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Handbook of mathematics for engineers and scienteists part 3

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This function is defined for all x, and its range coincides with the j/-axis. The arcsinh x is an odd. nonperiodic. unbounded function that crosses the axes Ox and Oy at the origin x = 0, y = 0. This is an increasing function on the entire real axis with no points of extremum. The graph of the function y = arcsinh x is given in Fig. 2.18. | 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. Painleve 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 Painleve Test and Truncated Expansions for Studying Nonlinear Equations . 750 15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painleve Test Using Truncated Expansions. 753 15.12. Methods of the Inverse Scattering Problem Soliton Theory . 755 15.12.1. Method Based on 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 .