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.
- Pub Date:
- April 2010
- Nonlinear Sciences - Pattern Formation and Solitons;
- Mathematics - Numerical Analysis
- 32 pages, 25 figures, Submitted for the conference proceeding "The Sixth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena" held at Athens, GA, March 23-26, 2009.