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.
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