Differential invariants of generic hyperbolic Monge-Ampère equations

Marvan, Michal; Vinogradov, A. M.; Yumaguzhin, V. A.

doi:10.2478/s11533-006-0043-4

Cited by 9 publications

(13 citation statements)

References 20 publications

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“…In these works, scalar differential invariants of hyperbolic and elliptic MA-equations are constructed indirectly as differential invariants of the associated e-structures. On the contrary, Definition 3.1 allows a direct construction of scalar differential invariants in terms of operators of the corresponding basic algebra, which leads to more complete and exact results (see [7,2,13,1]). …”

Section: Proofmentioning

confidence: 99%

Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

Vinogradov

2014

Int. J. Geom. Methods Mod. Phys.

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Section: Proofmentioning

confidence: 99%

Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

Vinogradov

2014

Int. J. Geom. Methods Mod. Phys.

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“…They are central to characteristics, Monge cones, geometric singularities of PDEs [11,7] and boundary conditions [9]. They have been used to find differential contact invariants of certain classes of PDEs [1,8]. Flags of integral elements appear in the context of the Cartan-Kähler theorem [5].…”

Section: Structure Of the Articlementioning

confidence: 99%

Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

Bächtold

2016

Journal of Differential Equations

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We study an intrinsic distribution, called polar, on the space of l-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a k th order jet with additional k + 1 st order information along l of the n possible directions. A choice of partial extensions of a jet into all possible l-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting l-directions.We further show that prolonging the polar distribution once more yields the space of (l, n)-dimensional integral flags with its double fibration distribution. When l > 1 the exterior differential system is holonomic, stabilizing after one further prolongation.The proof starts form the space of integral flags, constructing the tower of prolongations by reduction.

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“…These invariants are of second and third orders, i.e., depend on 2-nd and 3-rd order derivatives of coefficients of MAEs. This paper is a natural continuation of [7]. We construct simplest SDIs which are sufficient for solution of the equivalence problem for non-generic elliptic and hyperbolic MAEs.…”

Section: Introductionmentioning

confidence: 99%

“…This approach focuses on construction of scalar differential invariants (SDI) of MAEs which are also indispensable for the classification problem. In particular, simplest SDIs, sufficient for solution of the equivalence problem, were constructed in [7] for generic hyperbolic MAEs. These invariants are of second and third orders, i.e., depend on 2-nd and 3-rd order derivatives of coefficients of MAEs.…”

Section: Introductionmentioning

confidence: 99%

Scalar differential invariants of symplectic Monge-Ampère equations

Paris

Vinogradov

2011

Open Mathematics

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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 allows 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 uxy + f (x, y, ux, uy) = 0 and in particular find a simple linearization criterion.

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Differential invariants of generic hyperbolic Monge-Ampère equations

Abstract: Abstract. 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.

Cited by 9 publications

References 20 publications

Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

Some remarks on contact manifolds, Monge–Ampère equations and solution singularities

Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

Scalar differential invariants of symplectic Monge-Ampère equations

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