Bryant, Robert LHu, Yuhao2018-05-312018-05-312018https://hdl.handle.net/10161/16904<p>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.</p><p>My contribution through this thesis is in three aspects. </p><p>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. </p><p>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.</p><p>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.</p>MathematicsB\"acklund TransformationsDifferential geometryDissertation