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

Gino Biondini·Yuji Kodama
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Abstract

We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. 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.

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Gino Biondini, Yuji Kodama. 2003-06-03. On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy. https://doi.org/10.1088/0305-4470%2F36%2F42%2F008

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