Symmetry analysis of differential equations from a geometric point of view
Supervisor: Chris Athorne
The central theme of this work is the step by step integration of Frobenius and Darboux type differential problems by quadratures. Building on the concept of a solvable structure of vector fields introduced by Basarab-Horwath [Ukr. Jour. Math. 43 (1991) 1330-7; MathSciNet Review], we demonstrate the wider role played by chains of relatively closed ideals of vector fields or one-forms (under the Lie bracket or the exterior derivative respectively) in this context. We extend the solvable structure concept to one-forms and develop the notion of solvable structures compatible with a Poisson structure. In the preocess we shed light of certain types of so-called hidden symmetries and some other known types of symmetry reduction. In the introductory sections we also introduce a new way of defining the exterior derivative, slightly generalise the theorem of Frobenius and present a concise way of characterising Poisson tensors by giving an explicit expression for the auto-Schouten bracket of a two-vector. Also included is a classification result drawing attention to the wide scope of non-Lie point symmetries.