Rogue waves and lump-type solutions of a (3+1)-dimensional higher-order Kadomtsev-Petviashvili equation via bilinear neural network method

Rogue waves and lump solutions are crucial for describing extreme waves in higher dimensions. This paper reports the (3+1)-dimensional higher-order Kadomtsev-Petviashvili (hKP) equation, which governs the propagation of weakly nonlinear dispersive waves in shallow water. We present a bilinear neural network-based framework for symbolic computation to derive novel localized wave solutions. By configuring the network with a single hidden layer, we successfully obtain three distinct hybrid lump solutions consists of lump-sine, lump-exponential and lump-hyperbolic solutions. We reveal that each lump returns to its initial form, amplitude and velocity after a collision due to the elastic features exhibited in lump-sine interaction. Similarly, we establish a double hidden layer configuration to generate a rogue wave solution. The dynamic characteristics and evolutionary behaviors of the derived wave solutions are illustrated through surface, line and density plots, utilizing constraints and suitable selective parameter values. These solutions are reported for the first time by this method and enrich the analysis of hKP equation in fluid dynamics.