Geometric symmetry techniques for partial differential equations

The present work applies Vessiot theory and exterior calculus to fnd local solutions of partial differential equations (PDEs). We examine the classes of PDEs which are represented by vector felds forming the Vessiot distribution. In general the Vessiot distribution is not Frobenius integrable. We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution for PDEs. We develop and apply a method to fnd the largest integrable sub-distributions and hence (local) solutions of the PDEs. A more general class of symmetries known as solvable symmetry structures play an important role in our work. A novel feature of our approach is the formulation of the ‘quasi-group invariance method’. In particular, we show how the solvable symmetry structure of the original PDEs can be used to construct integrable sub-distributions leading to invariant solutions of the PDEs. We also extend this result to higher dimensional PDEs. We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. We also discuss ‘local’ conservations laws for evolution equations. Some consideration of the application of this technique to analyse Shannon entropy of evolution equations is taken at the end.

Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Engineering and Mathematical Sciences, Faculty of Science, Technology and Engineering, La Trobe University, Bundoora.