Twistor Theory and Differential Equations

Details Twistor Theory and Differential Equations Twistor Theory and Differential Equations

2009-02-02

arxiv.org/abs/0902.0274v2

J.Phys.A42:404004,2009

Physics

High Energy Physics

High Energy Physics - Theory

23 Pages, 9 Figures

Scientific paper

10.1088/1751-8113/42/40/404004

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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