Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets

Carlet, Guido; Casati, Matteo; Shadrin, Sergey

doi:10.1016/j.geomphys.2016.12.008

Cited by 11 publications

(43 citation statements)

References 19 publications

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“…Very recently it has been proved that the Poisson cohomology of multidimensional scalar brackets is extremely large [3], hence there are plenty of deformations of such structures. The bracket (23) is at the crossroads of the aforementioned cases.…”

Section: Discussionmentioning

confidence: 99%

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Casati¹

2016

Theor Math Phys

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Euler's equations for a two dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence-free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial.

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Section: Discussionmentioning

confidence: 99%

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Casati¹

2016

Theor Math Phys

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“…The structure of the Poisson cohomology for the stretched operator has a striking resemblance to the Poisson cohomology of constant higher order scalar differential Hamiltonian operators. In [2,Remark 15] the Poisson cohomology for the first order scalar differential operator is obtained with a similar approach to the one adopted in this paper. The procedure can be repeated for higher order differential operators of the form Q k (∂) = ∂ 2k+1 .…”

Section: The Poisson Cohomology For Stretched Hamiltonian Operatorsmentioning

confidence: 99%

“…The scalar differential case depending on several independent variables has been addressed in [2,3]: in this case the Poisson cohomology is infinitedimensional, however its highly non-trivial third group imposes enough constraints to classify all the compatible Hamiltonian operators up to an arbitrary order. A natural extension from the continuous to the discrete setting is to study differential-difference systems.…”

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confidence: 99%

A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators

Casati

Wang

2019

Commun. Math. Phys.

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In this paper we extend to the difference case the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in the algebra of difference functions, endowing the space of local functionals with the structure of a Lie algebra. Its Poisson-Lichnerowicz cohomology carries the information about the center, the symmetries and the admissible deformations of such algebra. The analogue notion for the differential case has been widely investigated: the first and most important result is the triviality of all but the lowest cohomology for first order Hamiltonian differential operators, due to Getzler. We study the Poisson-Lichnerowicz cohomology for the operator K0 = S − S −1 , which is the normal form for (−1, 1) order scalar difference Hamiltonian operators; we obtain the same result as Getzler did, namely H p (K0) = 0 ∀p > 1, and explicitly compute H 0 (K0) and H 1 (K0). We then apply our main result to the classification of lower order scalar Hamiltonian operators recently obtained by

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“…While the results already obtained by the author for two-component two-dimensional brackets of hydrodynamic type [6] did not allow to conclude anything about the triviality of the dispersive deformations of the bracket, they already showed that the first cohomology group is not vanishing in the second degree, as opposite as Getzler's result. Moreover, a careful study of one-component, two-dimensional brackets [4] tells that the cohomology groups (H ≥1 ) are infinite dimensional, in particular revealing the normal form of the finite Poisson brackets of arbitrary differential order [3].…”

Section: Cohomology Of a Pva And The Theory Of Deformationsmentioning

confidence: 99%

“…We can compare it with the dimension of the cohomology groups for the scalar two-dimensional bracket {u λ u} (sc) = λ y computed in [4] for general values of (p, d). We get…”

Section: Further Remarks On H P D (P 1 )mentioning

confidence: 99%

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

Casati

2018

Theor Math Phys

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The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with twodimensional, two-components Poisson brackets of hydrodynamic type at the third differential degree. This allows us to obtain their corresponding Poisson-Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of not equivalent infinitesimal deformation, nor in the third, corresponding to the obstructions to extend such deformations.

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Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets

Cited by 11 publications

References 19 publications

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Dispersive deformations of the Hamiltonian structure of Euler’s equations

A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

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