Existence theorem for Kruglov–Higgs vortices

Establish a rigorous existence theorem, comparable to Taubes’ classical theorem, for vortex solutions of the Kruglov–Higgs system satisfying the Bogomol'nyi equations and the prescribed boundary conditions.

Background

The constitutive-map analysis establishes that, under the stated parameter conditions, the algebraic relation between the auxiliary variable Y and the Higgs profile is real, continuous, and smooth across the relevant field interval. However, this result does not prove that the coupled first-order boundary-value problem admits vortex profiles satisfying the required core and asymptotic boundary conditions.

The paper explicitly identifies the absence of a Taubes-type existence result for the Kruglov–Higgs system. Numerical profiles are obtained for selected values of the nonlinear exponent and parameter, but a general rigorous existence theorem remains unresolved.