Throughout this talk, we are concerned with the long time behavior of
solutions to forced-damped Korteweg-de Vries equations.  When times
goes to the infinity, any trajectory converges towards a set included
in a proper subspace of the phase space under consideration.  This
regularity of the global attractor (or asymptotical smoothing effect)
is quite surprising, since these equations do not feature finite time
smoothing.  We also discuss the case of some Kadomstev-Petviashvili
equation. Part of these results were obtained in collaboration with
M. Abounouh and R. Rosa.