Instability of the soliton for the focusing, mass-critical generalized KdV equation

TL;DR

This paper proves that solitons for the focusing, mass-critical generalized KdV equation are unstable, showing solutions close to the soliton with smaller mass tend to diverge from it over time.

Contribution

It establishes the instability of solitons in the focusing, mass-critical generalized KdV equation for the first time.

Findings

Solutions with initial mass less than the soliton's mass move away from the soliton.

Proximity in $L^{2}$ norm does not guarantee stability for these solitons.

The result highlights the inherent instability in the focusing, mass-critical regime.

Abstract

In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in $L^{2}$ norm must eventually move away from the soliton.