Fission and annihilation phenomena of breather/rogue waves and interaction phenomena on nonconstant backgrounds for two KP equations

In this paper, we investigate various nonlinear phenomena of two Kadomtsev–Petviashvili (KP) equations. Based on bilinear neural network method, we present the fission and annihilation phenomena of breather waves and rogue waves on non-zero backgrounds of a (3+1)-dimensional KP equation by its reduced equation. Then with the aid of symmetry transformation and Hirota bilinear form, we obtain three interaction solutions on nonconstant backgrounds of a (2+1)-dimensional KP equation with variable coefficients. Also, we perform the analysis of the dynamic characteristics and evolution behaviors of the obtained solutions through three-dimensional animations with proper choices of different values for the parameters. This paper shows that the bilinear neural network method combined with symmetry analysis effectively solves high-dimensional differential equations with constant and variable coefficients.