Delta shock wave in the compressible Euler equations for a Chaplygin gas

We study the Riemann problem of the compressible Euler equations for the Chaplygin gas. With the analysis on the physically relevant region, we obtain two kinds of Riemann solutions by using the method of characteristic analysis. One composes of three contact discontinuities, and the other involves a delta shock wave in which both density and internal energy contain Dirac delta function simultaneously. We propose both generalized Rankine-Hugoniot relation and entropy condition for this type of delta shock wave. The numerical results coinciding with the theoretical analysis are also presented.