Solutions of the Bäcklund Problem

Abstract

In previous sections it has been assumed that a Bäcklund map is given, and taken for granted that its integrability conditions are interesting. We turn now to the main practical problem which is to find Bäcklund maps whose integrability conditions are a given system of partial differential equations. On the face of it, the solution to this problem, which we have called the Bäcklund problem, requires the determination of functions ψ_a^A such that $\tilde D_{\left[ a \right.}^{(h + 1)}{\mkern 1mu} \psi _{\left. b \right]}^A = 0$ (equations (4.20)) are combinations of the given equations. In practice, only special solutions of this problem have been found, and there is no obvious criterion for deciding which solutions are interesting ones. However, Wahlquist and Estabrook [72] have invented a very efficient way of finding Bäcklund transformations, which in effect yields solutions of the Backlund problem, although not in the language of jet bundles used here. Their method employs Cartan’s theory of exterior differential systems [8].