Integrable Systems and Twistors

AbstractFor the purpose of this chapter, integrable systems are special kinds of non-linear differential equations (although one should note that there are also integrable difference equations, and the study of these is increasingly prominent). For essentially any system of differential equations, one has local existence theorems, so existence of solutions is not the issue. The question, rather, is what the solutions are like. For a non-linear differential equation chosen at random, one has no hope of writing down explicit solutions in terms of known functions. The classification of a function as ‘known’ may seem artificial, having more to do with what we’ve bothered to learn than the function itself; but in fact there is more to it than that. The known functions (e.g. trigonometric functions or elliptic functions) are known because they have nice properties, and can be defined using only a finite amount of information. The functions involved in the solutions of a generic nonlinear differential equation are so awful as to be literally indescribable in any but a tautological way (that is, one can really only define them as solutions of the differential equation). An extreme example of this is chaotic systems.