Soliton resolution for equivariant self-dual Chern-Simons-Schrödinger equation in weighted Sobolev class

2022/02/15 by Kim, Kihyun, Kwon, Soonsik, Oh, Sung-Jin
#35B40 #35Q55 #37K40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.07314

Abstract

We consider the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariant symmetry, which is a L²-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 Schrödinger 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.

Related

• Rigidity of smooth finite-time blow-up for equivariant self-dual Chern-Simons-Schrödinger equation
• Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile
• On Threshold Solutions of equivariant Chern-Simons-Schrödinger Equation
• A Liouville theorem for the Chern--Simons--Schrödinger equation
• Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schrödinger equation