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

TL;DR

This paper investigates the stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity, revealing that stability depends on wave profile and parameters, and that traditional criteria are inconsistent.

Contribution

It provides a comprehensive numerical stability analysis of solitary waves in the nonlinear Dirac equation for arbitrary nonlinearity, challenging existing analytical criteria.

Findings

Stable waves are single-hump profiles.

Two-hump waves are always unstable.

Instability onset time increases exponentially with frequency.

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 Schr\"odinger 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 $\kappa$.