The geometrical study of differential equations has a long and distinguished history, dating back to the classical investigations of Sophus Lie, Gaston Darboux, and Elie Cartan. Currently, these ideas occupy a central position in several areas of pure and applied mathematics, including the theory of completely integrable evolution equations, the calculus of variations, and the study of conservation laws. In this book, the author gives an overview of a number of significant ideas and results developed over the past decade in the geometrical study of differential equations.Topics covered in the book include symmetries of differential equations and variational problems, the variational bi-complex and conservation laws, geometric integrability for hyperbolic equations, transformations of submanifolds and systems of conservation laws, and an introduction to the characteristic cohomology of differential systems. The exposition is sufficiently elementary so that non-experts can understand the main ideas and results by working independently. The book is also suitable for graduate students and researchers interested in the study of differential equations from a geometric perspective. It can serve nicely as a companion volume to ""The Geometrical Study of Differential Equations"", Volume 285 in the AMS series, ""Contemporary Mathematics"".
Differential equations and their geometry External and generalized symmetries Internal, external and generalized symmetries Transformations of surfaces Tranformations of submanifolds Hamiltonian systems of conservation laws The variational bicomplex The inverse problem of the calculus of variations Conservation laws and Darboux integrability Characteristic cohomology of differential systems Bibliography.