Stability of the Periodic Toda Lattice in the Soliton Region
  We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the periodic (and slightly more generally of the quasi-periodic finite-gap) Toda lattice for decaying initial data in the soliton region. In addition, we show how to reduce the problem in the remaining region to the known case without solitons. 
Helge Krueger and Gerald Teschl