Lie group classification of the N-th-order nonlinear evolution equations

In this paper, Lie group classification to the N-th-order nonlinear evolution equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t = u_{Nx} + F\left( {x,t,u,u_x , \ldots ,u_{\left( {N - 1} \right)x} } \right)$$\end{document} is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensional solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group so(3) as the symmetry group of the equation, and only two realizations of sl(2, ℝ) are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.