Numerical study of solitary wave stability in cubic nonlinear Dirac equations in 1D

Highlights: • Linear stability of massive Gross–Neveu soliton is confirmed by numerical simulations. • Combined nonreflecting and absorbing boundary conditions are essential for long-time simulations of fragile solitary waves. • Second-order split-step method and method of characteristics can simulate spinor solitons over thousands of time units. - Abstract: Recently there has occurred a controversy between the semi-analytical prediction of linear stability of the soliton of the massive Gross–Neveu model and direct numerical observations of its instability for small values of the frequency. We revisit the problem of numerical computation of this soliton, find a mechanism behind the numerical instability observed in earlier studies, and propose methods to stably compute the soliton over long times. Thus, we confirm the semi-analytical prediction of the soliton's being linearly stable.