Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

Abstract

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.

Self-dual gravity is completely integrable

“…These real Lax pairs are formed by the recursion operators for symmetries and operator A of the symmetry condition and so they are Lax pairs of the Olver-Ibragimov-Shabat type [14,15], which is different from the complex Lax pairs suggested by Mason and Newman [12,13] and Dunajski and Mason [8,9] and those that we used in [10,11] in relation to partner symmetries, even if we set our new Lax pairs in one-component forms. Furthermore, the commutator of the complex recursion operator of Mason-Dunajski with the operator of the symmetry condition, in a one-component form, reproduces the symmetry condition and not the original equation CMA [10].…”

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

“…It is easy to see that if ϕ satisfies (2.3), then ψ also satisfies (2.3) and so the potential ψ of a symmetry ϕ is itself a symmetry of CMA [6,8]. These ϕ and ψ are called partner symmetries.…”

“…Note that if |λ | = 1 the inverse transformation (2.7) reads 8) and then for HCMA there is a simplifying choice ψ = ϕ, when the transformation (2.4) coincides with its inverse (2.7) and becomes…”

“…It can be proved that if the functions b(z),b(z) are not constants, the formulas (6.4a)-(6.4c) yield non-invariant solutions of (5.2). As a consequence, by the reasoning similar to [8], the ultra-hyperbolic metrics governed by the potentials ψ in (6.4a)-(6.4c) have no Killing vectors [9] (though nonexistence of conformal Killing vectors was not proved).…”

“…Recently, we have developed method of partner symmetries that yields non-invariant solutions to the elliptic and hyperbolic CMA and to the second heavenly equation. Using them as metric potentials, we have obtained some heavenly metrics with no Killing vectors [6,7,8] (nonexistence of conformal Killing vectors was not proved).…”