Classification of the line-soliton solutions of KPII

TL;DR

This paper classifies line-soliton solutions of the KPII equation based on their asymptotic behavior at infinity, revealing connections to permutation theory and expanding understanding of these solitary wave solutions.

Contribution

It introduces a classification scheme for KPII line-solitons using asymptotic data, linking soliton structures to permutation theory, which was not previously established.

Findings

Classification of solutions by asymptotic behavior

Connection between soliton solutions and permutation theory

Enhanced understanding of KPII line-soliton structures

Abstract

In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190 (2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 (2006)), we found a large variety of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are solitary waves which decay exponentially in $(x,y)$-plane except along certain rays. In this paper, we show that those solutions are classified by asymptotic information of the solution as $|y| \to \infty$. Our study then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations.