We consider the classical equations of the Born-Infeld-Abelian-Higgs model (with and without coupling to gravity) in an axially symmetric ansatz. A numerical analysis of the equations reveals that the (gravitating) Nielsen-Olesen vortices are smoothly deformed by the Born-Infeld interaction, characterized by a coupling constant $beta^2$, and that these solutions cease to exist at a critical value of $beta^2$. When the critical value is approached, the length of the magnetic field on the symmetry axis becomes infinite.
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