Solitary waves in the Nonlinear Dirac Equation

TL;DR

This paper investigates the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation, providing explicit solutions in one dimension, numerical results in higher dimensions, and exploring special variants like PT-symmetric and discrete models.

Contribution

It offers new analytical and numerical insights into solitary waves in the nonlinear Dirac equation across multiple dimensions and variants.

Findings

Explicit solutions in one dimension

Numerical analogues in higher dimensions

Stability analysis and dynamics comparison

Abstract

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model.