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

Abstract

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g 2 κ + 1 ( ¯¯¯ 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 ω t 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 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 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 .