Lie Symmetries, Kac-Moody-Virasoro Algebras and Integrability of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

In this paper we study Lie symmetries, Kac-Moody-Virasoro algebras, similarity reductions and particular solutions of two different recently introduced (2+1)-dimensional nonlinear evolution equations, namely (i) (2+1)-dimensional breaking soliton equation and (ii) (2+1)-dimensional nonlinear Schrödinger type equation introduced by Zakharov and studied later by Strachan. Interestingly our studies show that not all integrable higher dimensional systems admit Kac-Moody-Virasoro type sub-algebras. Particularly the two integrable systems mentioned above do not admit Virasoro type subalgebras, eventhough the other integrable higher dimensional systems do admit such algebras which we have also reviewed in the Appendix. Further, we bring out physically interesting solutions for special choices of the symmetry parameters in both the systems.

On the Virasoro Structure of Symmetry Algebras of Nonlinear Partial Differential Equations

“…The precise relationship between symmetries and integrability is still far from being completely understood. As was noted in [7], not all integrable equations admit KMV symmetry algebras. On the other hand, all known non-integrable equations are invariant under finite or infinite-dimensional Lie point transformation groups without KMV structure.…”

“…The method for the determination of the coefficients of the vector field V is algorithmic [12]. Applying this to (6) gives an overdetermined set of linear partial differential equations for the coefficients ξ, η, τ and φ in equation (7). Solving this system we find that the vector field (7) should have the form…”

“…On the other hand, it should be mentioned that there are evolution type equations which are integrable, but do admit infinite-dimensional symmetry algebras without a KMV structure. For instance, the breaking soliton equation and Zakharov-Strachan equation [7] do not allow a KMV type symmetry algebra while they are integrable. This observation shows that the existence of a KMV symmetry algebra is not a necessary condition for integrability for a nonlinear evolution equation in 2+1 dimensions.…”

Infinite dimensional symmetry group, Kac-Moody-Virasoro algebras and integrability of Kac-Wakimoto equation

“…It is important to note that there are some integrable systems that are invariant under finite or infinite-dimensional Lie algebra without having Virasoro like structure (see ref. [20]), but all non-integrable system that are invariant finite or infinite-dimensional Lie algebra does not have Virasoro like structure. In this way, the presence of Virasoro like structure can be a week predictor of integrability and it could be used along with other strong predictors of integrability.…”

“…This supports the argument presented in the work [12] that Kac-Wakimoto equation is not integrable. Whereas the investigation of Zakharov-Strachan equation [20] reveals that it do not admit Virasoro algebra like structure but it is still integrable. However, the integrability of an equation can be judged from number of arguments in favour of or against it.…”