Externally driven nonlinear Dirac equation revisited: theory and simulations

The externally driven nonlinear Dirac (NLD) equation with scalar-scalar self-interaction studied in (2016 J. Phys. A: Math. Theor 49 065402) is revisited. By using a variational method and an ansatz with five collective coordinates, the dynamics of the NLD solitons is well described. It is shown that this new ansatz possesses certain advantages, namely the canonical momentum agrees with the field momentum, the energy associated to the collective coordinate equations agrees with the energy of the NLD soliton, whereas the ansatz with either three or four collective coordinates does not. Thus the study of the whole phase space of the system is enhanced. It is also shown that this approach is equivalent to the method of moments: the time variation of the charge, the momentum, the energy, and the first moment of the charge. The advantages of the new ansatz are illustrated by means of numerical simulations.