Multi-component Wronskian solution to the Kadomtsev–Petviashvili equation

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.