Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 014, 7 pages      nlin.SI/0602001      https://doi.org/10.3842/SIGMA.2006.014

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

Faruk Güngör
Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469, Istanbul, Turkey

Received November 30, 2005, in final form January 20, 2006; Published online January 30, 2006

Abstract
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 of the KP equation and contains 9 arbitrary functions of one and two arguments. The second one is a system of PDEs that depend on some physical parameters. We require that these PDEs are invariant under a Kac-Moody-Virasoro algebra. This leads to several limitations on the coefficients (either functions or parameters) under which equations are prime candidates for being integrable.

Key words: Kadomtsev-Petviashvili and Davey-Stewartson equations; symmetry group; Virasoro algebra.

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References

  1. David D., Kamran N., Levi D., Winternitz P., Subalgebras of loop algebras and symmetries of the Kadomtsev-Petviashvili equation, Phys. Rev. Let., 1985, V.55, N 20, 2111-2113.
  2. David D., Kamran N., Levi D., Winternitz P., Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra, J. Math. Phys., 1986, V.27, N 5, 1225-1237.
  3. Levi D., Winternitz P., The cylindrical Kadomtsev-Petviashvili equation, its Kac-Moody-Virasoro algebra and relation to the KP equation, Phys. Lett. A, 1988, V.129, 165-167.
  4. Orlov A.Yu., Winternitz P., Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev-Petviashvili hierarchy, J. Math. Phys., 1997, V.38, 4644-4674.
  5. Champagne B., Winternitz P., On the infinite dimensional symmetry group of the Davey-Stewartson equation, J. Math. Phys., 1988, V.29, 1-8.
  6. Martina L., Winternitz P., Analysis and applications of the symmetry group of the multidimensional three-wave resonant interaction problem, Ann. Phys., 1989, V.196, 231-277.
  7. Senthil Velan M., Lakshmanan M., Lie symmetries, Kac-Moody-Virasoro algebras and integrability of certain (2+1)-dimensional nonlinear evolution equations, J. Nonlinear Math. Phys., 1998, V.5, 190-211, math-ph/9804017.
  8. Babaoglu C., Erbay S., Two-dimensional wave packets in an elastic solid with couple stresses, Internat. J. Non-Linear Mech., 2004, V.39, 941-949.
  9. Winternitz P., Kac-Moody-Virasoro symmetries of integrable nonlinear partial differential equations, in Symmetries and Nonlinear Phenomena, Editors D. Levi and P. Winternitz, Singapore, World Scientific, 1988, 358-375.
  10. Winternitz P., Group theory and exact solutions of partially integrable differential systems, in Partially Integrable Evolution Equations in Physics (1989, Les Houches), Editors R. Conte and N. Boccara, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., V.310, Dordrecht, Kluwer Acad. Publ., 1990, 515-567.
  11. Güngör F., Winternitz P., Generalized Kadomtsev-Petviashvili equation with an infinite dimensional symmetry algebra, J. Math. Anal. Appl., 2002, V.276, 314-328, nlin.SI/0109009.
  12. Olver P.J., Applications of Lie groups to differential equations, New York, Springer, 1991.
  13. David D., Levi D., Winternitz P., Equations invariant under the symmetry group of the Kadomtsev-Petviashvili equation, Phys. Lett. A, 1988, V.129, 161-164.
  14. Lou S.Y., Tang X.Y., Equations of arbitrary order invariant under the Kadomtsev-Petviashvili symmetry group, J. Math. Phys., 2004, V.43, 1020-1030.
  15. Güngör F., Winternitz P., Equivalence classes and symmetries of the variable coefficient KP equation, Nonlinear Dynam., 2004, V.35, 381-396.
  16. Güngör F., Aykanat Ö., The generalized Davey-Stewartson equations, its Kac-Moody-Virasoro symmetry algebra and relation to Davey-Stewartson equations, J. Math. Phys., 2006, to appear.

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