On Kaup-Kupershchmidt type equations and their soliton solutions

TL;DR

This paper explores the Kaup-Kupershmidt equation's integrability, its relation to Kac-Moody algebras, and constructs explicit one-soliton solutions using the dressing method.

Contribution

It establishes the connection between scalar and matrix Lax operators for KKE and derives soliton solutions via the Zakharov-Shabat dressing method.

Findings

KKE's Lax representation is linked to Kac-Moody algebra structures.

Two types of one-soliton solutions are constructed and analyzed.

The spectral properties of the Lax operator are characterized.

Abstract

We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the MKdV equations gauge equivalent to the KKE. Next we outline the symmetry and the spectral properties of the relevant Lax operator. Using the dressing Zakharov-Shabat method we demonstrate that the MKdV and KKE have two types of one-soliton solutions and briefly comment on their properties.