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Homogeneous Hamiltonian operators and the theory of coverings. (English) Zbl 07341144

Summary: A new method (by Kersten, Krasil’shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps conserved quantities into symmetries of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin-Novikov homogeneous Hamiltonian operator the method yields conditions on the operator and the system that have interesting differential and projective geometric interpretations.

MSC:

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.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
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References:

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