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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.

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.

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.

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.

By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections.

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.