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

Abstract

We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar selfinteraction in the presence of external forces as well as damping of the form f (x, t) - ιμγͦψ, 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 a ected, because the soliton position oscillates around a mean trajectory. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and nd a good agreement, if we take into account a certain linear excitation with speci c wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved. We also solve explicitly the CC equations of the variational approximation in the non-relativistic regime for a homogeneous external force and obtain excellent agreement with the numerical solution of the CC equations.