Codimension one threshold manifold for the critical gKdV equation

TL;DR

This paper constructs a codimension-one threshold manifold near the soliton for the mass critical gKdV equation, distinguishing between blowup and global solutions, and showing the soliton's strong instability.

Contribution

It completes the construction of the threshold manifold for the critical gKdV, clarifying the boundary between blowup and global behaviors near the soliton.

Findings

The threshold manifold separates blowup and global solutions.

Solutions on the manifold converge locally to a soliton.

The soliton is strongly unstable by blowup.

Abstract

We construct the 'threshold manifold' near the soliton for the mass critical gKdV equation, completing results obtained in arXiv:1204.4625 and arXiv:1204.4624. In a neighborhood of the soliton, this C1 manifold of codimension one separates solutions blowing up in finite time and solutions in the 'exit regime'. On the manifold, solutions are global in time and converge locally to a soliton. In particular, the soliton behavior is strongly unstable by blowup.