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
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.
|Austrian Science Fund
|Research supported by the Austrian Science Fund (FWF) under Grants No. Y330, V120, and by the grant ”Network of Mathematical Research 2013–2015”.
PDF file (1055 kB)
MSC: Primary 37K40, 37K10; Secondary 37K60, 35Q15
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
\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.
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