Symmetry classification of KdV-type nonlinear evolution equations

Details Symmetry classification of KdV-type nonlinear evolution equations Symmetry classification of KdV-type nonlinear evolution equations

2002-01-30

arxiv.org/abs/nlin/0201063v2

J. Math. Phys. 45 (2004), 2280-2313

Nonlinear Sciences

Exactly Solvable and Integrable Systems

43 pages, AMSLaTeX, no figures, substantially revised

Scientific paper

10.1063/1.1737811

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.

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