 0895 Spyridon Kamvissis and Gerald Teschl
 Stability of the Periodic Toda Lattice: Higher Order Asymptotics
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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
RiemannHilbert 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" RiemannHilbert 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 RiemannHilbert factorization problem
on the complex plane. This result can then be pulled back to the hyperelliptic
curve.
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