Article

**Preprint**

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

### Iryna Egorova, Johanna Michor, and Gerald Teschl

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.

** MSC2000:** Primary 37K40, 37K10; Secondary 37K60, 35Q15

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

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