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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, astime goes to infinity, locally in an increasing-in-time region of space oforder $t^{1/2}$ around any compact set in space. This set is included in thelinearly dominated dispersive region $x\ll t$. Moreover, we prove this resultindependently of the well-known supercritical character of KdV scattering. Inparticular, no standing breather-like nor solitary wave structures exists inthis particular regime. For the proof, we make use of well-chosen weightedvirial identities. The main new idea employed here with respect to previousresults is the fact that the $L^1$ integral is subcritical with respect to theKdV scaling.

Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region