The Idea of a Lax Pair-Part II: Continuum Wave Equations.
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| Title: | The Idea of a Lax Pair-Part II: Continuum Wave Equations. |
|---|---|
| Authors: | Krishnaswami, Govind S.1 (AUTHOR) govind@cmi.ac.in, Vishnu, T. R.1 (AUTHOR) vishnu@cmi.ac.in |
| Source: | Resonance: Journal of Science Education. Feb2021, Vol. 26 Issue 2, p257-274. 18p. |
| Subject Terms: | Wave equation, Differential operators, Conserved quantity, Linear equations, Water waves, Conservation laws (Mathematics), Lax pair |
| Abstract: | In Part I [1], we introduced the idea of a Lax pair and explained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to continuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Kortewegde Vries (KdV) equation for water waves. Unlike the Lax matrices for systems of particles, here Lax pairs are differential operators. A key idea is to view the Lax equation as a compatibility condition between a pair of linear equations. This is used to obtain a geometric reformulation of the Lax equation as the condition for a certain curvature to vanish. This 'zero curvature representation' then leads to a recipe for finding (typically an infinite sequence of) conserved quantities. [ABSTRACT FROM AUTHOR] |
| Copyright of Resonance: Journal of Science Education is the property of Springer Nature 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: | Education Research Complete |
| FullText | Links: – Type: pdflink Text: Availability: 0 |
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| Header | DbId: ehh DbLabel: Education Research Complete An: 148887256 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Idea of a Lax Pair-Part II: Continuum Wave Equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Krishnaswami%2C+Govind+S%2E%22">Krishnaswami, Govind S.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> govind@cmi.ac.in</i><br /><searchLink fieldCode="AR" term="%22Vishnu%2C+T%2E+R%2E%22">Vishnu, T. R.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> vishnu@cmi.ac.in</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Resonance%3A+Journal+of+Science+Education%22">Resonance: Journal of Science Education</searchLink>. Feb2021, Vol. 26 Issue 2, p257-274. 18p. – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Wave+equation%22">Wave equation</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+operators%22">Differential operators</searchLink><br /><searchLink fieldCode="DE" term="%22Conserved+quantity%22">Conserved quantity</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+equations%22">Linear equations</searchLink><br /><searchLink fieldCode="DE" term="%22Water+waves%22">Water waves</searchLink><br /><searchLink fieldCode="DE" term="%22Conservation+laws+%28Mathematics%29%22">Conservation laws (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Lax+pair%22">Lax pair</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In Part I [1], we introduced the idea of a Lax pair and explained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to continuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Kortewegde Vries (KdV) equation for water waves. Unlike the Lax matrices for systems of particles, here Lax pairs are differential operators. A key idea is to view the Lax equation as a compatibility condition between a pair of linear equations. This is used to obtain a geometric reformulation of the Lax equation as the condition for a certain curvature to vanish. This 'zero curvature representation' then leads to a recipe for finding (typically an infinite sequence of) conserved quantities. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Resonance: Journal of Science Education is the property of Springer Nature 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: 10.1007/s12045-021-1124-1 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 257 Subjects: – SubjectFull: Wave equation Type: general – SubjectFull: Differential operators Type: general – SubjectFull: Conserved quantity Type: general – SubjectFull: Linear equations Type: general – SubjectFull: Water waves Type: general – SubjectFull: Conservation laws (Mathematics) Type: general – SubjectFull: Lax pair Type: general Titles: – TitleFull: The Idea of a Lax Pair-Part II: Continuum Wave Equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Krishnaswami, Govind S. – PersonEntity: Name: NameFull: Vishnu, T. R. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2021 Type: published Y: 2021 Identifiers: – Type: issn-print Value: 09718044 Numbering: – Type: volume Value: 26 – Type: issue Value: 2 Titles: – TitleFull: Resonance: Journal of Science Education Type: main |
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