Variational approach for (3+1)-dimensional shallow water wave equation

This paper focuses on the variational approach for solving the (3+1)-dimensional shallow water wave equation. Some families of soliton-type solutions and periodic wave solutions are given, and some remarks about the variational formulation for the special (2+1)-dimensional shallow water wave equation in [Gu, et al. Results in Physics, 45 (2023) 106213] are provided. Numerical results are presented to show the propagation behaviour