A goal of this preprint is to provide an analytical understanding of dispersive regularizations of hyperbolic shocks, in the context of completely integrable approximations to nonlinear hyperbolic Partial Differential Equations (PDEs) which exhibit shock formation. The fundamental analytical issue is to obtain a complete asymptotic description of continuum limits of integrable systems. Different families of completely integrable systems admit interpretation as semidiscrete approximations to hyperbolic PDEs - of these, the Toda lattice is a famous example. To investigate dispersive regularizations, it is required to carry out an asymptotic analysis of these systems in a very delicate continuum limit. Special attention will be paid to multidimensional (in space variables) generalizations.

Mathematical problems and theory of numerical methods