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

We consider the self-dual Chern--Simons--Schr\"{o}dinger 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 H1,1: 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\"{o}dinger 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 nonlocal nonlinearity.