Backlund transformation and L2-stability of NLS solitons

TL;DR

This paper proves L2-orbital stability of 1-solitons in a 1D cubic NLS equation using Backlund transformations, extending stability results to broader initial data classes.

Contribution

It establishes L2-orbital stability of 1-solitons for the cubic NLS and introduces a novel use of Backlund transformations for this purpose.

Findings

L2-orbital stability of 1-solitons proved for initial data close in L2

Solutions with additional H3 regularity stay near a specific soliton in L2

Backlund transformation is key to the stability proof

Abstract

Ground states of a L2-subcritical focusing nonlinear Schrodinger (NLS) equation are known to be orbitally stable in the energy class H1 thanks to its variational characterization. In this paper, we will show L2-orbital stability of 1-solitons to a one-dimensional cubic NLS equation for any initial data which are close to 1-solitons in L2. Moreover, we prove that if the initial data are in H3 in addition to being small in L2, then the solution remains in an L2-neighborhood of a specific 1-soliton solution for all the time. The proof relies on the Backlund transformation between zero and soliton solutions of this integrable equation.