Darboux transformation and multi-soliton solutions of Two-Boson hierarchy

TL;DR

This paper explores Darboux transformations for the Two-Boson hierarchy, revealing a modified approach that enables explicit multi-soliton solutions, extending the understanding of integrable models with $SL(2,R) imes U(1)$ symmetry.

Contribution

It introduces a modified Darboux transformation for the Two-Boson hierarchy, allowing for the construction of multi-soliton solutions in a novel integrable model.

Findings

Constructed a modified Darboux transformation with richer structure.

Explicitly derived one-soliton/kink solutions.

Demonstrated multi-soliton solutions via Wronskians.

Abstract

We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on $SL(2,R)$ within the AKNS framework, this model is based on $SL(2,R)\otimes U(1)$. The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on $SL(2,R)$ studied thus far. The conventional Darboux transformation is shown to be quite restricted in this model. We construct a modified Darboux transformation which has a much richer structure and which also allows for multi-soliton solutions to be written in terms of Wronskians. Using the modified Darboux transformations, we explicitly construct one soliton/kink solutions for the model.