Contact Structures of Partial Differential Equations

Abstract

We study the geometry of contact structures of partial differential equations. The main classes we study are first order systems of two equations in two independent and two dependent variables and the second order scalar equations in two independent variables. The contact distribution in these two cases is a rank 4 distribution on a manfifold of dimension 6 or 7. The contact structures for these two classes of equations have a very rich geometric structure. On the equation manifold we can find an almost complex or almost product structure depending on whether the system is elliptic or hyperbolic, respectively. The geometric structures allows us to study many geometric properties. A unifying theme for all theory is the concept of projections (or pseudosymmetries). Pseudosymmetries are a generalization of symmetries, a concept that has been introduced by Sophus Lie in studying partial differential equations. At the same time the method of Darboux is also a special example of a projection generated by pseudosymmetries. With the concept of pseudosymmetries in mind we can clarify and expand the existing theory on Darboux integrable systems, pseudoholomphic curves, Monge-Ampere equations and reductions of second order equations to first order systems.