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

TL;DR

This paper proves the uniqueness and non-degeneracy of positive solutions for a nuclear nonlinear Schrödinger equation, addressing challenges from mass dependence, and applies results to construct solutions in a coupled Dirac-Klein-Gordon model.

Contribution

It establishes fundamental properties of solutions to a nuclear NLS equation with mass dependence and constructs solutions for a coupled Dirac-Klein-Gordon system.

Findings

Proved uniqueness of positive solutions

Established non-degeneracy of solutions

Constructed solutions for the σ–ω model

Abstract

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).