Two-dimensional Riemann problem for Chaplygin gas dynamics in four pieces

In this paper we study the two-dimensional Riemann problem for Chaplygin gas with initial data being four states, and focus on the wave structures with fewer restriction on the initial data. For the solution made up of elementary waves and the density ρ satisfying 0<ρ<+∞, we mainly analyse the necessary and sufficient condition on the initial data as the self-similar solution can be constructed uniquely. In particular, if the initial densities are identical, we obtain a necessary and sufficient condition concretely and construct the solution explicitly.