Solitary Waves in a Discrete Nonlinear Dirac equation

TL;DR

This paper introduces a discrete nonlinear Dirac equation model inspired by the Gross-Neveu model, revealing novel solitary wave patterns, stability exchanges, and dynamical behaviors through theoretical analysis and simulations.

Contribution

It presents a new discrete formulation of the nonlinear Dirac equation with comprehensive stability analysis and dynamical simulations, expanding understanding of solitary waves in discrete relativistic systems.

Findings

Discovery of staggered solitary wave patterns from single-site excitation

Identification of stability exchanges between two- and three-site states

Observation of persistent instabilities over coupling strength and their dynamical evolution

Abstract

In the present work, we introduce a discrete formulation of the nonlinear Dirac equation in the form of a discretization of the Gross-Neveu model. The motivation for this discrete model proposal is both computational (near the continuum limit) and theoretical (using the understanding of the anti-continuum limit of vanishing coupling). Numerous unexpected features are identified including a staggered solitary pattern emerging from a single site excitation, as well as two- and three-site excitations playing a role analogous to one- and two-site, respectively, excitations of the discrete nonlinear Schr\"odinger analogue of the model. Stability exchanges between the two- and three-site states are identified, as well as instabilities that appear to be persistent over the coupling strength $\epsilon$, for a subcritical value of the propagation constant $\Lambda$. Variations of the propagation constant, coupling parameter and nonlinearity exponent are all examined in terms of their existence and stability implications and long dynamical simulations are used to unravel the evolutionary phenomenology of the system (when unstable).