Breathers in the elliptic sine-Gordon model

We provide new expressions for the scattering amplitudes in the soliton-antisoliton sector of the elliptic sine-Gordon model in terms of cosets of the affine Weyl group corresponding to infinite products of q-deformed Gamma functions. When relaxing the usual restriction on the coupling constants, the model contains additional bound states which admit an interpretation as breathers. These breather bound states are unavoidably accompanied by Tachyons. We compute the complete S-matrix describing the scattering of the breathers among themselves and with the soliton-antisoliton sector. We carry out various reductions of the model, one of them leading to a new type of theory, namely an elliptic version of the minimal D-n+1((1))-affine Toda field theory.