Scalar differential invariants of symplectic Monge-Ampère equations

• Autorzy: Alessandro Paris , Alexandre Vinogradov
• Języki publikacji: EN

Abstrakty

EN

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 allow 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 u xy + f(x, y, u x, u y) = 0 and in particular find a simple linearization criterion.

Słowa kluczowe

EN

Monge-Ampère equation; Scalar differential invariant; Symplectic manifold; Tangent distribution

Kategorie tematyczne

• 53D99: None of the above, but in this section
• 58J70: Invariance and symmetry properties

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 4
• Strony: 731-751

Daty

wydano

2011-08-01

online

2011-05-26

Twórcy

autor

Alessandro Paris

• Università di Napoli Federico II

autor

Alexandre Vinogradov

Bibliografia

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• [3] Ferraioli D.C., Vinogradov A.M., Differential invariants of generic parabolic Monge-Ampere equations, preprint available at http://arxiv.org/abs/0811.3947
• [4] CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it
• [5] Kruglikov B., Classification of Monge-Ampère equations with two variables, In: Geometry and Topology of Caustics - CAUSTICS’98, Warsaw, Banach Center Publ., 50, Polish Academy of Sciences, Warsaw, 1999, 179–194
• [6] Kushner A., Lychagin V., Rubtsov V., Contact Geometry and Non-Linear Differential Equations, Encyclopedia Math. Appl., 101, Cambridge University Press, Cambridge, 2007
• [7] Marvan M., Vinogradov A.M., Yumaguzhin V.A., Differential invariants of generic hyperbolic Monge-Ampère equations, Cent. Eur. J. Math., 2007, 5(1), 105–133 http://dx.doi.org/10.2478/s11533-006-0043-4
• [8] Vinogradov A.M., Scalar differential invariants, diffieties and characteristic classes, In: Mechanics, Analysis and Geometry: 200 years after Lagrange, North-Holland Delta Ser., North-Holland, Amsterdam, 1991, 379–414
• [9] Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001
• [10] Vinogradov A.M., On the geometry of second-order parabolic equations with two independent variables, Dokl. Akad. Nauk, 2008, 423(5), 588–591

Identyfikatory

DOI

10.2478/s11533-011-0046-7