Small BGK waves and nonlinear Landau damping

TL;DR

This paper investigates the existence of BGK waves and the validity of nonlinear Landau damping in various Sobolev spaces for the 1D Vlasov-Poisson system, revealing complex dynamics near homogeneous equilibria.

Contribution

It establishes the presence of nontrivial BGK waves in certain Sobolev neighborhoods and the absence in others, clarifying the nonlinear dynamics near homogeneous states.

Findings

Nontrivial BGK waves exist in Sobolev spaces with s<1+(1/p).

No nontrivial BGK waves in spaces with s>1+(1/p) under Penrose stability.

Linear damping persists in rough spaces for linearly stable states.

Abstract

Consider 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space W^{s,p} (p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and any spatial period. Indeed, in W^{s,p} (s<1+(1/p)) neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose's linear stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some W^{s,p} (p>1,s>1+(1/p)) neighborhood. Furthermore, when p=2,we prove that there exist no nontrivial invariant structures in the H^{s} (s>(3/2)) neighborhood of stable homogeneous states. These results suggest the long time dynamics in the W^{s,p} (s>1+(1/p)) and particularly, in the H^{s} (s>(3/2)) neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for initial perturbations in very rough spaces, for linearly stable homogeneous state. This suggests that the contrasting dynamics in W^{s,p} spaces with the critical power s=1+(1/p) is a trully nonlinear phenomena which can not be traced back to the linear level.