One usually considers two types of vortices in superconductors, Abrikosov vortices in the bulk and Josephson vortices inside the Josephson junction
or in strongly layered superconductors when the magnetic field is parallel to the layers. There are situations where vortices are something in between.
Away from the core they look like conventional Abrikosov vortices, but inside the core the order parameter is not fully suppressed. This incomplete
suppression of the order parameter in the core does not affect the properties of the vortex in the London region. Also if the order parameter drops, e.g.,
to the half of the equilibrium value within the same coherence length, pinning would be qualitatively the same. However, the low temperature dynamics
of such a vortex would be completely different. A non-vanishing order parameter induces an energy gap for quasi-particles and thus their absence at low
temperatures. As a result, the Bardeen Stephen dissipation vanishes when T -> 0. The dynamics then would be given by the electromagnetic vortex mass.
The current voltage curve would show hysteresis with a voltage jumping from zero below the critical current to a finite voltage value above it.