On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy

Abstract

We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_t+u_{xxx}+6u{u}_x{)}_x+3u_{yy}=0$. Those solutions also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for $y\to \pm \infty$, and we show that all the solutions (except the one-soliton solution) are of {\it resonant} type, consisting of arbitrary numbers of line solitons in both aymptotics; that is, arbitrary $N_{-}$ incoming solitons for $y\to -\infty$ interact to form arbitrary $N_{+}$ outgoing solitons for $y\to \infty$. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a {\it web-like} structure having $(N_{-}-1)(N_{+}-1)$ holes.

18 pages, 16 figures, submitted to JPA; Math. Gen