  In this paper our first aim is to identify a large class of non-linear
functions $\,f(\cdot)\,$ for which the IVP for the generalized Korteweg-de
Vries equation does not have breathers or "small" breathers solutions. Also 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.
