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On the local systems Hamiltonian in the weakly non-local Poisson brackets. (English) Zbl 0991.37041

The authors explain how weakly nonlocal Poisson brackets (PB) and symplectic structures appear from the theory of the famous completely integrable soliton systems like KdV and NLS. They show that all higher PB are weakly nonlocal for \(n\geq 0\). Moreover they prove that all higher symplectic structures are weakly nonlocal for \(n\leq 0\). The authors also deal with weakly nonlocal PB of hydrodynamic type associated with Riemannian geometry. They find the canonical forms, Casimirs and Hamiltonians for the structure flows.

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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