Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation

TL;DR

This paper explores the asymptotic and scattering behaviors of degenerate multi-solitons in the Hirota equation, introducing new solutions and analyzing their unique interaction properties.

Contribution

It constructs new degenerate multi-soliton solutions using Hirota's method and Darboux-Crum transformations, and analyzes their scattering behavior and asymptotic properties.

Findings

Degenerate multi-solitons exhibit distinct scattering characteristics from nondegenerate ones.

The scattering involves only absorb-emit type interactions.

Asymptotic displacements of solutions are computed and characterized.

Abstract

We construct all higher order conserved charges from a general two-dimensional zero curvature condition using a Gardner transformation. Employing two of those charges in the definition of a Hamiltonian allows to view the Hirota equations as an integrable $\mathcal{PT}$-symmetric extension of the nonlinear Schr\"{o}dinger equation. We construct new degenerate multi-soliton solutions from Hirota's direct method as well as Darboux-Crum transformations based on Jordan states. We study the properties of these solutions, computing their asymptotic time-dependent displacements and also show that their scattering process has a distinct characteristic behaviour different from the nondegenerate counterparts allowing only for interactions of absorb-emit type.