Long-time asymptotics for the Toda shock problem: non-overlapping spectra

I. Egorova, J. Michor, and G. Teschl

Keywords: Toda lattice, Riemann-Hilbert problem, steplike.

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 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.

MSC: Primary 37K40, 37K10; Secondary 37K60, 35Q15.

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