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In the present work, we explore a nonlinear Dirac equation motivated as thecontinuum limit of a binary waveguide array model. We approach the problem bothfrom a near-continuum perspective as well as from a highly discrete one.Starting from the former, we see that the continuum Dirac solitons can becontinued for all values of the discretization (coupling) parameter, down tothe uncoupled (so-called anti-continuum) limit where they result in a 9-siteconfiguration. We also consider configurations with 1- or 2-sites at theanti-continuum limit and continue them to large couplings, finding that theyalso persist. For all the obtained solutions, we examine not only theexistence, but also the spectral stability through a linearization analysis andfinally consider prototypical examples of the dynamics for a selected number ofcases for which the solutions are found to be unstable.

Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum