δ′-Shock waves as a new type of solutions to systems of conservation laws

A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called δ(n)-shock wave, where δ(n) is nth derivative of the Dirac delta function (n=1,2,…). In this paper the case n=1 is studied in details. We introduce a definition of δ′-shock wave type solution for the system Within the framework of this definition, the Rankine–Hugoniot conditions for δ′-shock are derived and analyzed from geometrical point of view. We prove δ′-shock balance relations connected with area transportation. Finally, a solitary δ′-shock wave type solution to the Cauchy problem of the system of conservation laws ut+(u2)x=0, vt+2(uv)x=0, wt+2(v2+uw)x=0 with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.