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 Zh. Mat. Fiz. Anal. Geom., 2018, Volume 14, Number 4, Pages 406–451 (Mi jmag706)

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

Iryna Egorovaa, Johanna Michorb, Gerald Teschlb

a B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
b Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

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 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.

Key words and phrases: Toda lattice, Riemann–Hilbert problem, shock wave.

 Funding Agency Grant Number Austrian Science Fund Y330V120 Research supported by the Austrian Science Fund (FWF) under Grants No. Y330, V120, and by the grant ”Network of Mathematical Research 2013–2015”.

DOI: https://doi.org/10.15407/mag14.04.406

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Document Type: Article
MSC: Primary 37K40, 37K10; Secondary 37K60, 35Q15
Language: English

Citation: Iryna Egorova, Johanna Michor, Gerald Teschl, “Long-time asymptotics for the Toda shock problem: non-overlapping spectra”, Zh. Mat. Fiz. Anal. Geom., 14:4 (2018), 406–451

Citation in format AMSBIB
\Bibitem{EgoMicTes18} \by Iryna~Egorova, Johanna~Michor, Gerald~Teschl \paper Long-time asymptotics for the Toda shock problem: non-overlapping spectra \jour Zh. Mat. Fiz. Anal. Geom. \yr 2018 \vol 14 \issue 4 \pages 406--451 \mathnet{http://mi.mathnet.ru/jmag706} \crossref{https://doi.org/10.15407/mag14.04.406}