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arXiv · 0902.0274

Twistor Theory and Differential Equations

Maciej Dunajski
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Abstract

This is an elementary and self--contained review of twistor theory as a geometric tool for solving non-linear differential equations. Solutions to soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or Sine-Gordon arise from holomorphic vector bundles over $T\CP^1$. A different framework is provided for the dispersionless analogues of soliton equations, like dispersionless KP or $SU(\infty)$ Toda system in 2+1 dimensions. Their solutions correspond to deformations of (parts of) $T\CP^1$, and ultimately to Einstein--Weyl curved geometries generalising the flat Minkowski space. A number of exercises is included and the necessary facts about vector bundles over the Riemann sphere are summarised in the Appendix.

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Maciej Dunajski. 2009-02-13. Twistor Theory and Differential Equations. https://doi.org/10.1088/1751-8113%2F42%2F40%2F404004

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