Hamiltonian theory over noncommutative rings and integrability in multidimensions

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  • I. Ya. Dorfman
    Institute of Chemical Physics, Academy of Sciences of the USSR, ul. Kosygina 4, Moscow, 117977, Russia
  • A. S. Fokas
    Department of Mathematics and Computer Science and Institute of Nonlinear Studies, Clarkson University, Potsdam, New York 13699-5815

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<jats:p>A generalization of the Adler–Gel’fand–Dikii scheme is used to generate bi-Hamiltonian structures in two spatial dimensions. In order to implement this scheme, a Hamiltonian theory is built over a noncommutative ring, namely the ring of formal pseudodifferential operators. Bi-Hamiltonian structures generated in this way can be used for the Kadomtsev–Petviashvili equation as well as other integrable equations in 2+1.</jats:p>

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