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

TL;DR

This paper proves that small solutions to KdV equations decay to zero locally in a region growing like the square root of time, regardless of the supercritical nature of scattering, using weighted virial identities.

Contribution

It introduces a novel approach leveraging the subcritical nature of the $L^1$ integral for KdV, proving decay without the need for soliton or breather structures.

Findings

Small solutions decay to zero in the linearly dominated region

Decay holds uniformly in time for solutions bounded in $L^1 \cap H^1$

No breather or solitary wave structures exist in this regime

Abstract

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.