08-95 Spyridon Kamvissis and Gerald Teschl

Stability of the Periodic Toda Lattice: Higher Order Asymptotics (56K, LaTeX2e) May 25, 08

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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.

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