Soliton resolution for equivariant self-dual Chern-Simons-Schr\"odinger equation in weighted Sobolev class

TL;DR

This paper proves that solutions to the equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev space decompose into a soliton and radiation, marking a first in non-integrable nonlinear Schrödinger equations.

Contribution

It establishes soliton resolution for the CSS equation with equivariant symmetry in a weighted Sobolev space, highlighting the role of self-duality and non-local nonlinearity.

Findings

Solutions decompose into at most one modulated soliton and radiation.

Nonscattering part must be a single modulated soliton.

First soliton resolution result for a non-integrable NLS-type equation.

Abstract

We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS) under equivariant symmetry, which is a $L^{2}$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton resolution for any solutions with equivariant data in the weighted Sobolev space $H^{1,1}$: every maximal solution decomposes into at most one modulated soliton and a radiation. A striking fact is that the nonscattering part must be a single modulated soliton. To our knowledge, this is the first result on soliton resolution in a class of nonlinear Schr\"odinger equations which are not known to be completely integrable. The key ingredient is the defocusing nature of the equation in the exterior of a soliton profile. This is a consequence of two distinctive features of (CSS): self-duality and non-local nonlinearity.