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In the present work, we introduce a discrete formulation of the nonlinearDirac equation in the form of a discretization of the Gross-Neveu model. Themotivation for this discrete model proposal is both computational (near thecontinuum limit) and theoretical (using the understanding of the anti-continuumlimit of vanishing coupling). Numerous unexpected features are identifiedincluding a staggered solitary pattern emerging from a single site excitation,as well as two- and three-site excitations playing a role analogous to one- andtwo-site, respectively, excitations of the discrete nonlinear Schr\"odingeranalogue of the model. Stability exchanges between the two- and three-sitestates are identified, as well as instabilities that appear to be persistentover the coupling strength $\epsilon$, for a subcritical value of thepropagation constant $\Lambda$. Variations of the propagation constant,coupling parameter and nonlinearity exponent are all examined in terms of theirexistence and stability implications and long dynamical simulations are used tounravel the evolutionary phenomenology of the system (when unstable).

Solitary Waves in a Discrete Nonlinear Dirac equation