Transverse instability of line solitons in massive Dirac equations

TL;DR

This paper analyzes the transverse stability of line solitons in two-dimensional massive Dirac equations, demonstrating analytically and numerically that these solitons are unstable under large-period transverse perturbations.

Contribution

It provides the first analytical proof of transverse instability for line solitons in these specific massive Dirac models, with numerical validation for various perturbation periods.

Findings

Line solitons are transversely unstable in both models.

Instability is driven by spatial translation and gauge rotation.

Numerical results show instability persists for all periods in one model and has a threshold in the other.

Abstract

Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross--Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the gauge rotation for the massive Gross--Neveu model. We also observe numerically that the instability holds for the transverse perturbations of any period in the massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the massive Gross--Neveu model.