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We consider the nonlinear Dirac (NLD) equation in 1+1 dimension withscalar-scalar self-interaction in the presence of external forces as well asdamping of the form $ f(x,t) - i \mu \gamma^0 \Psi$, where both $f$ and $\Psi$are two-component spinors. We develop an approximate variational approach usingcollective coordinates (CC) for studying the time dependent response of thesolitary waves to these external forces. This approach predicts intrinsicoscillations of the solitary waves, i.e. the amplitude, width and phase alloscillate with the same frequency. The translational motion is also affected,because the soliton position oscillates around a mean trajectory. We thencompare the results of the variational approximation with numerical simulationsof the NLD equation, and find a good agreement, if we take into account acertain linear excitation with specific wavenumber that is excited togetherwith the intrinsic oscillations such that the momentum in a transformed NLDequation is conserved. We also solve explicitly the CC equations of thevariational approximation in the non-relativistic regime for a homogeneousexternal force and obtain excellent agreement with the numerical solution ofthe CC equations.

Nonlinear Dirac equation solitary waves in the presence of external driving forces