Dirac solitons in one-dimensional nonlinear Schr\"odinger equations

TL;DR

This paper investigates Dirac solitons in one-dimensional nonlinear Schrödinger equations with periodic potentials, demonstrating how perturbations create spectral gaps and how the NLD equation effectively models these solitons.

Contribution

It provides a rigorous derivation of Dirac solitons in NLS equations with periodic potentials and justifies the NLD equation as an effective model for these phenomena.

Findings

Spectral gaps can be opened around Dirac points via periodic perturbations.

Dirac solitons are constructed as modulations of Bloch waves.

The NLD equation accurately models the leading-order behavior of these solitons.

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.