Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 045, 32 pages      arXiv:2012.12371      https://doi.org/10.3842/SIGMA.2021.045

How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave

Iryna Egorova a and Johanna Michor b
a) B. Verkin Institute for Low Temperature Physics and Engineering, 47, Nauky Ave., 61103 Kharkiv, Ukraine
b) Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received January 21, 2021, in final form April 26, 2021; Published online May 01, 2021

Abstract
We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].

Key words: Toda equation; Riemann-Hilbert problem; steplike; shock.

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