Stability of periodic soliton equations under short range perturbations

We consider the stability of (quasi-)periodic solutions of soliton equations under short range perturbations and give a complete description of the related long time asymptotics. So far, it is generally believed that a perturbed periodic integrable system splits asymptotically into a number of solitons plus a decaying radiation part, a situation similar to that observed for perturbations of the constant solution. We show here that this is not the case; instead the radiation part does not decay, but manifests itself asymptotically as a modulation of the periodic solution which undergoes a continuous phase transition in the isospectral class of the periodic background solution. We provide an explicit formula for this modulated solution in terms of Abelian integrals on the underlying hyperelliptic Riemann surface and provide numerical evidence for its validity. We use the Toda lattice as a model but the same methods and ideas are applicable to all soliton equations in one space dimension (e.g. the Korteweg–de Vries equation).