Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation

TL;DR

This paper develops multi-component Wronskian solutions for the KP equation using Darboux transformations, revealing complex soliton interactions and bound states, advancing the understanding of integrable systems.

Contribution

It introduces N-th iterated Darboux transformations for coupled AKNS systems to construct multi-component Wronskian solutions for the KP equation, including fully-resonant and bound state solitons.

Findings

Constructed multi-component Wronskian solutions for KP equations.

Derived fully-resonant line-soliton solutions with arbitrary asymptotic solitons.

Discovered bound states of parallel propagating KP line solitons.

Abstract

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 nonlinearization 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->\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.