Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dirac solitons in one-dimensional nonlinear Schrödinger equations

William Borrelli, Elena Danesi, Simone Dovetta, Lorenzo Tentarelli

TL;DR

The paper proves that Dirac solitons arise as robust, two-scale modulations of Bloch waves in a one-dimensional cubic NLS with a periodic potential, by opening a spectral gap at a Dirac point via a smooth perturbation. It first constructs a nonlinear Dirac spinor solving the effective NLD equation and then embeds it into a rigorous two-scale expansion to produce standing waves of the original NLS with a leading profile given by a Dirac-mode mixture, plus a controlled remainder. The main contributions are a rigorous derivation of the NLD as an effective model near Dirac points in 1D, and a complete fixed-point construction that yields Dirac solitons for a whole interval in the spectral gap, along with explicit decay and symmetry properties. This work provides a solid analytical link between nonlinear Dirac and nonlinear Schrödinger dynamics in periodic media, with potential applications to nonlinear optics and solid-state physics.

Abstract

In this paper we study a family of one-dimensional stationary cubic nonlinear Schrödinger (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.

Dirac solitons in one-dimensional nonlinear Schrödinger equations

TL;DR

The paper proves that Dirac solitons arise as robust, two-scale modulations of Bloch waves in a one-dimensional cubic NLS with a periodic potential, by opening a spectral gap at a Dirac point via a smooth perturbation. It first constructs a nonlinear Dirac spinor solving the effective NLD equation and then embeds it into a rigorous two-scale expansion to produce standing waves of the original NLS with a leading profile given by a Dirac-mode mixture, plus a controlled remainder. The main contributions are a rigorous derivation of the NLD as an effective model near Dirac points in 1D, and a complete fixed-point construction that yields Dirac solitons for a whole interval in the spectral gap, along with explicit decay and symmetry properties. This work provides a solid analytical link between nonlinear Dirac and nonlinear Schrödinger dynamics in periodic media, with potential applications to nonlinear optics and solid-state physics.

Abstract

In this paper we study a family of one-dimensional stationary cubic nonlinear Schrödinger (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.
Paper Structure (19 sections, 11 theorems, 286 equations, 1 figure)

This paper contains 19 sections, 11 theorems, 286 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Setting and main results
  3. Organization of the paper
  4. Features of periodic Schrödinger operators in dimension one
  5. Preliminaries and specific notation
  6. Fundamentals of Floquet--Bloch theory
  7. Dirac points of H
  8. Symmetries of Bloch waves
  9. Proof of Theorem \ref{['thm:main2']}
  10. Two-scale construction of Dirac solitons for (\ref{['NLS']})
  11. Consistency of U
  12. Conditions on U
  13. The equation for eta
  14. Proof of Theorem \ref{['thm:main1']}
  15. Finding far energy components
  16. ...and 4 more sections

Key Result

Theorem 1.6

Let $c_\sharp,\,\vartheta_\sharp,\,\beta_1\in\mathbb{R}\setminus\{0\}$, $\beta_2\in\mathbb{R}$, with $\beta_1\geq|\beta_2|$, and let $\mu_\sharp\in(-|\vartheta_\sharp|,|\vartheta_\sharp|)$. Then, equation NLD admits a non--trivial smooth solution of the form with Moreover, for every $k\in\mathbb{N}$ and every $\varepsilon\in(0,(\vartheta_\sharp^2-\mu_\sharp^2)/c_\sharp^2)$, such solution satisfi

Figures (1)

  • Figure 1: The energy level $H_0$, for $a=b=2$ and $\mu=0.5$.

Theorems & Definitions (46)

  • Remark 1.1
  • Remark 1.3
  • Remark 1.5
  • Theorem 1.6: Solutions to \ref{['NLD']}
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Dirac solitons for \ref{['NLS']}
  • Remark 1.10
  • Definition 2.1: Pseudoperiodic Lebesgue/Sobolev spaces
  • Remark 2.2
  • ...and 36 more