Affine Solitons: A Relation Between Tau Functions, Dressing and B\"acklund Transformations

TL;DR

This paper explores the construction of sine-Gordon solitons via dressing transformations, linking tau functions, B"acklund transformations, and affine Lie algebra representations to provide new insights into soliton solutions.

Contribution

It establishes a connection between dressing transformations, tau functions, and affine Lie algebra representations in constructing sine-Gordon solitons, offering new computational methods.

Findings

Solitons are in the orbit of the vacuum under dressing transformations.

Dressed tau-functions can be computed via affine Lie algebra actions.

Explicit formulas for tau-functions are derived using vertex operators.

Abstract

We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the $N$-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to built the $N$-soliton solutions from the vacuum solution. The dressed $\tau$-functions can be computed in two different ways~: either using adjoint actions in the affine Lie algebra $\hat {sl_2}$, and this gives the relation with the B\"acklund transformations, or using the level one representations of the affine Lie algebra $\widehat{sl_2}$, and this directly gives the formulae for the $\tau$-functions in terms of vertex operators.