Theory and classification of the reversible structures of discontinuities in hydrodynamic-type models

The results of an analysis of the numerical and analytical solutions of the partial differential equations for different models of continuum mechanics and also the solutions of the ordinary differential equations of the travelling waves for these models are presented. Typical models, the basic theses of the theory of discontinuities in models of the reversible and weakly dissipative type, the classification of time-invariant structures and time-varying ordered structures are considered. The theory includes elements such as the use of averaged equations, evolutionary conditions, conditions for the complete and partial reversibility of a discontinuity, conditions for the existence of a solution in the typical case obtained using dimensional analysis of the invariant manifolds and the number of additional parameters varied and the classification of the periodic waves, solitary waves and kinks with respect to the number of free parameters.