SIGMA 10 (2014), 037, 8 pages      arXiv:1401.0025      https://doi.org/10.3842/SIGMA.2014.037
Contribution to the Special Issue on Progress in Twistor Theory

Twistor Theory of the Airy Equation

Michael Cole and Maciej Dunajski
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Received November 28, 2013, in final form March 18, 2014; Published online March 29, 2014

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-Kähler metric in its conformal class which we construct explicitly.

Key words: twistor theory; Airy equation; self-duality.

pdf (695 kb)   tex (202 kb)

References

1. Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, Proc. Roy. Soc. London Ser. A 458 (2002), 1205-1222, math.DG/0102225.
2. Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
3. Eastwood M.G., Penrose R., Wells R.O., Cohomology and massless fields, Comm. Math. Phys. 78 (1981), 305-351.
4. Jeffreys H., Jeffreys B.S., Methods of mathematical physics, Cambridge University Press, New York, 1946.
5. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
6. John F., The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4 (1938), 300-322.
7. Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs. New Series, Vol. 15, The Clarendon Press, Oxford University Press, New York, 1996.
8. Maszczyk R., Mason L.J., Woodhouse N.M.J., Self-dual Bianchi metrics and the Painlevé transcendents, Classical Quantum Gravity 11 (1994), 65-71.
9. Penrose R., Solutions of the zero-rest-mass equations, J. Math. Phys. 40 (1969), 38-39.
10. Penrose R., Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31-52.
11. Shah M.R., Woodhouse N.M.J., Painlevé VI, hypergeometric hierarchies and Ward ansätze, J. Phys. A: Math. Gen. 39 (2006), 12265-12269.
12. Takasaki K., Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type, Asian J. Math. 2 (1998), 1049-1078, solv-int/9704004.
13. Ward R.S., On self-dual gauge fields, Phys. Lett. A 61 (1977), 81-82.
14. Woodhouse N.M.J., Contour integrals for the ultrahyperbolic wave equation, Proc. Roy. Soc. London Ser. A 438 (1992), 197-206.