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The Schouten Bracket and Hamiltonian operators
I. M. Gel'fand,I. Ya. Dorfman +1 more
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This article is published in Functional Analysis and Its Applications.The article was published on 1980-07-01. It has received 151 citations till now. The article focuses on the topics: Hamiltonian (quantum mechanics).read more
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Lie algebras and equations of Korteweg-de Vries type
TL;DR: A survey of the theory of Kats-Moody algebras is given in this paper, which contains a description of the connection between the infinite-dimensional Lie algebra of Kats and systems of differential equations generalizing the Korteweg-de Vries and sine-Gordon equations and integrable by the inverse scattering problem.
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Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation
TL;DR: In this paper, the Camassa-Holm equation is shown to be a different-factorization equation of the KdV, it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations.
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Modifying Lax equations and the second Hamiltonian structure
TL;DR: In this article, the authors define a differential operator whose coefficients are differential polynomials in u and its x-derivatives u (s~), where p+ is a P+ operator whose coefficient is a polynomial in u. (The subscript + may be ignored at this point: we introduce it so as not to conflict with the notation in the main body of the paper.)
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Recursion Operators and Bi-Hamiltonian Structures in Multidimensions. II
TL;DR: In this article, the authors present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions, and show that general classes of equations, which include the Kadomtsev-Petviashvili and the Davey-Stewartson equations, possess infinitely many commuting symmetries and infinitely many constants of motion under two distinct Poisson brackets.
References
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On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-devries type equations
TL;DR: In this article, the Lie geometric structure behind the Hamiltonian structure of the Korteweg deVries type equations was studied and the authors showed that it is the same as the Lie geometry behind the Lie structure of a Hamiltonian lattice.
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Some Remarks on the Nijenhuis Tensor
TL;DR: The Poincare lemma of as mentioned in this paper states that a differential form α of degree r on an n-manifold is exact if there exists a form β of degree β − 1 such that α = dβ and is closed if dα = 0.