  In this paper we prove that all small, uniformly in time $L^1\cap H^1$
bounded solutions to KdV and related perturbations must converge to zero, as
time goes to infinity, locally in an increasing-in-time region of space of
order $t^{1/2}$ around any compact set in space. This set is included in the
linearly dominated dispersive region $x\ll t$. Moreover, we prove this result
independently of the well-known supercritical character of KdV scattering. In
particular, no standing breather-like nor solitary wave structures exists in
this particular regime. For the proof, we make use of well-chosen weighted
virial identities. The main new idea employed here with respect to previous
results is the fact that the $L^1$ integral is subcritical with respect to the
KdV scaling.
