Uniqueness and non-degeneracy for a nuclear nonlinear Schr\"odinger equation

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).