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Journal of Lie Theory 20 (2010), No. 2, 375--392
Copyright Heldermann Verlag 2010

On a Lie Group Characterization of Quasi-Local Symmetries of Nonlinear Evolution Equations

Renat Zhdanov
BIO-key International, Eagan, MN 55123, U.S.A.

We develop an efficient algebraic approach to classifying nonlinear evolution equations in one spatial dimension that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable. It applies to evolution equations invariant under Lie point symmetries leaving the temporal variable invariant. We construct inequivalent realizations of two- and three-dimensional Lie algebras leading to evolution equations admitting quasi-local symmetries. Finally, we generalize the approach in question for the case of an arbitrary system of evolution equations in two independent variables.

Keywords: Quasi-local symmetry, nonlinear evolution equation, Lie algebra.

MSC: 35Q80, 58Z05, 58J70

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