Year: 2009
Type: article
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.
Author: Maciej Dunajski
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Award: G7498563395
OpenAlex ID: W2096814845