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

TL;DR

This paper rigorously analyzes how discrete spectrum and resonances affect the long-time asymptotics of Toda shock waves, especially the phase shift in the finite-gap solution, extending previous research.

Contribution

It provides a detailed mathematical description of the influence of discrete spectrum and resonances on the asymptotic behavior of Toda shock waves.

Findings

Discrete spectrum causes a phase shift in the theta-function representation.

Resonances at gap endpoints influence the phase and asymptotics.

Results extend previous asymptotic analysis of Toda shock waves.

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].