We explore a prototypical two-dimensional model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions consisting of a soliton in one component and a vortex in the other to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These results pave the way for a systematic stability and dynamics analysis of higher dimensional waveforms in a broad class of nonlinear Dirac models and a comparison revealing nontrivial differences with respect to their better understood non-relativistic analogue, the nonlinear Schrodinger equation.
Cuevas–Maraver, Jesús; Kevrekidis, P. G.; Saxena, Avadh; Comech, Andrew; and Lan, Ruomeng, "Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model" (2015). Mathematics and Statistics Department Faculty Publication Series. 1212.
Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/1212