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Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

Iryna Egorova, Johanna Michor, and Gerald Teschl

Abstract
We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann-Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to the free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.

MSC2000: Primary 37K40, 37K10; Secondary 37K60, 35Q15
Keywords: Toda lattice, Riemann-Hilbert problem, steplike

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