Non-topological solutions in a generalized Chern-Simons model on torus

TL;DR

This paper investigates the existence of non-topological doubly periodic vortex solutions in a generalized Chern-Simons model, demonstrating the construction of solutions with arbitrary bubble numbers and analyzing their structure.

Contribution

It provides the first proof of non-topological doubly periodic solutions in a quasi-linear elliptic equation from a generalized Chern-Simons-Higgs theory, including solutions with any number of bubbles.

Findings

Existence of only two solution types under doubly periodic conditions

Construction of non-topological solutions with arbitrary bubbles

Insight into the structure of the quasi-linear elliptic equation

Abstract

We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we show that under doubly periodic condition, there are only two types of solutions, topological and non-topological solutions as the coupling parameter goes to zero. Moreover, we succeed to construct non-topological solution with $k$ bubbles where $k\in\mathbb{N}$ is any given number. We believe that it is the first result for the existence of non-topological doubly periodic solution of the quasi-linear elliptic equation arising in a generalized self-dual Chern-Simons-Higgs gauge theory. To find a suitable approximate solution, it is important to understand the structure of quasi-linear elliptic equation.