Solitary waves in the nonlinear Dirac equation in the presence of external driving forces

We consider the nonlinear Dirac (NLD) equation in (1 + 1) dimensions with scalar–scalar self interaction $\frac{g^2}{\kappa +1}{\left(\overline{\Psi }\Psi \right)}^{\kappa +1}$ in the presence of external forces as well as damping of the form $f\left(x\right)-i\mu {\gamma }^0\Psi$, where both f and Ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also affected, because the soliton position oscillates around a mean trajectory. For $\kappa =1$ we solve explicitly the CC equations of the variational approximation for slow moving solitary waves in a constant external force without damping and find reasonable agreement with solving numerically the CC equations. We then compare the results of the variational approximation with no damping with numerical simulations of the NLD equation for $\kappa =1$, when the components of the external force are of the form $f_j=r_{j}exp(-iKx)$ and again find agreement if we take into account a certain linear excitation with specific wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. (paper)