Twistor theory and differential equations

Abstract

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 . 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) , 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.