ABSTRACT

This chapter is concerned with integrable systems and their twistor constructions. These are mostly based on R.S. Ward's correspondence. The most impressive applications of twistor theory arise in the analysis of nonlinear partial differential equations. The chapter The basic describes the idea of a harmonic morphism and its connections with the standard Kerr theorem. Integrable and soluble systems are systems of nonlinear differential equations that, despite their nonlinearity, are still relatively tractable, so one can obtain large families of exact solutions and precise analytic information about general classes of solutions. The nonlinear graviton construction indicates that the self-dual Einstein equations are also integrable and Penrose's method also describes their solutions in terms of free data.