Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associ- ated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical per- spective and then complemented with numerical computations. Finally, the dynam- ics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr¨odinger equation.AEI/FEDER, European Union project MAT2006-79866-