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Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds. (English) Zbl 1439.37071

The paper is focused on weakly nonlocal Poisson structures related to \((1+1)\)-dimensional Hamiltonian systems. The authors’ aim is to define the Schouten-Nijenhuis bracket for such integro-differential Poisson operators, thus extending well-known results for local Poisson structures. This allows them to prove in an alternative way the Hamiltonian property of the Poisson operators for the Krichever-Novikov equation, the modified KdV equation, and of certain first-order weakly nonlocal Poisson operators considered for the first time in [E. V. Ferapontov, Funct. Anal. Appl. 25, No. 3, 195–204 (1991, Zbl 0742.58018); translation from Funkts. Anal. Prilozh. 25, No. 3, 37–49 (1991)].
The approach used in the paper represents an alternative to those proposed in [A. De Sole and V. G. Kac, Jpn. J. Math. 8, No. 2, 233–347 (2013, Zbl 1286.37062); M. Casati et al., Three computational approaches to weakly nonlocal Poisson brackets, Preprint, arXiv:1903.08204].
By introducing extra odd (Grassmann) nonlocal variables, the weakly nonlocal Poisson operators are replaced by (super-)functions. In this way the original problem is reduced to a problem for superfunctions defined on superbundles.
The paper could be of some interest for experts in geometric theory of partial differential equations and integrable systems.

MSC:

37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
58A50 Supermanifolds and graded manifolds
46S60 Functional analysis on superspaces (supermanifolds) or graded spaces
35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:

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