  In this note we research the Abelian Higgs model subject to the Born-Infeld
theory of electrodynamics for which the BPS equations can be reduced into a
quasi-linear differential equation. We show that the equation exists a unique
solution under two interesting boundary conditions which realize the
corresponding phase transition. We construct the solution through a dynamical
shooting method for which the correct shooting slope is unique. We also obtain
the sharp asymptotic estimate for the solution at infinity.
