δ′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)δ(n)-shock wave, where δ(n)δ(n) is n  th derivative of the Dirac delta function (n=1,2,…n=1,2,…). In this paper the case n=1n=1 is studied in details. We introduce a definition of δ′δ′-shock wave type solution for the systemut+(f(u))x=0,vt+(f′(u)v)x=0,wt+(f″(u)v2+f′(u)w)x=0. Within the framework of this definition, the Rankine–Hugoniot conditions for δ′δ′-shock are derived and analyzed from geometrical point of view. We prove δ′δ′-shock balance relations connected with area transportation  . Finally, a solitary δ′δ′-shock wave type solution to the Cauchy problem of the system of conservation laws ut+(u2)x=0ut+(u2)x=0, vt+2(uv)x=0vt+2(uv)x=0, wt+2(v2+uw)x=0wt+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.