Journal
JOURNAL OF MATHEMATICAL PHYSICS ANALYSIS GEOMETRY
Volume 14, Issue 4, Pages 406-451Publisher
B VERKIN INST LOW TEMPERATURE PHYSICS & ENGINEERING
Keywords
Toda lattice; Riemann-Hilbert problem; shock wave
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Funding
- Austrian Science Fund (FWF) [Y330, V120]
- grant Network of Mathematical Research 2013-2015
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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.
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