  We study vortex solutions in the Born-Infeld theory coupled with a complex
scalar field. We show that for a specific form of the "Higgs" potential the
vortex satisfies a set of Bogomol'nyi-type equations. Another model, with
nonlinear interaction between gauge and Higgs fields, is also considered. We
show how it is derived from a supersymmetric extension of the Born-Infeld
theory with a minimally coupled complex scalar field.
