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

TL;DR

This paper develops a variational collective coordinate approach to analyze the dynamics of solitary waves in the nonlinear Dirac equation under external forces and damping, validating the method through numerical simulations.

Contribution

The paper introduces an approximate variational method using collective coordinates to study driven and damped nonlinear Dirac solitary waves, with validation against numerical simulations.

Findings

Intrinsic oscillations of solitary waves are predicted and observed.

The variational approach accurately captures wave dynamics when accounting for specific linear excitations.

Excellent agreement between analytical and numerical solutions in the non-relativistic regime.

Abstract

We consider the nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction in the presence of external forces as well as damping 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 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. We then compare the results of the variational approximation with numerical simulations of the NLD equation, and find a good 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. 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.