Orbital stability of KdV multisolitons in $H^{-1}$

TL;DR

This paper proves the orbital stability of multisoliton solutions to the KdV equation in the low-regularity space $H^{-1}$, using a novel variational approach that ensures convergence to the multisoliton manifold.

Contribution

It introduces a new variational characterization of multisolitons in $H^{-1}$ and demonstrates uniform initial proximity convergence, extending stability results to lower regularity.

Findings

Multisolitons are orbitally stable in $H^{-1}( )$.

A variational characterization remains valid at low regularity.

All optimizing sequences converge to the multisoliton manifold.

Abstract

We prove that multisoliton solutions of the Korteweg--de Vries equation are orbitally stable in $H^{-1}(\mathbb{R})$. We introduce a variational characterization of multisolitons that remains meaningful at such low regularity and show that all optimizing sequences converge to the manifold of multisolitons. The proximity required at the initial time is uniform across the entire manifold of multisolitons; this had not been demonstrated previously, even in $H^1$.