摘要:

We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar  self interaction $ \frac{ g^2}{ \kappa+1} ( {\bar \Psi} \Psi)^{ \kappa+1}$ and with mass $m$.    Using  the exact analytic form for  rest frame solitary waves of the form $\Psi(x,t) = \psi(x) e^{-i \omega t}$  for arbitrary $ \kappa$, we  discuss the validity  of various approaches to understanding stability that were successful for the nonlinear Schr\"odinger 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 $\kappa$ 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 $\kappa$.

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