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Abstract

We discuss the definitions of singular solutions (in the form of integral identities) to systems of conservation laws such as shocks, $\delta$-, $\delta'$-, and $\delta^{(n)}$-shocks ($n=2,3,\dots$). Using these definitions, the Rankine–Hugoniot conditions for $\delta$- and $\delta'$-shocks are derived. The weak asymptotics method for the solution of the Cauchy problems admitting $\delta$- and $\delta'$-shocks is briefly described. The algebraic aspects of such singular solutions are studied. Namely, explicit formulas for flux-functions of singular solutions are computed. Though the flux-functions are nonlinear, they can be considered as “right” singular superpositions of distributions, thus being well defined Schwartzian distributions. Therefore, singular solutions of Cauchy problems generate algebraic relations between their distributional components