Vortex Solutions for A Mixed Boundary-Value Problem in the Abelian-Higgs Model with A Neutral Scalar Field

Abstract: Vortices represent a class of topological solitons arising in gauge theories coupled with complex scalar fields, holding significant importance across various domains of modern physics. In this paper we establish the existence of vortex solutions for a mixed boundary-value problem derived from the Abelian-Higgs model incorporating a neutral scalar field, a system recently investigated by Eto, Peterson et al. [7]. By synergistically combining the shooting method with the Schauder fixed-point theorem, we derive sharp analytical criteria that delineate the Abelian vortex phase from the non-Abelian one. We also rigorously establish the monotonicity, uniform boundedness, and precise asymptotic behavior of the vortex profile functions. Our results provide rigorous confirmation of numerical observations regarding the phase boundary between these distinct vortex types.