The Riemann problem with delta initial data with Dirac delta function in both components for a pressureless gas dynamic model

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function in both components for a nonstrictly hyperbolic system of two conservation laws from zero-pressure gas dynamics by introducing the new state variabe. In solutions, we find another kind of delta shocks on which both state variables simultaneously contain the Dirac delta functions. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. This phenomenon is also captured numerically and experimentally by Kreji , Kruni and Nedeljkov [N. Kreji , T. Kruni , M. Nedeljkov, Numerical verification of delta shock waves for pressureless gas dynamics, J. Math. Anal. Appl. 345 (2008) 243-257]. Moreover, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity.