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 -NovikovVeselov 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 Schrödinger 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.

Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

“…It is interesting to note that a similar type of algebras also exist in other integrable systems mentioned in Introduction, namely, the Nizhnik-Novikov-Veselov equation, (2+1) dimensional nonlinear Schrödinger equation, and sine-Gordon equation [8].…”

“…(8) Now the reduced pde (8) in two independent variables can itself be further analyzed for its symmetry properties by looking at its own invariance property under the classical Lie algorithm again. In this case, we obtain the following five-parameter Lie symmetries,…”

“…Concentrating on Lie symmetries for the present, it has been realized that important classes of (2+1) dimensional extensions of soliton equations admit typically Lie symmetries involving infinite-dimensional symmetry algebras, often of the Kac-Moody-Virasoro type. Typical systems are the following: (i) Kadomtsev-Petviashvili equation [6], (ii) Davey-Stewartson equation [7], (iii) Nizhnik-Novikov-Veselov equation [8], (iv) nonlinear Schrödinger equation introduced by Fokas and the sine-Gordon equation [8]. However, there are certain integrable evolution equations which, though admit infinite-dimensional Lie algebras, do not seem to possess a Virasoro-type subalgebra [9].…”

Symmetries of the Classical Integrable Systems and 2-Dimensional Quantum Gravity: a `Map'

“…the Lie symmetries studied in depth in Ref. [89] at this meeting. Consequently in the context of 2-dimensional quantum gravity ( §3 next and the expressions (3) above) one can introduce the so-called "p-reduced" KP-I flows [72,81] which are constrained [7] by Douglas's 'string equation' so that the consequent infinite set of constraints satisfies the Weyl W ∞ algebra marked in Fig.1 (in the 'box' with contents (2)) in the form…”

“…The §4 following this §3 then reproduces the complete reference list of [87] (with some corrections) and so provides the reader with the opportunity, if he so wishes, to look again at the whole evolution of integrable systems during 1834-1995. The new references [87], [88] and [89] are added to the reference list in §4 as compared with that of [87].…”