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.

中文翻译：

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高阶模型的代数行波解的探索

目的

本文的目的是构造（3 +1）维修正的KdV-Zakharov-Kuznetsve（KdV-ZK）方程的代数行波解，该方程通常可以用来表示浅水波现象。

设计/方法/方法

作者应用平面动力学系统和不变代数固化方法来找到（3 +1）维修正KdV-ZK方程的代数行波解和有理解。同样，将平面动力学系统和不变代数固化方法应用于考虑的方程，以找到代数行波解。

发现

结果，作者发现积分常数为零和非零，代数行波解具有不同的演化过程。这些结果有助于更好地揭示浅水波现象的演化机制并找到内部联系。

研究局限/意义

本文提出，通过使用平面动力学系统和不变代数固化，作为强大的数学工具的已实现方法可处理（3 +1）维修正KdV-ZK方程。

实际影响

通过考虑代数行波解的重要特征，可以了解浅水波现象的演化机理并找到内部联系。

创意/价值

据作者所知，在其他地方还没有报道过代数行波解。最后，给出了代数行波解的非线性动力学行为。