93
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

Nonlocal symmetries of Plebański’s second heavenly equation

&
Pages 188-197 | Received 20 Jul 2017, Accepted 07 Nov 2017, Published online: 16 Mar 2018

References

  • M. Ablowitz, S. Chakravarty, and L. Takhtajan, A self-dual Yang–Mills hierarchy and its reductions to integrable systems in 1 + 1 and 2 + 1 dimensions, Comm. Math. Phys. 158 (1993) 289–314. doi: 10.1007/BF02108076
  • H. Baran, I.S. Krasil′shchik, O.I. Morozov, and P. Vojčák, Coverings over Lax integrable equations and their nonlocal symmetries, Theor. Math. Phys. 188 (2016) no. 3, 1273–1295. doi: 10.1134/S0040577916090014
  • H. Baran, I.S. Krasil′shchik, O.I. Morozov, and P. Vojčák, Nonlocal symmetries of Lax integrable equations: a comparative study, arXiv:1611.04938
  • H. Baran and M. Marvan, Jets: A software for differential calculus on jet spaces and diffeties, http://jets.math.slu.cz (2016).
  • P.M. Bies, P. Górka, E.G. Reyes, The dual modified Korteweg–de Vries–Fokas–Qiao equation: Geometry and local analysis, J. Math. Phys. 53 No 7 (2012) 073710 ( 19 pages) doi: 10.1063/1.4736845
  • M. Błaszak, Multi-Hamiltonian Theory of Dynamical Systems (Springer, Berlin, 1998).
  • M. Błaszak, Classical R-matrices on Poisson algebras and related dispersionless systems, Phys. Lett. A 297 (2002), 191–195. doi: 10.1016/S0375-9601(02)00421-8
  • G.W. Bluman, S. Kumei, Symmetries and Differential Equations. Applied Mathematical Sciences 81 (Springer, Berlin, 1989)
  • G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences 168 (Springer, Berlin, 2010)
  • A.V. Bocharov et al., Symmetries of Differential Equations in Mathematical Physics and Natural Sciences, (edited by A.M. Vinogradov and I.S. Krasil′shchik). (Factorial Publ. House, Moscow, 1997 (in Russian), English translation: Amer. Math. Soc., 1999)
  • B. Doubrov and E.V. Ferapontov, On the integrability of symplectic Monge–Ampère equations, J. Geom. Phys. 60 (2010) 1604–1616. doi: 10.1016/j.geomphys.2010.05.009
  • M. Dunajski, Solitons, Instantons and Twistors (Oxford University Press, Oxford, 2009)
  • J.D. Finley III and J.D. McIver, Non-Abelian infinite algebra of generalized symmetries for the SDiff(2) Toda equation, J. Phys. A 37 (2004) 5825–5847. doi: 10.1088/0305-4470/37/22/009
  • B. Fuchssteiner and A.S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D 4 (1981/82) 47–66. doi: 10.1016/0167-2789(81)90004-X
  • D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, (Consultants Bureau, New York, 1986).
  • R. Hernández Heredero, E.G. Reyes, Geometric Integrability of the Camassa–Holm Equation. II, Int. Math. Res. Notices 2012 No 13 (2012) 3089–3125 doi: 10.1093/imrn/rnr120
  • R. Hernández Heredero, E.G. Reyes, Nonlocal symmetries, compacton equations, and integrability, International Journal of Geometric Methods in Modern Physics 10 No 9 (2013) 1350046 ( 24 pages) doi: 10.1142/S0219887813500461
  • S. Igonin and M. Marvan, On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system, J. Geom. Phys. 85 (2014) 106–123. doi: 10.1016/j.geomphys.2014.05.017
  • I.S. Krasil'shchik, V.V. Lychagin, and A.M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, (Gordon and Breach, NY, 1986).
  • J. Krasil'shchik and A.M. Verbovetsky, Geometry of jet spaces and integrable systems, J. Geom. Phys. 61 (2011) 1633-1674 doi: 10.1016/j.geomphys.2010.10.012
  • I.S. Krasil'shchik, A.M. Verbovetsky and R. Vitolo, A unified approach to computation of integrable structures, Acta Appl. Math. 120 (2012) 199–218. doi: 10.1007/s10440-012-9699-x
  • I.S. Krasil'shchik and A.M. Vinogradov, Nonlocal symmetries and the theory of coverings, Acta Appl. Math. 2 (1984) 79–86. doi: 10.1007/BF01405492
  • I.S. Krasil'shchik and A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989) 161–209. doi: 10.1007/BF00131935
  • B. Kruglikov and O. Morozov, SDiff(2) and uniqueness of the Plebański equation, JMP 53 (2012) 083506 ( 11 p.) doi: 10.1063/1.4739749
  • B. Kruglikov and O. Morozov, Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations, Lett. Math. Phys. 105 (2015) 1703–1723. doi: 10.1007/s11005-015-0800-z
  • A.A. Malykh, Y. Nutku, and M.B. Sheftel, Partner symmetries and non-invariant solutions of four-dimensional heavenly equations, J. Phys. A: Math. Gen. 37 (2004), 7527–7545. doi: 10.1088/0305-4470/37/30/010
  • O.I. Morozov, Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations, J. Geom. Phys. 113 (2017) 215–225. doi: 10.1016/j.geomphys.2016.09.010
  • O.I. Morozov, Deformations of infinite-dimensional Lie algebras, exotic cohomology and integrable nonlinear partial differential equations, arXiv:1706.01090
  • O.I. Morozov and A. Sergyeyev, The four-dimensional Martínez Alonso–Shabat equation: Reductions and nonlocal symmetries, J. Geom. Phys. 85 (2014) 40–45. doi: 10.1016/j.geomphys.2014.05.025
  • F. Neyzi, Y. Nutku, and M.B. Sheftel, Multi-Hamiltonian Structure of Plebanski’s Second Heavenly Equation, J. Phys. A: Math. Gen. 38 (2005), 8473–8485. doi: 10.1088/0305-4470/38/39/012
  • P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, NY, 1993).
  • J.F. Plebański, Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975) 2395–2402. doi: 10.1063/1.522505
  • A. Sergyeyev, Infinite hierarchies of nonlocal symmetries of the Chen–Kontsevich–Schwarz type for the oriented associativity equations, J. Phys. A 42 (2009) 404017. ( p. 15) doi: 10.1088/1751-8113/42/40/404017
  • E.G. Reyes, Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations, Selecta Math. New Ser. 12 (2006) 241–270. doi: 10.1007/s00029-006-0024-2
  • D. Yazıcı and M.B. Sheftel, Symmetry Reductions of Second Heavenly Equation and (2 + 1)-Dimensional Hamiltonian Integrable Systems. JNMP 15 Suppl. 3 (2008), 417–425. doi: 10.2991/jnmp.2008.15.s3.40

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.