Methods of investigation, theory and classification of reversible shock structures in models of hydrodynamic type

This paper was written as a result of the analysis of numerical solutions of partial differential equations for various models of continuum mechanics and solutions of ordinary differential equations that describe travelling waves for these models. Examples of typical models, the basic principles of the theory of reversible shocks structures, classification of structures, typical of the types of solutions of an arbitrary discontinuity decay problem, the numerical methods used for analysis of solutions of ordinary differential equations, methods used for numerical solution of partial differential equations are given. The theory includes such elements as averaged equations, conditions of evolutionality, conditions of complete and partial reversibility, conditions of existence of solution in typical case that are based on analysis of dimensions of invariant manifolds and number of additional variations, classification of periodic waves, solitary waves and kinks based on number of free parameters.