<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Sihong Shao</style></author><author><style face="normal" font="default" size="100%">Niurka R. Quintero</style></author><author><style face="normal" font="default" size="100%">Franz G. Mertens</style></author><author><style face="normal" font="default" size="100%">Fred Cooper</style></author><author><style face="normal" font="default" size="100%">Avinash Khare</style></author><author><style face="normal" font="default" size="100%">Avadh Saxena</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">   Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity</style></title><secondary-title><style face="normal" font="default" size="100%">Physical Review E</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.032915</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">90</style></volume><pages><style face="normal" font="default" size="100%">032915</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">We consider the nonlinear Dirac equation in 1+1 dimension with scalar-scalar&amp;nbsp; self interaction $ \frac{ g^2}{ \kappa+1} ( {\bar \Psi} \Psi)^{ \kappa+1}$ and with mass $m$.&amp;nbsp; &amp;nbsp; Using&amp;nbsp; the exact analytic form for&amp;nbsp; rest frame solitary waves of the form $\Psi(x,t) = \psi(x) e^{-i \omega t}$&amp;nbsp; for arbitrary $ \kappa$, we &amp;nbsp;discuss the validity&amp;nbsp; of various approaches to understanding stability that were successful for the nonlinear Schr\&quot;odinger equation. In particular we study the validity of a version of&amp;nbsp; Derrick's theorem, the criterion of Bogolubsky&amp;nbsp; as well as&amp;nbsp; the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results.&amp;nbsp;Therefore, we study the stability by numerical&amp;nbsp; simulations using a recently developed 4th-order&amp;nbsp;operator splitting integration method. For different ranges of $\kappa$ we map&amp;nbsp;out the stability regimes in $\omega$.&amp;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.&amp;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&amp;nbsp;$t_c$ decreases monotonically with increasing $\kappa$.</style></abstract></record></records></xml>