Twistor Theory and Differential Equations

TL;DR

This paper reviews twistor theory as a geometric method for solving various non-linear differential equations, including soliton and dispersionless equations, linking solutions to holomorphic vector bundles and curved geometries.

Contribution

It provides an elementary, self-contained overview of how twistor theory applies to solving both classical and dispersionless integrable equations, highlighting new geometric frameworks.

Findings

Solutions to soliton equations arise from holomorphic vector bundles over $T ext{CP}^1$.

Dispersionless equations correspond to deformations of $T ext{CP}^1$ and Einstein--Weyl geometries.

The review includes exercises and summarizes vector bundle facts over the Riemann sphere.

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.