Method of group foliation, hodograph transformation and non-invariant solutions of the Boyer-Finley equation

TL;DR

This paper introduces a method combining group foliation and hodograph transformation to construct and analyze non-invariant solutions of the Boyer-Finley equation, relevant in gravitational instanton theory.

Contribution

It develops a novel approach integrating group foliation with hodograph transformation to find non-invariant solutions of the Boyer-Finley equation, including conditionally invariant solutions.

Findings

Derived non-invariant solutions using group foliation.

Demonstrated hodograph transformation yields known solutions.

Identified invariant relations for hodograph solutions.

Abstract

We present the method of group foliation for constructing non-invariant solutions of partial differential equations on an important example of the Boyer-Finley equation from the theory of gravitational instantons. We show that the commutativity constraint for a pair of invariant differential operators leads to a set of its non-invariant solutions. In the second part of the paper we demonstrate how the hodograph transformation of the ultra-hyperbolic version of Boyer-Finley equation in an obvious way leads to its non-invariant solution obtained recently by Manas and Alonso. Due to extra symmetries, this solution is conditionally invariant, unlike non-invariant solutions obtained previously. We make the hodograph transformation of the group foliation structure and derive three invariant relations valid for the hodograph solution, additional to resolving equations, in an attempt to obtain the orbit of this solution.