Article
J. Math. Phys. Anal. Geom. 19, 396-442 (2023)
[DOI: 10.15407/mag19.02.396]
Long-time asymptotics for Toda shock waves in the modulation region
Iryna Egorova, Johanna Michor, Anton Pryimak, and Gerald Teschl
We show that a Toda shock wave is asymptotically close to a modulated
finite gap solution in the right modulation region.
We previously derived formulas for the leading terms of the asymptotic
expansion of these shock waves in all five principal regions and conjectured that in two modulation regions the
next term is of order O(t-1). In the present paper we prove this fact and investigate
how resonances and eigenvalues influence the leading asymptotic behaviour.
Our main contribution is the solution of the local parametrix Riemann-Hilbert problems and a rigorous
justification of the analysis. In particular, this involves the construction of a proper singular matrix model solution.
MSC2020: Primary 37K40, 35Q53; Secondary 37K45, 35Q15
Keywords: Toda equation, Riemann-Hilbert problem, steplike, shock
Download