Analytic self-consistent condensates in quasi-1D superfluid fermi gases in the andreev approximation

We present an analytic method to approach Eilenberger equation and the associated Bogoliubov-de Gennes equation for quasi-1D fermionic gases. The problem of finding self-consistent inhomogeneous condensates is reduced to solving a certain class of nonlinear Schrödinger equations, whose most general solitonic solution is indeed available. Previously known solutions can be retrieved by taking appropriate limits in the parameters. The applicability of the method extends to systems with population imbalance and subject to external potential. In particular we show that fermionic zero-modes are robust against population imbalance.