On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike   Finite-Gap Initial Data I. Schwartz-Type Perturbations
  We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finite-gap potentials under the assumption that the respective spectral bands either coincide or are disjoint. 
Iryna Egorova, Katrin Grunert, and Gerald Teschl