Quasi-periodic traveling electron layers

We consider the one dimensional space-periodic Vlasov-Poisson equations and construct, close to symmetric flat velocity strips, small amplitude traveling quasi-periodic electron-layers, namely strip-shaped patches of electrons in the phase space. These solutions are found for most values of the strip area. The proof uses a Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. Thanks to a suitable linear transformation of the unknowns, we reveal a connection between the electron patch problem and the classical, physically relevant, electronic Euler-Poisson system with cubic pressure law. As a direct, non trivial consequence, we obtain small-amplitude quasi-periodic traveling waves for this model as well. To the best of our knowledge, this work provides the first rigorous example of construction of quasi-periodic solutions for kinetic models.