| Abstract: | By considering the one-dimensional model for describing long, smallamplitude waves in shallow water, a generalized fifth-orderevolution equation named the Olver water wave (OWW) equation isinvestigated by virtue of some new pseudo-potential systems. Byintroducing the corresponding pseudo-potential systems, the authorssystematically construct some generalized symmetries that considersome new smooth functions$left{X_{ibeta}right}^{i=1,2,cdots,n}_{beta=1,2,cdots,N}$depending on a finite number of partial derivatives of the nonlocalvariables $v^{beta}$ and a restriction$sumlimits_{i,alpha,beta}big(frac{partial xi^{i}}{partialv^{beta}}big)^{2}+big(frac{partial eta^{alpha}}{partialv^{beta}}big)^{2}neq 0$, i.e.,$sumlimits_{i,alpha,beta}big(frac{partialG^{alpha}}{partial v^{beta}}big)^{2}neq 0$. Furthermore, theauthors investigate some structures associated with the Olver waterwave (AOWW) equations including Lie algebra and Darbouxtransformation. The results are also extended to AOWW equations suchas Lax, Sawada-Kotera, Kaup-Kupershmidt, It^{o} andCaudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, thesymmetries are applied to investigate the initial value problems andDarboux transformations. |
