Nonlinear Schrödinger equation in cylindrical coordinates
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910459310899200 |
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| author | Krechetnikov, R. |
| author_facet | Krechetnikov, R. |
| contents | Nonlinear Schrödinger equation was originally derived in nonlinear optics as a model for beam propagation, which naturally requires its application in cylindrical coordinates. However, the derivation was done in the Cartesian coordinates with the Laplacian $Δ_{\perp} = \partial_{x}^{2} + \partial_{y}^{2}$ transverse to the beam $z$-direction tacitly assumed to be covariant. As we show, first, with a simple example and, next, with a systematic derivation in cylindrical coordinates, $Δ_{\perp} = \partial_{r}^{2} + \frac{1}{r} \partial_{r}$ must be amended with a potential $V(r)=-\frac{1}{r^{2}}$, which leads to a Gross-Pitaevskii equation instead. Hence, the beam dynamics and collapse must be revisited. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_15391 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonlinear Schrödinger equation in cylindrical coordinates Krechetnikov, R. Pattern Formation and Solitons Optics Nonlinear Schrödinger equation was originally derived in nonlinear optics as a model for beam propagation, which naturally requires its application in cylindrical coordinates. However, the derivation was done in the Cartesian coordinates with the Laplacian $Δ_{\perp} = \partial_{x}^{2} + \partial_{y}^{2}$ transverse to the beam $z$-direction tacitly assumed to be covariant. As we show, first, with a simple example and, next, with a systematic derivation in cylindrical coordinates, $Δ_{\perp} = \partial_{r}^{2} + \frac{1}{r} \partial_{r}$ must be amended with a potential $V(r)=-\frac{1}{r^{2}}$, which leads to a Gross-Pitaevskii equation instead. Hence, the beam dynamics and collapse must be revisited. |
| title | Nonlinear Schrödinger equation in cylindrical coordinates |
| topic | Pattern Formation and Solitons Optics |
| url | https://arxiv.org/abs/2209.15391 |