Stability of nonlinear Dirac solitons under the action of external potential

TL;DR

This paper investigates the stability of nonlinear Dirac solitons under external potentials, demonstrating their general stability except for low-frequency cases, and clarifies the nature of observed instabilities through numerical and analytical methods.

Contribution

It provides a detailed numerical and analytical study of Dirac soliton stability under external potentials, identifying conditions for stability and clarifying previous reports of instabilities.

Findings

Most Dirac solitons are numerically stable under external potentials.

Low-frequency solitons exhibit long-term oscillations and potential instabilities.

Reported instabilities in related models are shown to be spurious.

Abstract

The instabilities observed in direct numerical simulations of the Gross-Neveu equation under linear and harmonic potentials are studied. The Lakoba algorithm, based on the method of characteristics, is performed to numerically obtain the two spinor components. We identify non-conservation of energy and charge in simulations with instabilities and we find that all studied solitons are numerically stable, except the low-frequency solitons oscillating in the harmonic potential over long periods of time. These instabilities, as in the case of Gross-Neveu equation without potential, can be removed by imposing absorbing boundary conditions. The dynamics of the soliton is in perfect agreement with the prediction obtained using an ansatz with only two collective coordinates, namely the position and momentum of the center of mass. We use the temporal variation of both field energy and momentum to determine the evolution equations satisfied by the collective coordinates. By applying the same methodology, we also demonstrate the spurious character of the reported instabilities in the Alexeeva-Barashenkov-Saxena model under external potentials.