The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

Abstract

The approach by Painlevé to construct the general solutions of some nonlinear second-order ordinary differential equations is generalized for finding exact solutions of nonintegrable differential equations. We demonstrate that this approach generalizes some other well-known algorithms of finding exact solutions: the tanh-function method, the G′/G-expansion method, the Exp-function method and some variants of the simplest equation method. The method allows us to search for both solution in the form of solitary waves and periodic solution expressed through elliptic functions. We demonstrate possibilities of the method for finding exact solutions of nonintegrable ordinary differential equations which are obtained as reductions of the modified Korteweg–de Vries equation with source, the generalized Burgers equation, the generalized Fisher equation, the generalized Gardner equation and the generalized Kuramoto–Sivashinsky equation.