Lie's Structural Approach to PDE Systems (Encyclopedia of Mathematics and its Applications, Series Number 80)

First published in 2000, 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. 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. Olle Stormark. Includes Bibliographical References And Index.

Here is a lucid and comprehensive introduction to the differential geometric study of partial differential equations (PDE). The first book to present substantial results on local solvability of general and nonlinear PDE systems without using power series techniques, it describes a general approach to PDE systems based on ideas developed by Lie, Cartan and Vessiot. The central theme 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.

Cover; Half Title; Series Page; Title; Copyright; Contents; Preface; 1 Introduction and summary; 2 PDE systems, pfaffian systems and vector field systems; 2.1 ODE systems, vector fields and 1-parameter groups; 2.2 First order PDE systems in one dependent variable, pfaffian equations and contact transformations; 2.3 Jet bundles and contact pfaffian systems; 2.4 The theorem of Frobenius; 2.5 Mayer's blowing-up method for proving the Frobenius theorem; 3 Cartan's local existence theorem; 3.1 Involutions and characters; 3.2 From involutions to complete systems