Numerical study of the KP equation for non-periodic waves

TL;DR

This paper numerically investigates the KP equation's initial value problem with V- and X-shaped waves, demonstrating convergence to exact line-soliton solutions and relating findings to rogue wave phenomena.

Contribution

It provides the first detailed numerical analysis of non-periodic wave evolution in the KP equation, showing convergence to known soliton solutions.

Findings

Solutions converge asymptotically to exact solitons in $L^2$-sense.

V- and X-shaped initial waves evolve into stable line-solitons.

Results relate to rogue wave formation in shallow water.

Abstract

The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions in this paper. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. In this paper, we study the initial value problem of the KP equation with V- and X-shape initial waves consisting of two distinct line-solitons by means of the direct numerical simulation. We then show that the solution converges asymptotically to some of those exact soliton solutions. The convergence is in a locally defined $L^2$-sense. The initial wave patterns considered in this paper are related to the rogue waves generated by nonlinear wave interactions in shallow water wave problem.