Uniqueness of Transonic Shock Solutions to Euler-Poisson System with Varying Background Charges

In this paper, we investigate a one-dimensional Euler-Poisson system with varying background charges, which are two different constants when the flow speed is less than and greater than the sound speed. Using the shock matching method, we derive the properties of the solution trajectories and establish a monotonic relationship between the density at the nozzle exit and the shock position. This relationship demonstrates the existence and uniqueness of a transonic shock solution under suitable boundary conditions.