Breathers and interaction phenomena on the non-constant backgrounds for a (3+1)-dimensional generalized shallow water wave equation with variable coefficients

The analytic solutions of water wave equations on the non-constant backgrounds can better describe the complex marine and lake environment, including tidal effects, topographic changes and other factors. In this paper, a (3+1)-dimensional generalized shallow water wave equation with variable coefficients is investigated by the symmetry transformation and bilinear neural network method (BNNM). By constructing the "4-3-1" neural network models, various analytic solutions on the non-constant backgrounds of the equation are successfully obtained, including the breather wave solutions and interaction solutions. Then the dynamic characteristics of these analytic solutions are analyzed through selecting appropriate parameters and 3D animations. It is worth pointing out that the non-constant backgrounds have no effect on the evolutions of breather waves and interaction waves, which is useful for the study and modeling of the marine environments, lakes, and other problems related to water waves.