Small BGK waves and nonlinear Landau damping

Details Small BGK waves and nonlinear Landau damping Small BGK waves and nonlinear Landau damping

2010-03-15

arxiv.org/abs/1003.3005v1

Mathematics

Analysis of PDEs

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Small BGK waves and nonlinear Landau damping does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Small BGK waves and nonlinear Landau damping, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Small BGK waves and nonlinear Landau damping will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-215125