The dispersive self-dual Einstein equations and the Toda lattice

TL;DR

This paper introduces a dispersive version of the self-dual Einstein equations that encompasses the Toda lattice and reduces to the classical equations in the dispersionless limit, using a deformation of the algebra of area-preserving diffeomorphisms.

Contribution

It develops a novel dispersive deformation of the self-dual Einstein equations via an associative star-product, linking them to the Toda lattice.

Findings

Established a dispersive self-dual Einstein equation model.

Connected the model to the Toda lattice through deformation.

Provided a mathematical framework for dispersive deformations.

Abstract

The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionlesslimit of the $2d$-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative $\star$-product, of the algebra $sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless, equations.