Affine Solitons: A Relation Between Tau Functions, Dressing and B\"acklund 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 \(\tau\)-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.