The previous investigations on delta shock waves were mostly focused on those with Dirac delta function in only one state variable. In this paper, we obtain another kind from the nonlinear chromatography equations, in which the Dirac delta functions develop simultaneously in both state variables. It is strictly proved to satisfy the system in the sense of distributions. The generalized Rankine-Hugoniot relation and entropy condition are clarified. The numerical results completely coinciding with the theoretical analysis are presented.

The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.

This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

The so-called Cargo-LeRoux model is a hyperbolic system of conservation laws derived from the Euler equations with a constant gravity. This paper is concerned with such a model for an isentropic Chaplygin gas. The Riemann problem is solved with four kinds of structures. The first is composed of the contact discontinuities, the second contains an overcompressible δ-wave (δ-shock wave), and the third and fourth involve a contact discontinuity and a kind of non-overcompressible δ-wave. One of the highlights of this paper lies in the introduction of the non-overcompressible δ-waves.

This paper studies the Riemann problem of the isentropic relativistic Euler equations for a Chaplygin gas. The solutions exactly include five kinds. The first four consist of different contact discontinuities while the rest involves deltashock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.Mathematics Subject Classification (2010). 35L65 · 35L45.