Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves

TL;DR

This paper investigates how curvature and torsion influence soliton solutions of the nonlinear Dirac equation on planar and space curves, revealing that curvature can alter soliton profiles, with potential applications in curved Bose condensates.

Contribution

It introduces a method to analyze nonlinear Dirac solitons constrained to curved geometries, connecting curvature effects to soliton profile modifications.

Findings

Curvature affects soliton width, causing narrowing or expansion.

The arc variable simplifies the analysis of Dirac equations on curves.

Results are relevant for future curved Bose condensate experiments.

Abstract

We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature and then we can obtain the transformation for the 1+1 dimensional Dirac equation directly from the metric. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.