The Existence Of Ambj(?)rn-Nielsen-Olesen Vortex Solution

In this paper, we prove the existence for a class of Abelian vortex and non-Abelian vortex in the large flux limit and Ambj(?)rn-Nielsen-Olesen (ANO) magnetic instabilities.The first part is devoted to study a special structure of Abelian vortex which is called hole-vortex model. The BPS equations of the hole-vortex model can be changed to a second-order nonlinear ordinary differential equation. We establish the existence and uniqueness theorem for this problem by using a shooting method and present the gradual estimate at infinity under appropriate conditions. The second part is concentrated on the study of non-Abelian ordinary equations. We establish the existence theorems for the coupled second-order ordinary equations on the compact range [-R,R] and full line R.On the compact range [-R,R] we will use a direct minimization method to establish the existence of solutions. On the full line R , we will consider weighted space and use a constrained minimization method to establish the existence of solutions.