On a class of second-order PDEs admitting partner symmetries

TL;DR

This paper introduces a class of second-order PDEs with partner symmetries, including known heavenly equations, and demonstrates how to generate noninvariant solutions and related Ricci-flat metrics.

Contribution

It presents a general form of second-order PDEs with partner symmetries, including new equations, and shows how to derive noninvariant solutions and metrics from them.

Findings

Identified a class of PDEs with partner symmetries including heavenly equations.

Developed recursion relations for partner symmetries.

Constructed Ricci-flat self-dual metrics from solutions.

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.