On a class of second-order PDEs admitting partner symmetries

Abstract

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear equations which we call mixed heavenly equation and asymmetric heavenly equation. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.

LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions of the Legendre transformed mixed heavenly equation and Ricci-flat metrics governed by solutions of this equation are added. Eq. (6.10) on p. 14 is corrected

References in corpus (2)

• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics
• Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Cited by in corpus (6)

• On the integrability of symplectic Monge-Ampére equations
• Recursion operators for dispersionless integrable systems in any dimension
• General heavenly equation governs anti-self-dual gravity
• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Recursions of Symmetry Orbits and Reduction without Reduction
• Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors