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

@article{Velan1998LieSK, title={Lie Symmetries, Kac-Moody-Virasoro Algebras and Integrability of Certain (2+1)-Dimensional Nonlinear Evolution Equations}, author={M. Senthil Velan and Muthusamy Lakshmanan}, journal={Journal of Nonlinear Mathematical Physics}, year={1998}, volume={5}, pages={190-211} }

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 Schrudinger 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…

## 35 Citations

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

- Mathematics
- 2020

An eighth-order equation in (3+1)-dimension is studied for its integrability. Its symmetry group is shown to be infinite-dimensional and is checked for Virasoro like structure. The equation is shown…

Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa–Holm equation

- Mathematics, Physics
- 2000

Realizations of the Witt and Virasoro Algebras and Integrable Equations

- MathematicsJournal of Nonlinear Mathematical Physics
- 2019

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order…

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

- Mathematics
- 2006

We discuss Lie algebras of the Lie symmetry groups of two generically non- integrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization…

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

- Mathematics
- 2006

We discuss Lie algebras of the Lie symmetry groups of two generically nonintegrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of…

Transformation groups, Kac–Moody–Virasoro algebras and conservation laws of the Bogoyavlenskii–Kadomtsev–Petviashvili equation

- Mathematics
- 2017

New Solutions of the Schwarz-Korteweg-de Vries Equation in 2+1 Dimensions with the Gauge Transformation

- Physics
- 2014

In this paper, we find the relationship between the solution of (1+1)-dimensional Korteweg-de Vries (KdV) equation and the solution of (2+1)-dimensional integrable Schwarz-Korteweg-de Vries(SKdV)…

On realizations of the Witt algebra in $\mathbb{R}^3$

- Mathematics
- 2014

We obtain exhaustive classification of inequivalent realizations of the Witt and Virasoro algebras by Lie vector fields of differential operators in the space $\mathbb{R}^3$. Using this…

## References

SHOWING 1-10 OF 63 REFERENCES

Lie Symmetries, Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

- Mathematics
- 1996

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…

The cylindrical Kadomtsev-Petviashvili equation; its Kac-Moody-Virasoro algebra and relation to KP equation

- Mathematics
- 1988

Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra

- Mathematics
- 1986

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite‐dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie…

On generalized Loewner systems: Novel integrable equations in 2+1 dimensions

- Mathematics
- 1993

A reinterpretation and generalization of a class of infinitesimal Backlund transformations originally introduced in a gas‐dynamics context by Loewner in 1952 leads to a linear representation for a…

Some reductions of the self‐dual Yang–Mills equations to integrable systems in 2+1 dimensions

- Mathematics
- 1995

A reduction of the self‐dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one‐form belongs to a Lie algebra of formal…

Lie transformations, nonlinear evolution equations, and Painlevé forms

- Mathematics
- 1983

We present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one‐parameter continuous (Lie) group of transformations. It is…

The (2 + 1)-dimensional sine - Gordon equation; integrability and localized solutions

- Mathematics
- 1996

In this paper, the (2 + 1)-dimensional sine - Gordon equation (2DSG) introduced by Konopelchenko and Rogers is investigated and is shown to satisfy the Painleve property. A variable coefficient…

Singularity structure analysis and bilinear form of a (2+1) dimensional non-linear Schrodinger (NLS) equation

- Mathematics
- 1994

The integrable (2+1) dimensional generalization of the non-linear Schrodinger (NLS) equation discussed recently by Strachan is shown to admit the Painleve property. Further, we construct its bilinear…

On the infinite‐dimensional symmetry group of the Davey–Stewartson equations

- Mathematics
- 1988

The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions…