined Systems of Two Equations
Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems
Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction
First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type
Systems of Second-Order Equations of Reaction-Diffusion Type





SYMBOLIC AND NUMERICAL SOLUTIONS OF NONLINEAR PDES WITH MAPLE, MATHEMATICA, AND MATLAB
Nonlinear Partial Differential Equations with Maple
Introduction
Brief Introduction to Maple
Analytical Solutions and Their Visualizations
Analytical Solutions of Nonlinear Systems
Constructing Exact Solutions Using Symbolic Computation. What Can Go Wrong
Some Errors That People Commonly Do When Constructing Exact Solutions with the Use of Symbolic Computations
Numerical Solutions and Their Visualizations
Analytical-Numerical Solutions


Nonlinear Partial Differential Equations with Mathematica
Introduction
Brief Introduction to Mathematica
Analytical Solutions and Their Visualizations
Analytical Solutions of Nonlinear Systems
Numerical Solutions and Their Visualizations
Analytical-Numerical Solutions





Nonlinear Partial Differential Equations with MATLAB
Introduction
Brief Introduction to MATLAB
Numerical Solutions via Predefined Functions
Solving Cauchy Problems. Method of Characteristics
Constructing Finite-Difference Approximations





SUPPLEMENTS
Painlevé Transcendents
Preliminary Remarks. Singular Points of Solutions
First Painlevé Transcendent
Second Painlevé Transcendent
Third Painlevé Transcendent
Fourth Painlevé Transcendent
Fifth Painlevé Transcendent
Sixth Painlevé Transcendent
Examples of Solutions to Nonlinear Equations in Terms of Painlevé Transcendents


Functional Equations
Method of Differentiation in a Parameter
Method of Differentiation in Independent Variables
Method of Argument Elimination by Test Functions
Nonlinear Functional Equations Reducible to Bilinear Equations





Bibliography


Index