Dirac solitons in one-dimensional nonlinear Schrödinger equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912796873064448 |
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| author | Borrelli, William Danesi, Elena Dovetta, Simone Tentarelli, Lorenzo |
| author_facet | Borrelli, William Danesi, Elena Dovetta, Simone Tentarelli, Lorenzo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirac solitons in one-dimensional nonlinear Schrödinger equations Borrelli, William Danesi, Elena Dovetta, Simone Tentarelli, Lorenzo Analysis of PDEs Mathematical Physics Functional Analysis 35Q55, 35Q51, 35Q40, 81Q05, 35B32, 35B20 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. |
| title | Dirac solitons in one-dimensional nonlinear Schrödinger equations |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis 35Q55, 35Q51, 35Q40, 81Q05, 35B32, 35B20 |
| url | https://arxiv.org/abs/2512.24089 |