Spyridon Kamvissis and Gerald Teschl
Stability of the Periodic Toda Lattice: Higher Order Asymptotics
(56K, LaTeX2e)
ABSTRACT. In a recent paper we have considered the long time asymptotics
of the periodic Toda lattice under a short range perturbation and we
have proved that the perturbed lattice asymptotically approaches a
modulated lattice. In the present paper we capture the higher order
asymptotics, at least away from some resonance regions. In particular
we prove that the decay rate is $O(t^{-1/2})$.
Our proof relies on the asymptotic analysis of the associated
Riemann-Hilbert factorization problem, which is here set on a hyperelliptic
curve. As in previous studies of the free Toda lattice, the higher order
asymptotics arise from "local" Riemann-Hilbert factorization problems
on small crosses centered on the stationary phase points. We discover
that the analysis of such a local problem can be done in a chart around each
stationary phase point and reduces to a Riemann-Hilbert factorization problem
on the complex plane. This result can then be pulled back to the hyperelliptic
curve.