Asymptotic Stability for KdV Solitons in Weighted $H^s$ Spaces

TL;DR

This paper proves that KdV solitons are asymptotically stable in weighted Sobolev spaces below the energy level, showing exponential decay of perturbations over long times using advanced spectral and $I$-method techniques.

Contribution

It introduces a novel approach combining the $I$-method and spectral analysis to establish asymptotic stability of KdV solitons in weighted spaces below the energy space.

Findings

Perturbations decay exponentially in weighted spaces.

Stability holds for arbitrarily long times in weighted norms.

Global control of unweighted perturbations remains challenging.

Abstract

In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.