New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with Maple™, Mathematica®, and MATLAB® Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They outline the methods in a schematic, simplified manner and arrange the material in increasing order of complexity.

Table of Contents:
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