Scalar differential invariants of symplectic Monge-Ampère equations

Abstract

Abstract. All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allows us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form uxy + f (x, y, ux, uy) = 0 and in particular find a simple linearization criterion.

Geometry of Lagrangian Grassmannians and nonlinear PDEs

“…Due to the invariance of the framework, the sub-distribution X E of C can by all means replace E in the treatment of the equivalence problem. This point of view is at the basis of many works about invariants and classification of Goursat-type Monge-Ampère equations, see, e.g., [4,16,9,13,34,33].…”

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

“…For instance, the order of truncation is 0 in (3.7) and it is 1 in (3.10) and (3.11). It is convenient to set 12) where the total derivatives appearing in (3.12) are truncated to the (k − 1) st order. Indeed, both (3.9) and (3.11) simplify as…”

“…Much as in the geometric approach to classical MAEs it is convenient to exploit the contact/symplectic geometry underlying 2 nd order PDEs [4,5,12,20,22,26], to deal with 3 rd order MAEs we shall make use of the prolongation of a contact manifold, equipped with its Levi form (see, e.g., [27,Section 2]), a structure known as meta-symplectic, quickly reviewed below. It is worth stressing that our main gadget, i.e., the correspondence between E and V E , is just an adaptation of similar techniques traditionally found in other areas of modern mathematics 4 .…”

Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

“…In these works, scalar differential invariants of hyperbolic and elliptic MA-equations are constructed indirectly as differential invariants of the associated e-structures. On the contrary, Definition 3.1 allows a direct construction of scalar differential invariants in terms of operators of the corresponding basic algebra, which leads to more complete and exact results (see [7,2,13,1]). …”