Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory

This paper is devoted to a study of problems involving the propagation of one-dimensional non-linear waves of small amplitude in elastic media, using analytic and numerical methods. The equations of non-linear elasticity theory belong to the class of hyperbolic systems expressing conservation laws. For the unique construction of solutions it is necessary to supplement these equations with terms that make it possible to adequately describe actual small-scale phenomena, including the structure of the discontinuities that arise. The behaviour of non-linear waves is considered in two cases: when the small-scale processes are conditioned by viscosity, and when dispersion plays an essential role in addition to viscosity.