  Holographic superconductor model represents various inhomogeneous solutions
with homogeneous sources. In this paper, we study inhomogeneous structures in
the presence of the homogeneous current and the chemical potential. We find
single complex kink condensates, multiple complex kinks condensates and twisted
kink crystal condensates in which both the amplitude and the phase of the order
parameter modulate in space. Analyzing the gauge-invariant phase difference in
the single complex kink condensates, we find the non-monotonic behaviour with
respect to the current. We confirm that the multiple complex kinks condensates
and the twisted kink crystal condensates are well described by the analytic
solutions obtained from the Gross-Neveu model or the Nambu-Jona-Lasinio model.
We also analyze the thermodynamic stability of the complex kink(s) condensates
by computing the free energy. Our results imply that holographic superconductor
model provides the boundary physics which is effectively represented by the
Ginzburg-Landau theory with higher corrections.
