The focusing and defocusing fifth-order semi-discrete mKdV equation: bÄcklund transformation, soliton solution and continuous limit.
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| Title: | The focusing and defocusing fifth-order semi-discrete mKdV equation: bÄcklund transformation, soliton solution and continuous limit. |
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| Authors: | Gao, Ning-Ning1 (AUTHOR) gaonn2023@163.com, Fan, Fang-Cheng1 (AUTHOR) fanfc@mnnu.edu.cn |
| Source: | Reports on Mathematical Physics. Apr2026, Vol. 97 Issue 2, p201-218. 18p. |
| Subjects: | Bäcklund transformations, Solitons, Partial differential equations, Mathematical analysis, Korteweg-de Vries equation, Numerical analysis, Integrable systems |
| Abstract: | The integrable discretization of a nonlinear partial differential equation and the continuous limits of the relevant discrete integrability properties are one of the most important research subjects in soliton theory. In this paper we construct the spatial discrete version of the focusing and defocusing fifth-order mKdV equation and study the continuous limits of the related discrete integrable properties. Firstly, starting from a discrete spectral problem, we derive integrable equation hierarchies and then obtain the focusing and defocusing fifth-order semi-discrete mKdV equation by using a linear combination of them. Secondly, the Bäcklund transformation and soliton solutions of the equation are presented, the relationship between parameters and solutions' structures is discussed, some important physical quantities related to solutions are analyzed, the dynamics of soliton solutions are illustrated graphically. Thirdly, we show that the fifth-order mKdV theory including the Lax pairs, the Bäcklund transformation and soliton solutions is recovered through the continuous limits of corresponding theory for the fifth-order semi-discrete mKdV equation. As a conclusion, we believe that the focusing and defocusing fifth-order semidiscrete mKdV equation which we construct in this paper is an extremely useful model for the numerical analysis for considering the Cauchy problem with a general initial data of the fifth-order mKdV equation. [ABSTRACT FROM AUTHOR] |
| Copyright of Reports on Mathematical Physics is the property of Pergamon Press - An Imprint of Elsevier Science 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 |
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| Items | – Name: Title Label: Title Group: Ti Data: The focusing and defocusing fifth-order semi-discrete mKdV equation: bÄcklund transformation, soliton solution and continuous limit. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Gao%2C+Ning-Ning%22">Gao, Ning-Ning</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> gaonn2023@163.com</i><br /><searchLink fieldCode="AR" term="%22Fan%2C+Fang-Cheng%22">Fan, Fang-Cheng</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> fanfc@mnnu.edu.cn</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Reports+on+Mathematical+Physics%22">Reports on Mathematical Physics</searchLink>. Apr2026, Vol. 97 Issue 2, p201-218. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Bäcklund+transformations%22">Bäcklund transformations</searchLink><br /><searchLink fieldCode="DE" term="%22Solitons%22">Solitons</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Korteweg-de+Vries+equation%22">Korteweg-de Vries equation</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Integrable+systems%22">Integrable systems</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The integrable discretization of a nonlinear partial differential equation and the continuous limits of the relevant discrete integrability properties are one of the most important research subjects in soliton theory. In this paper we construct the spatial discrete version of the focusing and defocusing fifth-order mKdV equation and study the continuous limits of the related discrete integrable properties. Firstly, starting from a discrete spectral problem, we derive integrable equation hierarchies and then obtain the focusing and defocusing fifth-order semi-discrete mKdV equation by using a linear combination of them. Secondly, the Bäcklund transformation and soliton solutions of the equation are presented, the relationship between parameters and solutions' structures is discussed, some important physical quantities related to solutions are analyzed, the dynamics of soliton solutions are illustrated graphically. Thirdly, we show that the fifth-order mKdV theory including the Lax pairs, the Bäcklund transformation and soliton solutions is recovered through the continuous limits of corresponding theory for the fifth-order semi-discrete mKdV equation. As a conclusion, we believe that the focusing and defocusing fifth-order semidiscrete mKdV equation which we construct in this paper is an extremely useful model for the numerical analysis for considering the Cauchy problem with a general initial data of the fifth-order mKdV equation. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Reports on Mathematical Physics is the property of Pergamon Press - An Imprint of Elsevier Science 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: 10.1016/S0034-4877(26)00024-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 201 Subjects: – SubjectFull: Bäcklund transformations Type: general – SubjectFull: Solitons Type: general – SubjectFull: Partial differential equations Type: general – SubjectFull: Mathematical analysis Type: general – SubjectFull: Korteweg-de Vries equation Type: general – SubjectFull: Numerical analysis Type: general – SubjectFull: Integrable systems Type: general Titles: – TitleFull: The focusing and defocusing fifth-order semi-discrete mKdV equation: bÄcklund transformation, soliton solution and continuous limit. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Gao, Ning-Ning – PersonEntity: Name: NameFull: Fan, Fang-Cheng IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00344877 Numbering: – Type: volume Value: 97 – Type: issue Value: 2 Titles: – TitleFull: Reports on Mathematical Physics Type: main |
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