Self-consistent multiple complex-kink solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems

We exhaust all exact self-consistent solutions of complex-valued fermionic condensates in the 1+1 dimensional Bogoliubov-de Gennes and chiral Gross-Neveu systems under uniform boundary conditions. We obtain $n$ complex (twisted) kinks, or grey solitons, with $2n$ parameters corresponding to their positions and phase shifts. Each soliton can be placed at an arbitrary position while the self-consistency requires its phase shift to be quantized by $\pi \mathrm{/}N$ for $N$ flavors.