Transverse Instability of Line Solitary Waves in Massive Dirac Equations

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 solitary waves are spectrally unstable with respect to periodic transverse perturbations of large periods. The spectral instability is induced by the spatial translation for the generalized massive Thirring model and by the gauge rotation for the generalized massive Gross-Neveu model. We also observe numerically that the spectral instability holds for the transverse perturbations of any period in the generalized massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the generalized massive Gross-Neveu model.