Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Application of the shallow water waves in environmental engineering and hydraulic engineering is seen. In this paper, a (3+1)-dimensional generalized nonlinear evolution equation (gNLEE) for the shallow water waves is investigated. The Nth-order Wronskian, Gramian and Pfaffian solutions are proved, where N is a positive integer. Soliton solutions are constructed from the Nth-order Wronskian, Gramian and Pfaffian solutions. Moreover, we analyze the second-order solitons with the influence of the coefficients in the equation and illustrate them with graphs. Through the Hirota-Riemann method, one-periodic-wave solutions are derived. Relationship between the one-periodic-wave solutions and one-soliton solutions is investigated, which shows that the one-periodic-wave solutions can approach to the one-soliton solutions under certain conditions. We reduce the (3+1)-dimensional gNLEE to a two-dimensional planar dynamic system. Based on the qualitative analysis, we give the phase portraits of the dynamic system.

Automated summary

In this paper, a (3+1)-dimensional generalized nonlinear evolution equation for shallow water waves is investigated. Nth-order solutions are proved to exist and soliton solutions are constructed. One-periodic-wave solutions are also derived, and the relationship between these solutions is explored. The equation is further simplified to a two-dimensional dynamic system, and the phase portraits of the system are given based on a qualitative analysis.