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 $\frac{g^2}{\kappa +1}(\bar{\Psi }\Psi {)}^{\kappa +1}$ 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 Schrödinger 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 $\kappa$.

35 pages, 13 figures

References in corpus (2)

• Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity
• Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates

Cited by in corpus (4)

• Solitary Waves in a Discrete Nonlinear Dirac equation
• Study of instability of the Fourier split-step method for the massive Gross--Neveu model
• Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains
• Three discontinuous Galerkin methods for one- and two-dimensional nonlinear Dirac equations with a scalar self-interaction