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Long-time asymptotics for Toda shock waves in the modulation region

We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion of these shock waves in all principal regions and conjectured that in the modulation region 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.