Book contents
- Frontmatter
- 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
11 - Second order PDEs in one dependent and two independent variables
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- 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
Summary
By Drach's classification an arbitrary PDE system can be reduced to a first or second order system in one unknown, and first order systems in one unknown are completely understood thanks to the theory in chapter 6. So the real difficulties start with second order systems in one unknown.
In this chapter we begin the study of a single PDE in one dependent and two independent variables with some generalities from [Vessiot 1924] and [Vessiot 1936]. Chapters 12-13 are then devoted to hyperbolic PDEs, and chapters 16-17 to parabolic PDEs. Finally, we study PDE systems in one dependent and three independent variables in chapter 18—thus leaving the investigation of PDE systems in one dependent and more than three independent variables in suspense.
Second order PDEs and associated vector field systems
A second order PDE in one dependent and two independent variables is equivalent to a hypersurface in the jet bundle J2(R2,R). The latter has the dimension 8, and is classically endowed with the following coordinates:
independent variables x and y,
dependent variable z,
first order jet coordinates p ↔ ∂z/∂x and q ↔ ∂z/∂y,
second order jet coordinates r ↔ ∂2z/∂2x, s ↔ ∂2z/∂x∂y and t ↔ ∂2z/∂y2.
- Type
- Chapter
- Information
- Lie's Structural Approach to PDE Systems , pp. 270 - 295Publisher: Cambridge University PressPrint publication year: 2000