New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its B\"acklund transformations

TL;DR

This paper develops new soliton solutions and Bäcklund transformations for the Chen-Lee-Liu and Burgers hierarchies using Riemann-Hilbert methods, classifying solutions via vertex operators and analyzing integrable defects.

Contribution

It introduces a novel framework for constructing and classifying soliton solutions and Bäcklund transformations for these hierarchies using dressing methods and algebraic techniques.

Findings

Explicit multi-soliton solutions for Burgers hierarchy derived.

Classification of solutions based on vertex operator types.

Development of gauge Bäcklund transformations for solution generation.

Abstract

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-B\"acklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.