On the stability of the compacton waves for the degenerate KdV and NLS models.
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| Title: | On the stability of the compacton waves for the degenerate KdV and NLS models. |
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| Authors: | Hakkaev, Sevdzhan1 (AUTHOR), Ramadan, Abba2 (AUTHOR), Stefanov, Atanas G.3 (AUTHOR) |
| Source: | Quarterly of Applied Mathematics. Sep2022, Vol. 80 Issue 3, p507-528. 22p. |
| Subjects: | Korteweg-de Vries equation, Standing waves, Traveling waves (Physics) |
| Abstract: | In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p >8. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p < 8. [ABSTRACT FROM AUTHOR] |
| Copyright of Quarterly of Applied Mathematics is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 156938184 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: On the stability of the compacton waves for the degenerate KdV and NLS models. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Hakkaev%2C+Sevdzhan%22">Hakkaev, Sevdzhan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Ramadan%2C+Abba%22">Ramadan, Abba</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Stefanov%2C+Atanas+G%2E%22">Stefanov, Atanas G.</searchLink><relatesTo>3</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Quarterly+of+Applied+Mathematics%22">Quarterly of Applied Mathematics</searchLink>. Sep2022, Vol. 80 Issue 3, p507-528. 22p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Korteweg-de+Vries+equation%22">Korteweg-de Vries equation</searchLink><br /><searchLink fieldCode="DE" term="%22Standing+waves%22">Standing waves</searchLink><br /><searchLink fieldCode="DE" term="%22Traveling+waves+%28Physics%29%22">Traveling waves (Physics)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bell-shaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p >8. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p < 8. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Quarterly of Applied Mathematics is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1090/qam/1616 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 22 StartPage: 507 Subjects: – SubjectFull: Korteweg-de Vries equation Type: general – SubjectFull: Standing waves Type: general – SubjectFull: Traveling waves (Physics) Type: general Titles: – TitleFull: On the stability of the compacton waves for the degenerate KdV and NLS models. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Hakkaev, Sevdzhan – PersonEntity: Name: NameFull: Ramadan, Abba – PersonEntity: Name: NameFull: Stefanov, Atanas G. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2022 Type: published Y: 2022 Identifiers: – Type: issn-print Value: 0033569X Numbering: – Type: volume Value: 80 – Type: issue Value: 3 Titles: – TitleFull: Quarterly of Applied Mathematics Type: main |
| ResultId | 1 |
