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

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.

39 pages

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Cited by in corpus (3)

• Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
• Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
• Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation