Lie's Structural Approach to PDE Systems

This book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations.

Olle Stormark (Author)

9781107403321, Cambridge University Press

Paperback / softback, published 26 January 2012

590 pages
23.4 x 15.6 x 3 cm, 0.82 kg

Review of the hardback: '… a worthwhile and successful attempt to introduce the ideas of Sophus Lie.' H. Boseck, Zentralblatt für Mathematik

This book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations. It was the first book to present substantial results on local solvability of general and, in particular, nonlinear PDE systems without using power series techniques. The book describes a general approach to systems of partial differential equations based on ideas developed by Lie, Cartan and Vessiot. The most basic question is that of local solvability, but the methods used also yield classifications of various families of PDE systems. The central idea is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail. This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.

Preface
1. Introduction and summary
2. PDE systems, pfaffian systems and vector field systems
3. Cartan's local existence theorem
4. Involutivity and the prolongation theorem
5. Drach's classification, second order PDEs in one dependent variable and Monge characteristics
6. Integration of vector field systems n satisfying dim n' = dim n + 1
7. Higher order contact transformations
8. Local Lie groups
9. Structural classification of 3-dimensional Lie algebras over the complex numbers
10. Lie equations and Lie vector field systems
11. Second order PDEs in one dependent and two independent variables
12. Hyperbolic PDEs with Monge systems admitting 2 or 3 first integrals
13. Classification of hyperbolic Goursat equations
14. Cartan's theory of Lie pseudogroups
15. The equivalence problem
16. Parabolic PDEs for which the Monge system admits at least two first integrals
17. The equivalence problem for general 3-dimensional pfaffian systems in five variables
18. Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
Bibliography
Index.

Subject Areas: Calculus & mathematical analysis [PBK]