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D.E. Pelinovsky and Y. Shimabukuro

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Transverse instability of line solitons in massive Dirac equations

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**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.

**Keywords**:

nonlinear Dirac equations, transverse stability of solitary waves,
massive Thirring model, massive Gross-Neveu model, perturbation theory for eigenvalues, numerical approximations