On the Yamazaki-Hori solution of the Ernst equation

TL;DR

This paper introduces a new family of solutions to the Ernst equation that generalizes the Yamazaki-Hori and Tomimatsu-Sato solutions, connecting them with Vein's formulation and Cosgrove's nonlinear differential equation.

Contribution

It presents a unified family of solutions to the Ernst equation that encompasses known solutions and links to other nonlinear differential equations.

Findings

Recovers Yamazaki-Hori solution in a specific limit

Includes Vein's formulation as a special case

Establishes a connection to Cosgrove's nonlinear differential equation

Abstract

A family of solutions to the Ernst equation is presented, which, in a certain limit, recovers the Yamazaki-Hori solution - an extension of the Tomimatsu-Sato solutions for all integer values of the deformation parameter $\delta$. Our solution also recovers, as a special case, the formulation given by Vein, which is equivalent to the Yamazaki-Hori solution. Furthermore, our construction establishes a connection to a nonlinear differential equation proposed by Cosgrove, which is also associated with the Ernst equation.