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

TL;DR

This paper analyzes the long-time behavior of the Toda shock problem, revealing a five-region structure with solutions described by free backgrounds and modulated two-band solutions, confirming a 1991 conjecture.

Contribution

It provides the first detailed asymptotic description of the Toda shock problem using nonlinear steepest descent, confirming a longstanding conjecture about solution structure.

Findings

Solution regions split into five main parts.

Middle region described by a two band solution.

Separating regions have slowly modulated solutions.

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 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.