Symmetry classification of KdV-type nonlinear evolution equations

Abstract

Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation. *

Time-dependent exact solutions for Rosenau–Hyman equations with variable coefficients

“…We have found three Rosenau-Hyman-like equations with variable coefficients (vcKðm; nÞ) [38]. Once Lie approach is a useful tool to search for solutions [26,34,[39][40][41][42][43][44], we also employed it here to obtain Lie-symmetry invariant time-dependent solutions for suited values of parameters m and n of vcKðm; nÞ equations.…”

Preliminary group classification of quasilinear third-order evolution equations

“…In the case of F u = F ux = F uxx = 0 and G u = G ux = G uxx = 0, the complete group classification of (1) was performed by Gazeau and Winternitz [13] . Recently, Güngör, Lahno, and Zhdanov [14] gave a complete group classification of the most general linear third-order evolution equation,…”

Symmetry approach and the generalized Korteweg-de Vries equation with variable coefficients

“…A lot of effort has been carried out in order to understand the nonlinear mechanism that underlies processes described by K(m, n) equations [3,4,5], including an analogous generalization of the Sine-Gordon equation [6]. Lie symmetry methods have also been used for this purpose, and a partial symmetry classification of K(m, n) equations has been achieved [7,8,9].…”