Power series reversion and exact solutions of nonlinear mathematical physics equations

A technique for constructing exact solutions of nonlinear mathematical physics equations is proposed, based on the reversion of a partial sum of a perturbation method series. The latter is represented as a power series in powers of an exponential function, which is a solution of the linearized equation. The rational generating function of a sequence of coefficients of the power series is an exact solution of the original equation. The method is based on the property that the reverted power series for soliton-like solutions terminate, starting from a degree at least one greater than the order of the solution’s pole. The effectiveness of the method is demonstrated by constructing exact localized solutions of a non-integrable Korteweg-de Vries-Burgers equation, as well as nonlinear integrable differential-difference equations.