Josephson-like transport in overlapping relativistic superconducting vortices

We develop a relativistic gauge-invariant scalar-electrodynamics framework describing coherent transport generated by overlapping superconducting vortex modes embedded within a finite condensate background under crossed electromagnetic fields. The model extends the relativistic Abelian Higgs theory through a non-minimal condensate-dependent electromagnetic coupling, allowing the local electromagnetic response to depend dynamically on the condensate density. Using a variational two-mode overlap ansatz consisting of localized Gaussian vortex excitations superimposed on a homogeneous condensate component, we derive explicit expressions for the charge density, the radial and azimuthal supercurrents generated by coherent vortex interference. The resulting transport exhibits an intrinsic Josephson-like structure arising internally within a continuous superconducting medium without requiring an insulating junction. In particular, the radial current contains both an overlap-induced interference contribution and an electrically driven transport term generated by nonlinear electromagnetic backreaction. The dynamics are governed by the relative vortex phase, producing oscillatory internal charge exchange, phase-sensitive current modulation, and collective transport behavior while remaining consistent with current conservation. The homogeneous condensate background further generates coherent coupling between the localized vortex excitations and the surrounding macroscopic condensate. The framework suggests possible extensions to dense vortex ensembles, nonequilibrium superconducting transport, relativistic condensate analogues, and strongly confined superconducting heterostructures.