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Contact Geometry of Hyperbolic Equations of Generic Type
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2008, том 4
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Contact Geometry of Hyperbolic Equations of Generic Type
Інший ID:
2000 Mathematics Subject Classification: 35A30; 35L70; 58J70
Посилання:
Contact Geometry of Hyperbolic Equations of Generic Type / D. The // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 26 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. It is my pleasure to thank Niky Kamran for his lucid explanations of exterior dif ferential systems and the Cartan equivalence method, his guidance while studying [11], and for bringing to my attention Vranceanu’s work [25]. Many of the calculations in this paper were either facilitated by or rechecked using the DifferentialGeometry, LieAlgebras, and JetCalculus packages (in Maple v.11) written by Ian Anderson. I would also like to thank Thomas Ivey and the three anonymous referees for their comments and corrections to help improve the exposition of this article. This work was supported by funding from NSERC and McGill University.
Дата:
2008
Переглядів:
518
Завантажень:
237
Анотація:
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampère (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) generic hyperbolic structures is also given.
