Lie's structural approach to PDE systems / Olle Stormark.

Includes bibliographical references and index.

Print version record.

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.

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

3.3 How general is the general solution?3.4 Cauchy characteristics; 3.5 Maximal involutions and integrable vector field systems; 4 Involutivity and the prolongation theorem; 4.1 Independence condition and involutivity; 4.2 Prolongations; 4.3 Explanation of the prolongation theorem; 5 Drach's classification, second order PDEs in one dependent variable, and Monge characteristics; 5.1 The classification of Drach; 5.2 Second order PDEs in one unknown and their singular vector fields; 5.3 Monge characteristic subsystems; 6 Integration of vector field systems V satisfying dim V' = dim V +1

6.1 Maximal involutions6.2 Complete subsystems; 6.3 The generalized contact bracket; 6.4 Reduction to a canonical form and systems of contact coordinates; 6.5 How to find all maximal complete subsystems of V; 6.6 Contact transformations and Lie pseudogroups; 7 Higher order contact transformations; 7.1 Lie's rectification theorem for first order PDE systems in one dependent variable; 7.2 Backlund's theorems; 7.3 Contact prolongations of local diffeomorphisms; 8 Local Lie groups; 8.1 The parameter group and its structure constants; 8.2 The left- and right-invariant parameter groups

8.3 Left- and right-invariant vector fields and their dual Maurer-Cartan forms8.4 One-parameter subgroups and the exponential mapping; 8.5 The first and second fundamental theorems; 8.6 The third fundamental theorem; 8.7 Local transformation groups; 9 Structural classification of 3-dimensional Lie algebras over the complex numbers; 9.1 The classification; 9.2 Realizations as transformation groups; 10 Lie equations and Lie vector field systems; 10.1 Characterization of ODE systems with fundamental solutions; 10.2 Lie vector field systems associated to Lie groups

11 Second order PDEs in one dependent and two independent variables11.1 Second order PDEs and associated vector field systems; 11.2 Monge systems; 11.3 A connection with line geometry; 11.4 Darboux's method for hyperbolic PDEs; 12 Hyperbolic PDEs with Monge systems admitting two or three first integrals; 12.1 First integrals of the first order; 12.2 Two first integrals for each Monge system; 12.3 How to find integral manifolds; 12.4 Integrable systems; 12.5 Two first integrals for one Monge system and three for the other; 12.6 Three first integrals for each Monge system

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