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Abstract. For two classes of hyperbolic systems of conservation laws new definitions of a δ-shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of δ-shocks. The weak asymptotics method developed by the authors is used to describe the propagation of δ-shock waves to the three types of systems of conservation laws and derive the corresponding Rankine-Hugoniot conditions for δ-shocks.

We describe δ-shock wave generation from continuous initial data in the case of triangular conservation law system arising from "generalized pressureless gas dynamics model." We use smooth approximations in the weak sense that are more general than small viscosity approximations.In this paper we investigate formation of δ-shock wave in the case of triangular system of conservation laws:v t + vg(u) x = 0.(2)

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