Exploration of the algebraic traveling wave solutions of a higher order model. Issue 2 (9th July 2020) - Liu, Jian-Gen; Feng, Yi-Ying; Zhang, Hong-Yi
Abstract : Purpose: The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena. Design/methodology/approach: The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions. Findings: As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections. Research limitations/implications: The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure. Practical implications: By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections. Originality/value: To the best of the authors' knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.