KdV solitons in Einstein's vacuum field equations

TL;DR

This paper demonstrates a specific metric in Einstein's vacuum equations that leads to the KdV equation, linking soliton solutions to solutions of Einstein's equations, and analyzes the properties of the one-soliton metric.

Contribution

It introduces a novel metric in Einstein's vacuum equations that produces the KdV equation, establishing a new connection between soliton theory and general relativity.

Findings

The metric generates the KdV equation from Einstein's vacuum equations.

The one-soliton metric is non-singular and Lorentzian.

The metric is classified as Petrov type N.

Abstract

We present a metric for which Einstein's field equations in vacuum generate the Kortweg-de Vries (KdV) equation and thus its $N$-soliton solutions solve the vacuum equations. The metric of the one soliton solution has been investigated and is a non-singular, Lorentzian metric of type N in the Petrov classification.