Dirac solitons in one-dimensional nonlinear Schr\"odinger equations
Abstract
In this paper we study a family of one-dimensional stationary cubic nonlinear Schr\"odinger (NLS) equations with periodic potentials and linear part displaying Dirac points in the dispersion relation. By introducing a suitable periodic perturbation, one can open a spectral gap around the Dirac-point energy. This allows to construct standing waves of the NLS equation whose leading-order profile is a modulation of Bloch waves by means of the components of a spinor solving an appropriate cubic nonlinear Dirac (NLD) equation. We refer to these solutions as Dirac solitons. Our analysis thus provides a rigorous justification for the use of the NLD equation as an effective model for the original NLS equation.
Cite
@article{arxiv.2512.24089,
title = {Dirac solitons in one-dimensional nonlinear Schr\"odinger equations},
author = {William Borrelli and Elena Danesi and Simone Dovetta and Lorenzo Tentarelli},
journal= {arXiv preprint arXiv:2512.24089},
year = {2026}
}
Comments
39 pages, 1 Figure. Keywords: Dirac solitons, Dirac points, NLS equations, NLD equations, standing waves, periodic Schr\"odinger operators