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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.
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]
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Database: Engineering Source
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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]
ISSN:00344877
DOI:10.1016/S0034-4877(26)00024-8