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.
@article{bwmeta1.element.doi-10_2478_s11533-011-0046-7,
author = {Alessandro Paris and Alexandre Vinogradov},
title = {Scalar differential invariants of symplectic Monge-Amp\`ere equations},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {731-751},
zbl = {1252.58022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0046-7}
}
Alessandro Paris; Alexandre Vinogradov. Scalar differential invariants of symplectic Monge-Ampère equations. Open Mathematics, Tome 9 (2011) pp. 731-751. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0046-7/
[1] Alekseevskij D.V., Vinogradov A.M., Lychagin V.V., Basic Ideas and Concepts of Differential Geometry, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991
[2] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999
[3] Ferraioli D.C., Vinogradov A.M., Differential invariants of generic parabolic Monge-Ampere equations, preprint available at http://arxiv.org/abs/0811.3947 | Zbl 1245.53017
[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 | Zbl 1122.53044
[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 | Zbl 1129.58015
[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 | Zbl 1152.58308
[10] Vinogradov A.M., On the geometry of second-order parabolic equations with two independent variables, Dokl. Akad. Nauk, 2008, 423(5), 588–591