Localized excitation on the Jacobi elliptic periodic background for the (n+1)-dimensional generalized Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem, which associated with the (n+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, with the Jacobi elliptic function as the external potential is investigated based on the Lamé function, from which some novel local nonlinear wave solutions on the Jacobi elliptic function have been obtained by Darboux transformation, and the corresponding dynamics have also been discussed. The degenerate solutions of the nonlinear wave solutions on the Jacobi function background for the gKP equation are constructed by taking the modulus of the Jacobi function to be 0 and 1. The findings indicate that there can be various types of nonlinear wave solutions with different ranges of spectral parameters, including soliton and breather waves. Furthermore, the interplay between nonlinearity and dispersion is found to have observable effects on the propagation dynamics of breather waves. These results will be useful for elucidating and predicting nonlinear phenomena in related physical fields, such as fluid mechanics and physical ocean.

💡 Research Summary

This paper investigates the linear spectral problem associated with the (n + 1)-dimensional generalized Kadomtsev‑Petviashvili (gKP) equation when the external potential is taken to be a Jacobi elliptic function. By recognizing that the spatial part of the Lax pair reduces to a Lamé equation in Jacobi form, the authors construct explicit eigenfunctions in terms of Jacobi theta functions and determine the corresponding eigenvalues as functions of the elliptic modulus k and a spectral parameter λ.

The seed solution for the gKP equation is obtained via a traveling‑wave ansatz u = U(η) with η = x₁ + c t. Balancing the highest‑order derivative and nonlinear terms yields a periodic solution expressed as a quadratic combination of sn(η,k). After fixing the wave speed c = 4β(1 + k²) the simplest seed reduces to U(η) = −2k² sn²(η,k).

With this background, the authors develop the Darboux transformation (DT) for the gKP equation. The one‑fold DT is written as u