Covariant Gross-Pitaevskii-like equation for relativistic fermions

The covariant Gross-Pitaevskii-like equation for studying systems of interacting relativistic fermions is proposed. The Lagrangian generating such an equation is derived. The considered equation is examined in studying the fermi-liquid, superconductivity and the confinement problem in QCD. The Migdal's step in the fermi-liqiud model, the excitation spectrum of superconductive fermion system are derived in the context of the proposed Gross-Pitaevskii-like equation. We show that the quark fields governing such an equation turn out to be confined in the (3+1) space time.