Geodesic
Twistor Theory of the Airy Equation
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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-Kähler metric in its conformal class which we construct explicitly.
Keywords: twistor theory; Airy equation; self-duality. 
@article{SIGMA_2014_10_a36,
     author = {Michael Cole and Maciej Dunajski},
     title = {Twistor {Theory} of the {Airy} {Equation}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a36/}
}
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Michael Cole; Maciej Dunajski. Twistor Theory of the Airy Equation. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a36/

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