Toda shock waves: justification of asymptotics

This paper is a continuation of research started in arXiv:1406.0720, where formulas for the leading terms of the long-time asymptotics for the Toda shock wave were derived by the method of nonlinear steepest descent, in all principal domains of the space-time half plane. In the present paper we study the same Riemann--Hilbert problem in more detail, including investigations of the influence of resonances and eigenvalues on the asymptotics in the elliptic zone. Our main task was to solve the local parametrix RH problems and to implement a rigorous conclusive analysis (an analog of "small norm" arguments) which justifies the asymptotics obtained previously.