Waveguide arrays, their underlying Dirac equations and solitons living in the Dirac gap

De Jager, Johannes Louis Wilhelmus

Waveguide arrays, their underlying Dirac equations and solitons living in the Dirac gap

Thesis / Dissertation

2026

Publisher

Unversity of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

This thesis is a mathematical study of optical waveguide arrays, their underlying spinor equations, and soliton solutions. We discuss the origin and fundamental properties of the linear and nonlinear Dirac equations, including their invariances and the absence of analytical stability criteria. Equipped with an understanding of the properties of spinor solitons, we consider two novel waveguide structures. The first of these is a one-dimensional nonlinear waveguide array consisting of a pair of coupled periodic binary chains with a periodic variation of the propagation constant. This system is shown to reduce to the one-dimensional massive Thirring model. We present the derivation of soliton solutions of this system and examine their properties numerically. In the two-dimensional setting, we start by establishing the existence of Dirac equations in hexagonal lattices. Results of this analysis are used to study a novel waveguide array consisting of two hexagonal lattices stacked on top of each other with a unit offset and which has alternating gain and loss. We derive the stationary multivortex solutions of the two-dimensional Dirac equation arising in the array's continuum limit. Stability of these vortices is examined numerically by means of the Chebyshev spectral methods.

Keywords

Mathematics

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