**Preprint**

## Long-time asymptotics for the Toda shock problem: non-overlapping spectra

### I. Egorova, J. Michor, and G. Teschl

**Keywords:** Toda lattice, Riemann-Hilbert problem, steplike.

**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.

**MSC:**
Primary 37K40, 37K10; Secondary 37K60, 35Q15.

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