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-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {Xiβ}β=1,2,··· ,Ni=1,2,··· ,n depending on a finite number of partial derivatives of the nonlocal variables vβ and a restriction i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{i,\alpha ,\beta } {\left( {\tfrac{{\partial \xi ^i }} {{\partial v^\beta }}} \right) + \left( {\tfrac{{\partial \eta ^\alpha }} {{\partial v^\beta }}} \right)} \ne 0$$\end{document} ≠ 0, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{i,\alpha ,\beta } {\left( {\tfrac{{\partial G^\alpha }} {{\partial v^\beta }}} \right)} \ne 0$$\end{document}. 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.