On a class of initial value problems and solitons for the KP equation: A numerical study

Abstract

Recent studies show that the Kadomtsev–Petviashvili (KP) equation admits a large class of exact solutions, referred to as the KP solitons, which are solitary waves localized along distinct rays and form web-like patterns in the xy-plane. It is also shown that each KP soliton can be uniquely parametrized by a specific element of the symmetric group of permutations. This paper presents a numerical study of the initial value problem of the KP equation, for certain classes of initial data which are not a small perturbation of any KP soliton. The numerical simulations demonstrate that the initial condition evolves to certain types of KP solitons whose permutations can be found from the initial data, as well as dispersive waves. In this sense, the KP scenario is analogous to its one-dimensional counterpart, namely, the Korteweg–de Vries (KdV) equation. The solution of the KdV equation is known to evolve asymptotically into a sum of individual solitons and dispersive radiation. Although some numerical studies on KP were reported earlier in connection with the Mach reflection phenomena, the present study includes a much larger class of initial data than those discussed in the previous studies.