Bäcklund transformations for Darboux integrable differential systems: Examples and applications

In this article we demonstrate a new symmetry based method for constructing Bäcklund transformations by finding explicit Bäcklund transformations between Darboux integrable systems. This results in a number of new examples of Bäcklund transformations which are quite different in character than those typically found in the literature. The relation between the intermediate integrals for Darboux integrable systems and the differential invariants of the Vessiot group is also illustrated. We then show that a well known class of Bäcklund transformations between a Darboux integrable Monge–Ampère system and the wave equation always arises by this method. The results of this paper build upon the presentation of Darboux integrable systems as quotients of differential systems by symmetry groups.