Periodic dynamics of transonic shocks in Euler-Poisson flows

This paper studies the existence and dynamical stability of time-periodic transonic shock solutions to the one-dimensional Euler--Poisson equations under periodic boundary perturbations. The analysis relies on two key mechanisms: the structural dissipation induced by the transonic shock and the stabilization effect of the positive electric field on the shock position. Using a reformulation of the Rankine-Hugoniot conditions and uniform estimates on the flow and shock perturbations, we establish the existence of a time-periodic transonic shock solution. We further prove its dynamical stability under small perturbations of the initial data.