We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction $\frac{g^2}{\kappa +1}{\left(\overline{\Psi }\Psi \right)}^{\kappa +1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form $\Psi (x,t)=\psi \left(x\right)e^{-i\omega t}$ for arbitrary $\kappa$, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of $\kappa$ we map out the stability regimes in $\omega$. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time $t_c$, it takes for the instability to set in, is an exponentially increasing function of $\omega$ and $t_c$ decreases monotonically with increasing $\kappa$.

• Received 23 May 2014

DOI:https://doi.org/10.1103/PhysRevE.90.032915