Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation

Abstract

Two exact stationary soliton solutions are found in the parametrically driven and damped nonlinear Dirac equation. The parametric force considered is a complex ac force. The solutions appear when their frequencies are locked to half the frequency of the parametric force, and their phases satisfy certain conditions depending on the force amplitude and on the damping coefficient. Explicit expressions for the charge, the energy, and the momentum of these solutions are provided. Their stability is studied via a variational method using an ansatz with only two collective coordinates. Numerical simulations confirm that one of the solutions is stable, while the other is an unstable saddle point. Consequently, the stabilization of damped Dirac solitons can be achieved via time-periodic parametric excitations.