numerical study of the kp equation for non-periodic waves

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 l²-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.