Local Dynamics Near Unstable Branches of NLS Solitons

TL;DR

This paper analyzes the local dynamics near unstable solitons of the nonlinear Schrödinger equation, showing solutions either converge to a soliton or diverge, with implications for understanding blowup phenomena.

Contribution

It characterizes the qualitative behavior of solutions near unstable NLS solitons under symmetry and spectral assumptions, extending understanding of their stability and blowup.

Findings

Solutions near unstable solitons either converge or diverge transversally.

The dynamics are independent of whether blowup occurs.

Provides insights into blowup behavior in supercritical NLS.

Abstract

Consider a branch of unstable solitons of NLS whose linearized operators have one pair of simple real eigenvalues in addition to the zero eigenvalue. Under radial symmetry and standard assumptions, solutions to initial data from a neighbourhood of the branch either converge to a soliton, or exit a larger neighbourhood of the branch transversally. The qualitative dynamic near a branch of unstable solitons is irrespective of whether blowup eventually occurs, which has practical implications for the description of blowup of NLS with supercritical nonlinearity.