Plasma echoes near stable Penrose data

In this paper we construct particular solutions to the classical Vlasov-Poisson system near stable Penrose initial data on $$\mathbb{T}$ \times $\mathbb{R}$$ that are a combination of elementary waves with arbitrarily high frequencies. These waves mutually interact giving birth, eventually, to an infinite cascade of echoes of smaller and smaller amplitude. The echo solutions do not belong to the analytic or Gevrey classes studied by Mouhot and Villani, but do, nonetheless, exhibit damping phenomena for large times.