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Differential invariants of generic hyperbolic Monge-Ampère equations

100%
Marvan M., Vinogradov A., Yumaguzhin V.
Open Mathematics
|
2007
|
tom 5
|
nr 1
105-133
EN
In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.
2
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Fundamental solutions of the complex Monge-Ampère equation

100%
Celik H., Poletsky E.
Annales Polonici Mathematici
|
1997
|
tom 67
|
nr 2
103-110
EN
We prove that any positive function on ℂℙ¹ which is constant outside a countable $G_δ$-set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.
3
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Scalar differential invariants of symplectic Monge-Ampère equations

100%
Paris A., Vinogradov A.
Open Mathematics
|
2011
|
tom 9
|
nr 4
731-751
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.
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