Doubly Periodic Self-Dual Vortices for a Relativistic Non-Abelian Chern--Simons Model

TL;DR

This paper proves the existence of at least two doubly periodic solutions for a nonlinear elliptic system modeling self-dual non-Abelian Chern--Simons vortices, depending on the coupling parameter.

Contribution

It establishes the existence of multiple solutions for the system using variational methods, revealing solution behavior as the coupling parameter varies.

Findings

At least two solutions exist for small coupling parameter

No solutions exist for large coupling parameter

One solution is gauge-equivalent to the vacuum as coupling tends to zero

Abstract

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions for a $2\times2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern--Simons vortices. We show that the given system admits at least two solutions when the Chern--Simons coupling parameter $\kappa>0$ is sufficiently small; while no solutions exist for $\kappa>0$ sufficiently large. As in [36] we use a variational formulation of the problem. Thus, we obtain a first solution via a (local) minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as $\kappa\to 0$. Then we obtain the second solution by a min-max procedure of "mountain pass" type.