Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 3, pp. 369-387   https://doi.org/10.15407/mag17.03.369     ( to contents , go back )

### Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

Bo Xu

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
School of Educational Science, Bohai University, Jinzhou 121013, China
E-mail: bxu@bhu.edu.cn

Sheng Zhang

School of Mathematics and Physics, Bohai University, Jinzhou 121013, China
E-mail: szhangchina@126.com

Received May 19, 2020, revised June 14, 2020.

Abstract

In this paper, a direct method called negative power expansion (NPE) method is presented and extended to construct exact solutions of nonlinear mathematical physical equations. The presented NPE method is also effective for the coupled, variable-coefficient and some other special types of equations. To illustrate the effectiveness, the (2 + 1)-dimensional dispersive long wave (DLW) equations, Maccari’s equations, Tzitzeica–Dodd–Bullough (TDB) equation, Sawada–Kotera (SK) equation with variable coefficients and two lattice equations are considered. As a result, some exact solutions are obtained including traveling wave solutions, non-traveling wave solutions and semi-discrete solutions. This paper shows that the NPE method is a simple and effective method for solving nonlinear equations in mathematical physics.

Mathematics Subject Classification 2010: 35Q51, 35J99, 68W30
Key words: exact solution, NPE method, (2+1)-dimensional DLW equations, Maccari’s equations, TDB equation, SK equation with variable coefficients, lattice equations