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

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.


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


Full Text: DOI arXiv


[1] Alekseevskij D.V., Vinogradov A.M., Lychagin V.V., Basic Ideas and Concepts of Differential Geometry, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991 · Zbl 0675.53001
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.