Author： Sheftel M. B.
Publisher： Springer Publishing Company
ISSN： 0040-5779
Source： Theoretical and Mathematical Physics, Vol.137, Iss.3, 2003-12, pp. : 1743-1752
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
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.
Related content
Access to resources
Recommend
Lorentz transformation with noninvariant c.
By Rasul M.
Physics Essays, Vol. 22, Iss. 4, 2009-12 ,pp. :
A Hodograph Transformation Applicable to a Large Class of PDEs
By Mañas M.
Theoretical and Mathematical Physics, Vol. 137, Iss. 2, 2003-11 ,pp. :