A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2006 To book contents

15. Nonlinear Partial Differential Equations

• 15.1. Classification of Second-Order Nonlinear Equations
  • 15.1.1. Classification of Semilinear Equations in Two Independent Variables
  • 15.1.2. Classification of Nonlinear Equations in Two Independent Variables
• 15.2. Transformations of Equations of Mathematical Physics
  • 15.2.1. Point Transformations: Overview and Examples
  • 15.2.2. Hodograph Transformations (Special Point Transformations)
  • 15.2.3. Contact Transformations. Legendre and Euler Transformations
  • 15.2.4. Backlund Transformations. Differential Substitutions
  • 15.2.5. Differential Substitutions
• 15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. Similarity Method
  • 15.3.1. Preliminary Remarks
  • 15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations
  • 15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations
  • 15.3.4. Equations Invariant Under Combinations of Translation and Scaling Transformations, and Their Solutions
  • 15.3.5. Generalized Self-Similar Solutions
• 15.4. Exact Solutions with Simple Separation of Variables
  • 15.4.1. Multiplicative and Additive Separable Solutions
  • 15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations
  • 15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations
• 15.5. Method of Generalized Separation of Variables
  • 15.5.1. Structure of Generalized Separable Solutions
  • 15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions
  • 15.5.3. Solution of Functional Differential Equations by Differentiation
  • 15.5.4. Solution of Functional-Differential Equations by Splitting
  • 15.5.5. Titov--Galaktionov Method
• 15.6. Method of Functional Separation of Variables
  • 15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities
  • 15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions
  • 15.6.3. Differentiation Method
  • 15.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications
• 15.7. Direct Method of Symmetry Reductions of Nonlinear Equations
  • 15.7.1. Clarkson--Kruskal Direct Method
  • 15.7.2. Some Modifications and Generalizations
• 15.8. Classical Method of Studying Symmetries of Differential Equations
  • 15.8.1. One-Parameter Transformations and Their Local Properties
  • 15.8.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition
  • 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions
  • 15.8.4. Some Generalizations. Higher-Order Equations
• 15.9. Nonclassical Method of Symmetry Reductions
  • 15.9.1. Description of the Method. Invariant Surface Condition
  • 15.9.2. Examples: The Newell--Whitehead Equation and a Nonlinear Wave Equation
• 15.10. Differential Constraints Method
  • 15.10.1. Description of the Method
  • 15.10.2. First-Order Differential Constraints
  • 15.10.3. Second- and Higher-Order Differential Constraints
  • 15.10.4. Connection Between the Differential Constraints Method and Other Methods
• 15.11. Painleve Test for Nonlinear Equations of Mathematical Physics
  • 15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method Description
  • 15.11.2. Examples of Performing the Painleve Test and Truncated Expansions for Studying Nonlinear Equations
  • 15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painleve Test, Using Truncated Expansions
• 15.12. Methods of the Inverse Scattering Problem (Soliton Theory)
  • 15.12.1. Method Based on Using Lax Pairs
  • 15.12.2. Method Based on a Compatibility Condition for Systems of Linear Equations
  • 15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method
• 15.13. Conservation Laws and Integrals of Motion
  • 15.13.1. Basic Definitions and Examples
  • 15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries
• 15.14. Nonlinear Systems of Partial Differential Equations
  • 15.14.1. Overdetermined Systems of Two Equations
  • 15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems
  • 15.14.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction
  • 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type
  • 15.14.5. Systems of Second-Order Equations of Reaction-Diffusion Type
• References for Chapter 15

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