On the Lie Algebras, Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water
By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave （OWW） equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potenti...
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| Published in | Chinese annals of mathematics. Serie B Vol. 36; no. 4; pp. 543 - 560 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2015
Department of Mathematics,China University of Mining and Technology,Xuzhou 221116,Jiangsu,China%School of Mathematics and Information Sciences,Weifang University,Weifang 261061,Shandong,China%School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,Liaoning,China |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0252-9599 1860-6261 |
| DOI | 10.1007/s11401-015-0908-6 |
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| Abstract | By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave （OWW） equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {Xiβ}β=1,2…,N^i=1,2…,n depending on a finite number of partial derivatives of the nonlocal variables vβ and a restriction i,α,β∑（ ξi/ vβ）^2＋（ ηα/ vβ）^2≠0,ie.,i,α,β∑（ ξi/ vβ）^2≠0. Furthermore, i,a,B i,a,~ the authors investigate some structures associated with the Olver water wave （AOWW） equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, It6 and Caudrey-Dodd-Cibbon-Sawada-Kotera equations, et al. Finally, the symmetries are ap- plied to investigate the initial value problems and Darboux transformations. |
|---|---|
| AbstractList | By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave (OWW) equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {
X
iβ
}
β
=1,2,··· ,
N
i
=1,2,··· ,
n
depending on a finite number of partial derivatives of the nonlocal variables
v
β
and a restriction i.e.,
≠ 0, i.e.,
. Furthermore, the authors investigate some structures associated with the Olver water wave (AOWW) equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, Itˆo and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations. By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave （OWW） equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {Xiβ}β=1,2…,N^i=1,2…,n depending on a finite number of partial derivatives of the nonlocal variables vβ and a restriction i,α,β∑（ ξi/ vβ）^2＋（ ηα/ vβ）^2≠0,ie.,i,α,β∑（ ξi/ vβ）^2≠0. Furthermore, i,a,B i,a,~ the authors investigate some structures associated with the Olver water wave （AOWW） equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, It6 and Caudrey-Dodd-Cibbon-Sawada-Kotera equations, et al. Finally, the symmetries are ap- plied to investigate the initial value problems and Darboux transformations. |
| Author | Shoufu TIAN Yufeng ZHANG Binlu FENG Hongqing ZHANG |
| AuthorAffiliation | Corresponding author. Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China School of Mathematics and Information Sciences, Weifang University, Weifang 261061, Shandong, China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China |
| AuthorAffiliation_xml | – name: Department of Mathematics,China University of Mining and Technology,Xuzhou 221116,Jiangsu,China%School of Mathematics and Information Sciences,Weifang University,Weifang 261061,Shandong,China%School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,Liaoning,China |
| Author_xml | – sequence: 1 givenname: Shoufu surname: Tian fullname: Tian, Shoufu email: sftian@cumt.edu.cn organization: Department of Mathematics, China University of Mining and Technology – sequence: 2 givenname: Yufeng surname: Zhang fullname: Zhang, Yufeng organization: Department of Mathematics, China University of Mining and Technology – sequence: 3 givenname: Binlu surname: Feng fullname: Feng, Binlu organization: School of Mathematics and Information Sciences, Weifang University – sequence: 4 givenname: Hongqing surname: Zhang fullname: Zhang, Hongqing organization: School of Mathematical Sciences, Dalian University of Technology |
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| Copyright | Fudan University and Springer-Verlag Berlin Heidelberg 2015 Copyright © Wanfang Data Co. Ltd. All Rights Reserved. |
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| Keywords | Darboux transformations 74J35 68W30 Analytical solutions 35C99 35Q53 35Q51 Generalized symmetries |
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| Notes | By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave （OWW） equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {Xiβ}β=1,2…,N^i=1,2…,n depending on a finite number of partial derivatives of the nonlocal variables vβ and a restriction i,α,β∑（ ξi/ vβ）^2＋（ ηα/ vβ）^2≠0,ie.,i,α,β∑（ ξi/ vβ）^2≠0. Furthermore, i,a,B i,a,~ the authors investigate some structures associated with the Olver water wave （AOWW） equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, It6 and Caudrey-Dodd-Cibbon-Sawada-Kotera equations, et al. Finally, the symmetries are ap- plied to investigate the initial value problems and Darboux transformations. 31-1329/O1 Generalized symmetries, Darboux transformations, Analytical solutions |
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| PublicationYear | 2015 |
| Publisher | Springer Berlin Heidelberg Department of Mathematics,China University of Mining and Technology,Xuzhou 221116,Jiangsu,China%School of Mathematics and Information Sciences,Weifang University,Weifang 261061,Shandong,China%School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,Liaoning,China |
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| Title | On the Lie Algebras, Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water |
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