
SIGMA 7 (2011), 043, 13 pages arXiv:1005.0153
https://doi.org/10.3842/SIGMA.2011.043
Contribution to the Special Issue “Symmetry, Separation, Superintegrability and Special Functions (S^{4})”
Recursions of Symmetry Orbits and Reduction without Reduction
Andrei A. Malykh ^{a} and Mikhail B. Sheftel ^{b}
^{a)} Department of Numerical Modelling, Russian State
Hydrometeorlogical University, Malookhtinsky pr. 98, 195196 St. Petersburg, Russia
^{b)} Department of Physics, Bogazici University 34342 Bebek, Istanbul, Turkey
Received January 29, 2011, in final form April 25, 2011; Published online April 29, 2011
Abstract
We consider a fourdimensional PDE possessing partner symmetries
mainly on the example of complex MongeAmpère 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 eightdimensional
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 noninvariant 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 fivedimensional space which are obtained
by a threedimensional Legendre transformation of the reduced extended system. We present
algebraic and exponential examples of such solutions that govern Legendretransformed Ricciflat
Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation.
Key words:
MongeAmpère equation; partner symmetries; symmetry reduction; noninvariant solutions; antiselfdual gravity; Ricciflat metric.
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