From the Painlevet test to methods for constructing analytical solutions of nonlinear ODEs

Abstract: The application of the Painlevet test to analyze nonlinear ordinary differential equations is discussed. A brief review of classical works by S. V. Kovalevskaya on solving the problem of motion of a rigid body with a fixed point and works by P. Penleve on the classification of one class of second-order equations is given. The well-known example of the Korteweg–de Vries equation taking into account the traveling wave solutions illustrates the Painlevet property for a nonlinear oscillator. Special attention is paid to non-integrable partial differential equations such as the Korteweg–de Vries–Burgers equation and the Kuramoto–Sivashinsky equation. Using traveling wave solutions, the construction of analytical solutions to these equations is illustrated. Possible applications of the simplest equations method for constructing analytical solutions of non-integrable differential equations are discussed. The application of the method for constructing optical solitons of a generalized nonlinear Schrodinger equation of unrestricted order with nonlinearity in the form of a polynomial is illustrated.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods