Fully localized solitary waves for the forced Kadomtsev–Petviashvili equation

Abstract

The Kadomtsev–Petviashvili (KP) equation is a generalized form of the Korteweg–de Vries equation for a three-dimensional channel flow. Three-dimensional fully localized solitary wave solutions can be obtained when the effect of surface tension is significant; we refer to this as the KP-I equation. An exact solution of a three-dimensional fully localized stationary solution for the KP-I equation has been found in the absence of any obstacle on the rigid channel bottom. However, three-dimensional fully localized stationary solutions for the KP-I equation have not been found in the presence of a forcing neither analytically nor numerically. This forcing term comes from some obstacles in the rigid channel bottom. In this work, we focus on the fully localized solitary wave solutions when a positive bump or a negative hole is given as the rigid bottom configuration. The forced KP-I equation is defined in an infinite domain and it is reduced to a finite computational domain by introducing artificial boundary conditions. Interestingly, there are at least two distinct stationary lump-type solutions for a positive bump, while there are at least three distinct stationary lump-type solutions for a negative hole. Furthermore, we investigate their numerical stability by solving the forced time-dependent KP-I equation using them as initial conditions. Our numerical results confirm that there exists a stable stationary lump-type solitary wave solution for both a positive bump and a negative hole when they evolve in time.