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- Title
Method of Group Foliation, Hodograph Transformation, and Noninvariant Solutions of the Heavenly Equation.
- Authors
Sheftel, M. B.
- Abstract
We present the method of group foliation for constructing noninvariant solutions of partial differential equations on an important example of the “heavenly equation” from the theory of gravitational instantons. We show that the constraint of commutativity of a pair of invariant differential operators leads to a set of noninvariant solutions of the heavenly equation. In the second part of the paper, we demonstrate how the noninvariant solution of the ultrahyperbolic heavenly equation recently obtained by Mañas and Martínez Alonso becomes obvious after hodograph transformation of the heavenly equation. Because of extra symmetries, this solution is conditionally invariant, unlike noninvariant solutions obtained previously. We make the hodograph transformation of the group foliation structure and derive two invariant relations valid for the hodograph solution, in addition to resolving equations in an attempt to obtain the orbit of this solution.
- Subjects
HODOGRAPH; KINEMATICS; BESSEL functions; RENORMALIZATION (Physics); GAUGE field theory; WAVE equation
- Publication
Theoretical & Mathematical Physics, 2003, Vol 137, Issue 3, p1743
- ISSN
0040-5779
- Publication type
Article
- DOI
10.1023/B:TAMP.0000007922.86197.3e