The localized excitation on the Weierstrass elliptic function periodic background for the (<i>n</i>+1)-dimensional generalized Kadomtsev–Petviashvili equation

Abstract In order to investigate physically meaning localized nonlinear waves on the periodic background defined by Weierstrass elliptic ℘ -function for the ( n + 1)-dimensional generalized Kadomtsev–Petviashvili equation by Darboux transformation, the associated linear spectral problem with the Weierstrass function as the external potential is studied by utilizing the Lamé function. The degenerate solutions of the nonlinear waves have also been obtained by approaching the limits of the half-periods ω 1 and ω 2 of ℘ ( x ). At the same time, the evolution and nonlinear dynamics of various nonlinear waves under different parameter regimes are systematically discussed. The findings may open avenues for related experimental investigations and potential applications in various nonlinear science domains, such as nonlinear optics and oceanography.