  The generalized time-dependent Ginzburg-Landau (GTDGL) theory was first
proposed to describe better gap superconductors and the phenomenon of thermal
phase-slips (PSs) in defect-free systems. However, there is a lack of
information about studies involving PSs in mesoscopic superconductors with
surface defects. Thus, in this work, we simulated samples with two co-linear
surface defects consisting of a lower $T_c$ superconductor narrowing the sample
in its central part. The non-linear GTDGL equations were solved
self-consistently under variable applied currents and by considering both
gapless and gap-like superconductors. In such systems, the currents passing by
the constriction induce the appearance of kinematic vortices even in the
gapless sample. The dynamics always occur with a pair forming at opposite edges
of the sample and annihilating in the center. It is noticed that the resistive
state appears at distinct values of the applied current density for different
samples, and the critical current presents a tiny difference between gapless
and gap-like samples. It is worth mentioning that parameters such as the size
of electrical contacts and constriction affect the critical current and the
average velocity of the kinematic vortices.
