Year: 2026
Type: preprint
Abstract: The existence and structure of non-simple blow-up are fundamental questions for the self-dual Chern--Simons--Higgs equation. Physically, this phenomenon describes magnetic-flux concentration in several cores whose mutual distances are much larger than their widths, although all cores approach the same vortex point. Despite substantial progress for singular Liouville equations, the existence of non-simple blow-up solutions to the full Chern--Simons--Higgs equation on closed surfaces has remained open in both the Chern--Simons (CS) and mean-field regimes. We prove that the cores of every non-simple cluster have equal limiting masses, share a common profile type, and form an asymptotically regular polygon. These identities yield an explicit finite list of admissible cluster masses and a four-unit mass-gap criterion. We also construct non-simple blow-up solutions on closed Riemannian surfaces in both regimes. Together, these results establish a sharp arithmetic criterion for single-point non-simple CS blow-up on each fixed flat torus when a suitable exterior divisor may be chosen, and show that vortex blow-up need not be simple, answering Li--Liu's question in the negative.
Source: arXiv (Cornell University)
Authors: Lee, Youngae, Zhang, Lei
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213606202