Shao, S. H.,Quintero, N. R.,Mertens, F. G.,Cooper, F.,Khare, A.,Saxena, A.
We consider the nonlinear Dirac equation in 1+ 1 dimension with scalar- scalar self interaction g(2)/k+1((Psi) over bar Psi)(k+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Psi(x, t) = psi(x)e(-lwt) for arbitrary k, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrodinger 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 k 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 k.