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Lie's Structural Approach to PDE Systems

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: January 2012
  • availability: Available
  • format: Paperback
  • isbn: 9781107403321

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  • 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.

    • A lucid and comprehensive introduction to the differential geometric study of partial differential equations
    • This was the first book to present substantial results on local solvability of general (and in particular nonlinear) PDE systems without using power series techniques
    • This book emphasises the importance of infinite dimensional Lie pseudogroups
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    Reviews & endorsements

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

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    Product details

    • Date Published: January 2012
    • format: Paperback
    • isbn: 9781107403321
    • length: 590 pages
    • dimensions: 234 x 156 x 30 mm
    • weight: 0.82kg
    • availability: Available
  • Table of 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 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.

  • Author

    Olle Stormark, Royal Institute of Technology, Stockholm

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