Page 15 -

xiv CONTENTS

15.6.3. Differentiation Method .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 700
15.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and
Their Applications .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 704
15.7. Direct Method of Symmetry Reductions of Nonlinear Equations . . . . . . . . . . . . . . . . . . 708
15.7.1. Clarkson–Kruskal Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
15.7.2. Some Modifications and Generalizations .. .. .. .. .. ... .. .. .. .. .. .. .. .. 712
15.8. Classical Method of Studying Symmetries of Differential Equations . . . . . . . . . . . . . . . 716
15.8.1. One-Parameter Transformations and Their Local Properties . . . . . . . . . . . . . . 716
15.8.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition . . . . 719
15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant
Solutions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 724
15.8.4. Some Generalizations. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . 730
15.9. Nonclassical Method of Symmetry Reductions . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 732
15.9.1. Description of the Method. Invariant Surface Condition . . . . . . . . . . . . . . . . . 732
15.9.2. Examples: The Newell–Whitehead Equation and a Nonlinear Wave Equation 733
15.10. Differential Constraints Method .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 737
15.10.1. Description of the Method .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 737
15.10.2. First-Order Differential Constraints . .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 739
15.10.3. Second- and Higher-Order Differential Constraints . . . . . . . . . . . . . . . . . . . 744
15.10.4. Connection Between the Differential Constraints Method and Other
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
15.11. Painlev´ e Test for Nonlinear Equations of Mathematical Physics . . . . . . . . . . . . . . . . . 748
15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method
Description . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 748
15.11.2. Examples of Performing the Painlev´ e Test and Truncated Expansions for
Studying Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlev´ e
Test,Using Truncated Expansions .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 753
15.12. Methods of the Inverse Scattering Problem (Soliton Theory) . . . . . . . . . . . . . . . . . . . . 755
15.12.1. Method Basedon Using Lax Pairs .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 755
15.12.2. Method Based on a Compatibility Condition for Systems of Linear
Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 757
15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760
15.13. Conservation Laws and Integrals of Motion . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 766
15.13.1. Basic Definitions and Examples . .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 766
15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries . . . 767
15.14. Nonlinear Systems of Partial Differential Equations .. .. .. .. ... .. .. .. .. .. .. .. .. 770
15.14.1. Overdetermined Systems of Two Equations .. .. .. ... .. .. .. .. .. .. .. .. 770
15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined
Systems . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 772
15.14.3. Systems of First-Order Equations Describing Convective Mass Transfer
with Volume Reaction . .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 775
15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of
Conservation Laws of Gas Dynamic Type .. .. .. .. ... .. .. .. .. .. .. .. .. 780
15.14.5. Systems of Second-Order Equations of Reaction-Diffusion Type . . . . . . . . 796
References for Chapter 15 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 798