We derive the long-time asymptotics for the Toda shock problem using thenonlinear steepest descent analysis for oscillatory Riemann--Hilbertfactorization problems. We show that the half plane of space/time variablessplits into five main regions: The two regions far outside where the solutionis close to free backgrounds. The middle region, where the solution can beasymptotically described by a two band solution, and two regions separatingthem, where the solution is asymptotically given by a slowly modulated two bandsolution. In particular, the form of this solution in the separating regionsverifies a conjecture from Venakides, Deift, and Oba from 1991.
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