Landau damping on the torus for the Vlasov-Poisson system with massless electrons

This paper studies the nonlinear Landau damping on the torus ${\mathbb{T}}^d$ for the Vlasov-Poisson system with massless electrons (VPME). We consider solutions with analytic or Gevrey ($\gamma >1\mathrm{/}3$) initial data, close to a homogeneous equilibrium satisfying a Penrose stability condition. We show that for such solutions, the corresponding density and force field decay exponentially fast as time goes to infinity. This work extends the results for Vlasov-Poisson on the torus to the case of ions and, more generally, to arbitrary analytic nonlinear couplings.