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. . | xiv Contents . Differentiation Method . 700 . Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications . 704 . Direct Method of Symmetry Reductions of Nonlinear Equations. 708 . Clarkson-Kruskal Direct Method . 708 . Some Modifications and Generalizations . 712 . Classical Method of Studying Symmetries of Differential Equations. 716 . One-Parameter Transformations and Their Local Properties . 716 . Symmetries of Nonlinear Second-Order Equations. Invariance Condition . 719 . Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions . 724 . Some Generalizations. Higher-Order Equations . 730 . Nonclassical Method of Symmetry Reductions. 732 . Description of the Method. Invariant Surface Condition . 732 . Examples The Newell-Whitehead Equation and a Nonlinear Wave Equation 733 . Differential Constraints Method . 737 . Description of the Method. 737 . First-Order Differential Constraints. 739 . Second-and Higher-Order Differential Constraints . 744 . Connection Between the Differential Constraints Method and Other Methods . 746 . Painleve Test for Nonlinear Equations of Mathematical Physics . 748 . Solutions of Partial Differential Equations with a Movable Pole. Method Description. 748 . Examples of Performing the Painleve Test and Truncated Expansions for Studying Nonlinear Equations . 750 . Construction of Solutions of Nonlinear Equations That Fail the Painleve Test Using Truncated Expansions. 753 . Methods of the Inverse Scattering Problem Soliton Theory . 755 . Method Based on Using Lax Pairs. 755 . Method Based on a Compatibility Condition for Systems of Linear Equations . 757 . Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 . Conservation Laws and Integrals of Motion . 766 . Basic Definitions and Examples . 766 .
