Affine Solitons: A Relation Between Tau Functions, Dressing and Bäcklund Transformations

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 $τ$-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 Bäcklund transformations, or using the level one representations of the affine Lie algebra $\widehat{sl_2}$, and this directly gives the formulae for the $τ$-functions in terms of vertex operators.

33 pages. SPhT-92-055; LPTHE-92-16