Asymptotic stability of Toda lattice solitons

We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of a linearized Bäcklund transformation whose domain has codimension one. Combining a linear stability result with a general theory of nonlinear stability by Friesecke and Pego for solitary waves in lattice equations, we conclude that all solitons in the Toda lattice are asymptotically stable in an exponentially weighted norm. In addition, we determine the complete spectrum of an operator naturally associated with the Floquet theory for these lattice solitons