NONLINEAR WAVES. SOLITONS. AUTOWAVES. SELF-ORGANIZATION

Power series reversion and exact solutions of nonlinear mathematical physics equations

A. I. Zemlyanukhin, N. A. Artamonov, A. V. Bochkarev, V. I. Bezlyudny

Yuri Gagarin State Technical University of Saratov, Russia

Abstract: Purpose. Develop a new method for finding exact solutions to equations of nonlinear mathematical physics. Methods. The partial sum of a perturbation series, written for the original nonlinear equation, is represented as a power series in powers of the exponential function, which is the solution of the linearized equation. The rational generating function of the sequence of coefficients of the power series represents the exact solution of the original equation. The method is based on the property that inverted power series for soliton-like solutions terminates at powers at least one greater than the order of the pole of the solution. Results. The effectiveness of the method is demonstrated in constructing exact localized solutions of the nonintegrable Korteweg–de Vries–Burgers equation, as well as nonlinear integrable differential-difference equations. Conclusion. The proposed method is applicable to solving integrable and non-integrable differential equations with constant coefficients, as well as integrable differential-difference equations.

Keywords: power series reversion, generating function, nonlinear differential equations, differential-difference equations, exact solutions.

UDC: 530.182, 517.912, 517.929

Received: 22.04.2025
Revised: 28.11.2025
Accepted: 22.05.2025

DOI: 10.18500/0869-6632-003180