@article {26631,
	title = {   Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity},
	journal = {Physical Review E},
	volume = {90},
	year = {2014},
	pages = {032915},
	abstract = {We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar\&nbsp; self interaction $ \frac{ g^2}{ \kappa+1} ( {\bar \Psi} \Psi)^{ \kappa+1}$ and with mass $m$.\&nbsp; \&nbsp; Using\&nbsp; the exact analytic form for\&nbsp; rest frame solitary waves of the form $\Psi(x,t) = \psi(x) e^{-i \omega t}$\&nbsp; for arbitrary $ \kappa$, we \&nbsp;discuss the validity\&nbsp; 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\&nbsp; Derrick{\textquoteright}s theorem, the criterion of Bogolubsky\&nbsp; as well as\&nbsp; the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results.\&nbsp;Therefore, we study the stability by numerical\&nbsp; simulations using a recently developed 4th-order\&nbsp;operator splitting integration method. For different ranges of $\kappa$ we map\&nbsp;out the stability regimes in $\omega$.\&nbsp;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.\&nbsp;We also find that the time $t_c$, it takes for the instability to set in, is an exponentially increasing function of $\omega$ and\&nbsp;$t_c$ decreases monotonically with increasing $\kappa$.},
	url = {https://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.032915},
	author = {Sihong Shao and Niurka R. Quintero and Franz G. Mertens and Fred Cooper and Avinash Khare and Avadh Saxena}
}
