Twistor theory and differential equations

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