Existence of Doubly Periodic Vortices in a Generalized Chern--Simons Model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern--Simons model. We show that there exists a critical value of the coupling parameter such that there exits self-dual doubly periodic vortex solutions for the generalized self-dual Chern--Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern--Simons equation we can also construct doubly periodic vortex solutions to a generalized self-dual Abelian Higgs equation.