Scalar differential invariants of symplectic Monge-Ampère equations. (English) Zbl 1252.58022

De Paris, Alessandro; Vinogradov, Alexandre M.

Cent. Eur. J. Math. 9, No. 4, 731-751 (2011).

Summary: 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 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of the Monge-Ampère equations. As an example, we study equations of the form \(u_{xy} + f(x,y,u_{x},u_{y}) = 0\) and, in particular, find a simple linearization criterion.

MSC:

58J70	Invariance and symmetry properties for PDEs on manifolds
58J45	Hyperbolic equations on manifolds
35L70	Second-order nonlinear hyperbolic equations
53D12	Lagrangian submanifolds; Maslov index
35J96	Monge-Ampère equations

Keywords:

Monge-Ampère equation; scalar differential invariant; symplectic manifold; tangent distribution; equivalence problem; elliptic; symplectic hyperbolic

Software:

CoCoA

References:

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