Lie Symmetries, infinite-dimensional Lie algebras and similarity reductions of certain (2+1)-dimensional nonlinear evolution equations

Abstract

The Lie point symmetries associated with a number of (2 +1)-dimensional generalizations of soliton equations are investigated. These include the Niznik - Novikov - Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrodinger equation by Fokas as well as the (2+1)- dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.