On the Quasi-Periodic Wave Solutions and Asymptotic Analysis to a (3+1)-Dimensional Generalized Kadomtsev—Petviashvili Equation

Abstract

In this paper, a (3+1)-dimensional generalized Kadomtsev—Petviashvili (GKP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the generalized Bell's polynomials, we succinctly construct the Hirota's bilinear equation to the GKP equation. By virtue of multidimensional Riemann theta functions, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta function periodic waves (quasi-periodic waves) for the (3+1)-dimensional GKP equation. Interestingly, the one-periodic waves are well-known cnoidal waves, which are considered as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional that they have two independent spatial periods in two independent horizontal directions. Finally, we analyze asymptotic behavior of the multiperiodic periodic waves, and rigorously present the relationships between the periodic waves and soliton solutions by a limiting procedure.