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Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar self interaction $ \frac{g^2}{κ+1} ({\bar Ψ} Ψ)^{κ+1}$ and with mass $m$. Using the exact analytic form for rest frame solitary waves of the form $Ψ(x,t) = ψ(x) e^{-i ωt}$ for arbitrary $ κ$, 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, 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 4th-order operator splitting integration method. For different ranges of $κ$ we map out the stability regimes in $ω$. 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 $ω$ and $t_c$ decreases monotonically with increasing $κ$.