Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 058, 52 pages      arXiv:0804.1559      http://dx.doi.org/10.3842/SIGMA.2008.058
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Contact Geometry of Hyperbolic Equations of Generic Type

Dennis The
McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

Received April 10, 2008, in final form August 11, 2008; Published online August 19, 2008

Abstract
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.

Key words: contact geometry; partial differential equations; hyperbolic; generic; nonlinear.

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