CONTACT GEOMETRY OF THIRD-ORDER PARTIAL DIFFERENTIAL EQUATIONS WITH TWO INDEPENDENT VARIABLES

Abstract

We develop rigorously a geometric theory of third-order partial differential equations for a scalar function. Under our framework, we can define a notion of nilpotent graded Lie algebras as an invariant useful to study geometry of third-order equations. In terms of these graded Lie algebras, we provide a classification for some classes of third-order equations under a contact equivalence. By this classification, together with model equations, we also clarify several aspects for each subcategory of equations