Adiabatic Perturbation Theory for the Soliton of the Nonlinear Dirac Equation in 1D

ABSTRACT We derive equations for the slow changes of parameters of the soliton of the 1D nonlinear Dirac, or Gross–Neveu, equation under the action of a small perturbation. Our perturbation theory uses the neutral modes of the linearized operator of the nonlinear Dirac equation. In addition to the conventional soliton parameters such as its frequency (related to the amplitude and width), velocity, and shifts of the center and phase, we also account for an additional parameter, related the so‐called Bogoliubov symmetry of the Dirac equation, which was first pointed out almost half a century ago and rediscovered in the last decade. This aspect of our theory allows one to explain both asymmetric changes of the soliton profile and large growth of the soliton amplitude, which was observed in previous studies via numerical simulations.