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dc.contributor.author Malykh, A.A.
dc.contributor.author Sheftel, M.B.
dc.date.accessioned 2019-02-11T17:51:07Z
dc.date.available 2019-02-11T17:51:07Z
dc.date.issued 2011
dc.identifier.citation Recursions of Symmetry Orbits and Reduction without Reduction / M.B. Sheftel, A.A. Malykh // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. uk_UA
dc.identifier.issn 1815-0659
dc.identifier.other 2010 Mathematics Subject Classification: 35Q75; 83C15
dc.identifier.other DOI:10.3842/SIGMA.2011.043
dc.identifier.uri http://dspace.nbuv.gov.ua/handle/123456789/146860
dc.description.abstract We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation. uk_UA
dc.description.sponsorship This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html. The authors thank M. Dunajski for an interesting discussion about the problem of performing the Legendre transformation of a metric with respect to a symmetry group parameter. The research of MBS was supported in part by the research grant from Bogazici University Scientific Research Fund, research project No. 07B301. AAM is grateful to Feza Gursey Institute for the support of his stay at Istanbul where this work was finalized. uk_UA
dc.language.iso en uk_UA
dc.publisher Інститут математики НАН України uk_UA
dc.relation.ispartof Symmetry, Integrability and Geometry: Methods and Applications
dc.title Recursions of Symmetry Orbits and Reduction without Reduction uk_UA
dc.type Article uk_UA
dc.status published earlier uk_UA

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