Geodesic
Long-time asymptotics for the Toda shock problem: non-overlapping spectra
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018), pp. 406-451.

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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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Iryna Egorova; Johanna Michor; Gerald Teschl. Long-time asymptotics for the Toda shock problem: non-overlapping spectra. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018), pp. 406-451. http://geodesic.mathdoc.fr/item/JMAG_2018_14_a1/

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