Rank 2 Bäcklund Transformations of Hyperbolic Monge-Ampère Systems

Abstract This article studies rank 2 Backlund transformations of hyperbolic Monge-Ampere systems using Cartan's method of equivalence. Such Backlund transformations have two main types, which we call Type A and Type B . For Type A , we completely determine a subclass whose local invariants satisfy a specific but simple algebraic constraint. We show that such Backlund transformations are parametrized by a finite number of constants; in a subcase of maximal symmetry, we determine the coordinate form of the underlying PDEs, which turn out to be Darboux integrable. For Type B , we present an invariantly formulated condition that determines whether a Backlund transformation is one that, under suitable choices of local coordinates, relates solutions of two PDEs of the form z x y = F ( x , y , z , z x , z y ) and preserves the x , y variables on solutions.