Solutions to a nonlinear Dirac–Maxwell system: from periodic waves to soliton-like waves

Abstract

Using variational methods, we study the existence of stationary solutions to a periodic nonlinear Dirac–Maxwell system. Firstly, we prove the existence of n-periodic solutions to the system for all positive integers n. Then we use the concentration-compactness argument and the periodic approximation method for obtaining soliton-like solutions. We also investigate the existence of periodic ground states and soliton-like ground states, as well as their connection. Besides, results for the regularity of solutions and decay rate of the soliton-like solutions are given.