Consider electrostatic plasmas described by 1D Vlasov-Poisson
with a fixed ion background. In 1946, Landau discovered the linear decay
of electric field near a stable homogeneous state. The nonlinear Landau
damping was recently proved for analytic perturbations by Villani and
Mouhot, but for general perturbations it is still largely open. With
Chongchun Zeng at Georgia Tech, we construct nontrivial traveling waves
(BGK waves) with any spatial period which are arbitrarily near any
homogeneous state in H^s (s<3/2) Sobolev norm of the distribution
function. Therefore, the nonlinear Landau damping is NOT true in H^s
(s<3/2) spaces. We also showed that in small H^s (s>3/2) neighborhoods
of linearly stable homogeneous states, there exist no nontrivial
invariant structures. This suggests that the long time dynamics near
stable homogeneous states in H^s (s>3/2) spaces might be much simpler
and the nonlinear damping might be possible. We also obtained similar
results for the problem of nonlinear inviscid damping of Couette flow,
for which the linear decay was first observed by Orr in 1907.