Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Abstract

We study the geometry of multidimensional scalar 2 nd order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M (1) over a (2n + 1)-dimensional contact manifold (M, C). We develop the theory of characteristics of the equation E in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of E. After specifying the results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat in [11], i.e. MAEs of the formWe show that any MAE of the aforementioned class is associated with an n-dimensional subdistribution D of the contact distribution C, and viceversa. We characterize this Goursat-type equations together with its intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method of solutions of a Cauchy problem, provided the existence of a suitable number of intermediate integrals.

Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

“…Returning to the notion of prolongation of a distribution (M, E), one defines the manifold M (1) to consist of all (dim N)-dimensional π-horizontal integral elements of E. The prolonged distribution E (1) on M (1) is then defined to be the lift of the tautological relative distribution along the natural projection π (1) :…”

“…They are central to characteristics, Monge cones, geometric singularities of PDEs [11,7] and boundary conditions [9]. They have been used to find differential contact invariants of certain classes of PDEs [1,8]. Flags of integral elements appear in the context of the Cartan-Kähler theorem [5].…”

Meta-Symplectic Geometry of 3rdOrder Monge-Ampère Equations and their Characteristics

“…We keep the same symbol L for both the tautological bundles over M (1) and M (2) , since it will be clear from the context which is which.…”

“…Then, by definition, a Lagrangian plane of M (1) is a 2D subspace which is π-horizontal 5 and such that all the forms belonging to the differential ideal generated by θ (1) , vanish on it. In analogy with (2.1), we define the 2 nd prolongation M (2) of a contact manifold (M, C) as the first prolongation of M (1) , that is…”

“…A generic point of M (resp., M (1) , M (2) ) is denoted by m (resp., m 1 , m 2 ). As a rule, when both m 1 and m 2 (resp.…”