On the Lie Algebras, Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water

• Published in: Chinese annals of mathematics. Serie B Vol. 36; no. 4; pp. 543 - 560
• Main Authors: Tian, Shoufu, Zhang, Yufeng, Feng, Binlu, Zhang, Hongqing
• 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: Applications of Mathematics; Darboux变换; Mathematics; Mathematics and Statistics; Sawada-Kotera方程; 一维模型; 五阶; 发展方程; 广义对称性; 李代数; 浅水; Darboux transformations; 74J35; 68W30; Analytical solutions; 35C99; 35Q53; 35Q51; Generalized symmetries
• ISSN: 0252-9599; 1860-6261
• DOI: 10.1007/s11401-015-0908-6

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.