Twistor Theory of the Airy Equation
High Energy Physics - Theory
2014-04-03 v3 Complex Variables
Differential Geometry
Exactly Solvable and Integrable Systems
Abstract
We demonstrate how the complex integral formula for the Airy functions arises from Penrose's twistor contour integral formula. We then use the Lax formulation of the isomonodromy problem with one irregular singularity of order four to show that the Airy equation arises from the anti-self-duality equations for conformal structures of neutral signature invariant under the isometric action of the Bianchi II group. This conformal structure admits a null-K\"ahler metric in its conformal class which we construct explicitly.
Keywords
Cite
@article{arxiv.1401.0025,
title = {Twistor Theory of the Airy Equation},
author = {Michael Cole and Maciej Dunajski},
journal= {arXiv preprint arXiv:1401.0025},
year = {2014}
}
Comments
Special Issue of SIGMA on Progress in Twistor Theory