# Scalar differential invariants of symplectic Monge-Ampère equations

Alessandro Paris; Alexandre Vinogradov

Open Mathematics (2011)

- Volume: 9, Issue: 4, page 731-751
- ISSN: 2391-5455

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topAlessandro Paris, and Alexandre Vinogradov. "Scalar differential invariants of symplectic Monge-Ampère equations." Open Mathematics 9.4 (2011): 731-751. <http://eudml.org/doc/269253>.

@article{AlessandroParis2011,

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 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.},

author = {Alessandro Paris, Alexandre Vinogradov},

journal = {Open Mathematics},

keywords = {Monge-Ampère equation; Scalar differential invariant; Symplectic manifold; Tangent distribution; scalar differential invariant; symplectic manifold; tangent distribution; equivalence problem; elliptic; symplectic hyperbolic},

language = {eng},

number = {4},

pages = {731-751},

title = {Scalar differential invariants of symplectic Monge-Ampère equations},

url = {http://eudml.org/doc/269253},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Alessandro Paris

AU - Alexandre Vinogradov

TI - Scalar differential invariants of symplectic Monge-Ampère equations

JO - Open Mathematics

PY - 2011

VL - 9

IS - 4

SP - 731

EP - 751

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/269253

ER -

## References

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- [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 Zbl1129.58015
- [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 Zbl1152.58308
- [10] Vinogradov A.M., On the geometry of second-order parabolic equations with two independent variables, Dokl. Akad. Nauk, 2008, 423(5), 588–591

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