Abstract
We report for the first time exact solutions of a completely integrable nonlinear lattice system for which the dynamical variables satisfy a q-deformed Lie algebra - the Lie-Poisson algebra suq(2). The system considered is a q-deformed lattice for which in the continuum limit the equations of motion become the envelope Maxwell-Bloch (or SIT) equations describing the resonant interaction of light with a nonlinear dielectric. Thus the N-soliton solutions we report here are the natural q-deformations, necessary for a lattice, of the well known multi-soliton and breather solutions of self-induced transparency (SIT). The method we use to find these solutions is a generalization of the Darboux-Bäcklund dressing method. The extension of these results to quantum solitons is sketched.