  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
Lame 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. At the same time, 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. These results will be useful for
elucidating and predicting nonlinear phenomena in related physical fields, such
as physical ocean and Bose-Einstein condensates.
