Seminarium Gamma

In these two seminar talks I will survey the Lie group theoretic approach to the classical integration method of Darboux -- based on the symmetry reduction of pairs of differential systems admitting a common symmetry group. Topics include:Differential Equations via Differential SystemsFundamental ConstructionsSymmetry Reduction of Differential Systems.Lie EquationsSymbolic ToolsExamplesDarboux Integrable (DI) SystemsThe Equivalence Problem for DI Systems ClassificationsFurther ApplicationsReferences The book by O. Stomark, Lie's Structural Approach to PDE Systems is a good introduction to the classical literature. Here are references to my work on superposition principles with Mark Fels, Peter Vassiliou, and Brandon Ashley. Exterior Differential Systems with Symmetry Acta. Appl. Math., 87 (2005) 3-31. Superposition Formulas for Darboux Integrable Exterior Differential Systems, Advances in Math., 221 (2009) 1910--1963Transformations of Darboux Integrable Equations,} Proceeding of the 2008 Abel Conference, Tromso, Norway. Springer.Backlund transformations for Darboux integrable differential systems} Selecta Math. New Series 21 (2014) 379-448.Backlund transformations for Darboux integrable differential systems: Examples and Applications Journal of Geometry and Physics 102 (2016) 1-31The Cauchy Problem for Darboux Integrable Systems and Non Linear d’Alembert Formulas, SIGMA 9 (2017) Darboux Integrable f-Gordon Equations and Rank 2 Distributions in Five Variables (in preparation)