Journal of Mathematical Physics, Analysis, Geometry
2018, vol. 14, No 4, pp. 406-451   https://doi.org/10.15407/mag14.04.406     ( to contents , go back )
https://doi.org/10.15407/mag14.04.406

Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

Iryna Egorova

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: iraegorova@gmail.com

Johanna Michor

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
E-mail: Johanna.Michor@univie.ac.at

Gerald Teschl

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
E-mail: Gerald.Teschl@univie.ac.at

Received January 6, 2018.

To Vladimir Aleksandrovich Marchenko with deep admiration on the occasion of his 95th birthday

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.

Mathematics Subject Classification 2000: Primary 37K40, 37K10; Secondary 37K60, 35Q15.
Key words: Toda lattice, Riemann-Hilbert problem, shock wave.

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