NLS approximation for wavepackets in periodic cubically nonlinear wave problems in $\mathbb{R}^d$ Article Swipe
Tomáš Dohnal
,
Daniel Rudolf
·
YOU?
·
· 2017
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1710.07077
YOU?
·
· 2017
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1710.07077
The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schrödinger equation (NLS) as an effective model for the wavepacket envelope. We provide a detailed proof of this approximation result for the Gross-Pitaevskii equation (GP) and a semilinear wave equation, both with periodic coefficients in $\mathbb{N}\ni d$ spatial dimensions and with cubic nonlinearities. The proof is carried out in Bloch expansion variables with estimates in an $L^1$-type norm, which translates to an estimate of the supremum norm of the error. The regularity required from the periodic coefficients in order to ensure a small residual and a small error is discussed. We also present a numerical example in two spatial dimensions confirming the approximation result and presenting an approximate traveling solitary wave in the GP with periodic coefficients.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/1710.07077
- https://arxiv.org/pdf/1710.07077
- OA Status
- green
- References
- 20
- Related Works
- 20
- OpenAlex ID
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https://openalex.org/W2765676883Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.1710.07077Digital Object Identifier
- Title
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NLS approximation for wavepackets in periodic cubically nonlinear wave problems in $\mathbb{R}^d$Work title
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2017Year of publication
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2017-10-19Full publication date if available
- Authors
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Tomáš Dohnal, Daniel RudolfList of authors in order
- Landing page
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https://arxiv.org/abs/1710.07077Publisher landing page
- PDF URL
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https://arxiv.org/pdf/1710.07077Direct link to full text PDF
- Open access
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/1710.07077Direct OA link when available
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Wave packet, Mathematical analysis, Mathematics, Norm (philosophy), Nonlinear system, Infimum and supremum, Residual, Uniform norm, Envelope (radar), Physics, Mathematical physics, Quantum mechanics, Law, Algorithm, Political science, Radar, Computer science, TelecommunicationsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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20Other works algorithmically related by OpenAlex
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| abstract_inverted_index.equation, | 52 |
| abstract_inverted_index.estimates | 76 |
| abstract_inverted_index.expansion | 73 |
| abstract_inverted_index.nonlinear | 7, 22 |
| abstract_inverted_index.numerical | 117 |
| abstract_inverted_index.traveling | 131 |
| abstract_inverted_index.variables | 74 |
| abstract_inverted_index.$L^1$-type | 79 |
| abstract_inverted_index.confirming | 123 |
| abstract_inverted_index.dimensions | 61, 122 |
| abstract_inverted_index.discussed. | 112 |
| abstract_inverted_index.presenting | 128 |
| abstract_inverted_index.regularity | 94 |
| abstract_inverted_index.semilinear | 50 |
| abstract_inverted_index.structures | 12 |
| abstract_inverted_index.translates | 82 |
| abstract_inverted_index.wavepacket | 32 |
| abstract_inverted_index.approximate | 130 |
| abstract_inverted_index.coefficient | 21 |
| abstract_inverted_index.wavepackets | 5 |
| abstract_inverted_index.Schrödinger | 23 |
| abstract_inverted_index.approximated | 17 |
| abstract_inverted_index.coefficients | 56, 99 |
| abstract_inverted_index.approximation | 41, 125 |
| abstract_inverted_index.coefficients. | 139 |
| abstract_inverted_index.$\mathbb{N}\ni | 58 |
| abstract_inverted_index.nonlinearities. | 65 |
| abstract_inverted_index.Gross-Pitaevskii | 45 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |