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Lie's Structural Approach to PDE Systems
About this book
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.
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Table of contents
- Cover
- Half Title
- Series Page
- Title
- Copyright
- Contents
- 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 V satisfying dim V' = dim V +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 two or three first integrals
- 13 Classification of hyperbolic Goursat equations
- 14 Cartan's theory of Lie pseudogroups
- 15 The equivalence problem
- 16 Parabolic PDEs and associated PDE systems
- 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