  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.
