  We prove the uniqueness and non-degeneracy of positive solutions to a cubic
nonlinear Schr\"odinger (NLS) type equation that describes nucleons. The main
difficulty stems from the fact that the mass depends on the solution itself. As
an application, we construct solutions to the $\sigma$--$\omega$ model, which
consists of one Dirac equation coupled to two Klein-Gordon equations (one
focusing and one defocusing).
