Non-crossing permutations for the KP solitons under the Gel'fand-Dickey reductions and the vertex operators

TL;DR

This paper classifies regular soliton solutions of the KP hierarchy under Gel'fand-Dickey reductions using permutations, demonstrating the role of non-crossing permutations in ensuring solution regularity and systematically constructing solutions via vertex operators.

Contribution

It introduces a permutation-based classification of KP solitons under Gel'fand-Dickey reductions and links non-crossing permutations to solution regularity, providing a systematic construction method.

Findings

Regular solitons classified by permutations.

Non-crossing permutations ensure solution regularity.

Explicit construction of solutions via vertex operators.

Abstract

We give a classification of the $regular$ soliton solutions of the KP hierarchy, referred to as the $KP solitons$, under the Gel'fand-Dickey $\ell$-reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have $at ~most$ one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the $\ell$-reductions using the vertex operators. In particular, we show that the $non-crossing$ permutation gives the regularity condition for the soliton solutions.