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

Abstract

In this paper, Lie group classification to the N -th-order nonlinear evolution equation t, u, ux, . . . , u 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.

Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness

“…In order to construct solutions of MHD, many effective methods have been put forward, such as the inverse scattering method, Backlund transformation, Hirota method, and homogeneous balance method [3]. In the branches of mathematics and physics, Lie Group theory [9][10][11] was often used extensively. Ever since the 1970s Bluman and Col proposed similarity theory for differential equations, the Lie Group theory has been developed indifferential equations.…”