Multi-component Wronskian solution to the Kadomtsev–Petviashvili equation

Tao Xu   ; Fu-Wei Sun   ; Yi Zhang   ; Juan Li

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 1

Résumé

It is known that the Kadomtsev–Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy by the binary non-linearization of Lax pairs. In this paper, we construct the $N$-th iterated Darboux transformation (DT) for the second- and third-order $m$-coupled AKNS systems. By using together the $N$-th iterated DT and Cramer’s rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as $y\to\mp\infty$ to the KPII equation, and the ordinary $N$-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision.

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