Geometry of Bäcklund Transformations

Abstract

This thesis is a study of B\"acklund transformations using geometric methods. A B\"acklund transformation is a way to relate solutions of two PDE systems. If such a relation exists for a pair of PDE systems, then, using a given solution of one system, one can generate solutions of the other system by solving only ODEs.My contribution through this thesis is in three aspects. First, using Cartan's Method of Equivalence, I prove the generality result: a generic rank-1 B\"acklund transformation relating a pair of hyperbolic Monge-Amp\`ere systems can be uniquely determined by specifying at most 6 functions of 3 variables. In my classification of a more restricted case, I obtain new examples of B\"acklund transformations, which satisfy various isotropy conditions. Second, by formulating the existence problem of B\"acklund transformations as the integration problem of a Pfaffian system, I propose a method to study how a B\"acklund transformation relates the invariants of the underlying hyperbolic Monge-Amp\`ere systems. This leads to several general results.Third, I apply the method of equivalence to study rank-$2$ B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems and partially classify those that are homogeneous. My classification so far suggests that those homogeneous B\"acklund transformations (relating two hyperbolic Monge-Amp\`ere systems) that are genuinely rank-2 are quite few.