Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics

Abstract

In the present work, we give a comparative summary of different recent contributions to the theme of the linear stability and nonlinear dynamics of solitary waves in the nonlinear Dirac equation in the form of the Gross-Neveu model. We indicate some of the key controversial statements in publications within the past few years and we attempt to address them to the best of our current understanding. The conclusion that we reach is that the solitary wave solution of the model is spectrally stable in the cubic nonlinearity case, however, it may become unstable through an instability amounting to the violation of the Vakhitov-Kolokolov criterion for higher exponents. We find that for the Dirac model, the interval of instability is narrower. A fundamental numerical finding of our work is that, contrary to what is the case in the nonlinear Schrodinger analogue of the model, the unstable dynamical evolution, does not lead to collapse (blowup) and hence it appears that the relativistic nature of the model mitigates the collapse instability. Various issues associated with different numerical schemes are highlighted and some possibilities for future alleviation of these is suggested.