Contact Maps and Induced Differential Equations

Abstract

The goal of this paper is to examine constructive ways of generating solutions of partial differential equations (pde’s) by the use of associated ordinary differential equations (ode’s). Specifically, the classical method of Cauchy characteristics for first-order pde’s is examined from the Cartan geometric viewpoint, i.e., as a method of extension via a contact (or Cauchy-characteristic) vector field on a submanifold determined by the equation in an appropriate jet bundle. Contact vector fields generate contact transformations, which are self maps of a jet bundle with induced cotangent space maps preserving the contact structure. Contact maps, or immersions, from one jet bundle to another with induced maps carrying contact structures in a specified manner are introduced. Contact transformations transform pde’s to pde’s; a contact map can induced an ordinary differential equation from a pde. For example, a contact map that induces a sixth-order, linear ode from the classical Burgers’ pde is constructed. The final goal is to examine extending Cauchy data for second- or higher-order pde’s via nonautonomous “Cauchy vector fields” on the jet bundles of the equations induced by contact maps.