Singular Degree of a Rational Matrix Pseudodifferential Operator

Carpentier, Sylvain; Sole, Alberto De; Kač, Victor G.

doi:10.1093/imrn/rnu093

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…The Fréchet derivative (7) is an example of a difference operator of order (p, q) and total order Ord a * = q − p. For an element a ∈ F the order and total order are defined as ord a * and Ord a * respectively.…”

Section: Note That Every Element Of K and F Depends On A Finite Numbe...mentioning

confidence: 99%

“…While difference operators act naturally on elements of the field F, rational operators cannot be a priori applied to elements of F. Similarly to the theory of rational differential operators [16] for L = AB −1 ∈ Q and a, b ∈ F we define the correspondence a = Lb when there exists c ∈ F such that a = Ac and b = Bc.…”

Section: Definition 2 Rational Pseudo-difference Operators Are Elemen...mentioning

confidence: 99%

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PreHamiltonian and Hamiltonian operators for differential-difference equations

2020

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In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo-difference Hamiltonian operator can be represented as a ratio AB −1 of two difference operators with coefficients from a difference field F where A is preHamiltonian. A difference operator A is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on F. We show that a skew-symmetric difference operator is Hamiltonian if and only if it is preHamiltonian and satisfies simply verifiable conditions on its coefficients. We show that if H is a rational Hamiltonian operator, then to find a second Hamiltonian operator K compatible with H is the same as to find a preHamiltonian pair A and B such that AB −1 H is skew-symmetric. We apply our theory to non-trivial multi-Hamiltonian structures of Narita-Itoh-Bogayavlensky and Adler-Postnikov equations.

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Section: Note That Every Element Of K and F Depends On A Finite Numbe...mentioning

confidence: 99%

Section: Definition 2 Rational Pseudo-difference Operators Are Elemen...mentioning

confidence: 99%

PreHamiltonian and Hamiltonian operators for differential-difference equations

2020

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“…Throughout this section, unless otherwise specified, we consider instead of V its field of fractions K = F rac(V). Recall that any ℓ × ℓ matrix differential operator H ∈ (Note that we can always construct a differential field extension K of K with the same field of constants such that, in this extension, dim C (Ker H) = deg(H), see [CDSK13,Lem.4.3], but, in general, this extension K will not be an algebra of differential functions.) For a more detailed exposition on the Dieudonnè determinant, see [CDSK14].…”

Section: Poisson Structuresmentioning

confidence: 99%

On integrability of some bi-Hamiltonian two field systems of partial differential equations

Sole

Kač²,

Turhan

2015

Journal of Mathematical Physics

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We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H 0 , H 1 ), where H 0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H 0 , H 1 ) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.

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“…Using the results of [CDSK13c] one can show that, in fact, the naive argument used to "prove" the association relation h n−1…”

Section: Dirac Reduction For (Non-local) Poisson Structures and Hamilmentioning

confidence: 99%