A Geometry for SecondOrder PDEs and their Integrability, Part I
Abstract
For the purpose of understanding secondorder scalar PDEs and their hydrodynamic integrability, we introduce Gstructures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of secondorder jet space) and are defined by nondegenerate scalar secondorderonly (Hessian) PDEs in any number of variables. The fiber group is a conformal orthogonal group that acts on the space of independent variables, and it is a subgroup of the conformal orthogonal group for a semiRiemannian metric that exists on the PDE. These Gstructures are automatically compatible with the definition of hydrodynamic integrability, so they allow contactinvariant analysis of integrability via moving frames and the CartanKaehler theorem. They directly generalize the GL(2)structures that arise in the case of Hessian hyperbolic equations in three variables as well as several related geometries that appear in the literature on hydrodynamic integrability. Part I primarily discusses the motivation, the definition, and the solution to the equivalence problem, and Part II will discuss integrability in detail.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 arXiv:
 arXiv:1010.6010
 Bibcode:
 2010arXiv1010.6010S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 58A15;
 37K10
 EPrint:
 25 pages