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Recursion operators and bi-Hamiltonian structures in multidimensions. I

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Abstract

The algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied. In particular, the factorization of certain operators, called recursion operators, establishes the bi-Hamiltonian nature of all these equations. Recently, we have presented the recursion operator and the bi-Hamiltonian formulation of the Kadomtsev-Petviashvili equation, a two spatial dimensional analogue of the Korteweg-deVries equation. Here we present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions. As an application we 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 in involution under two distinct Poisson brackets. Furthermore, we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems.

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  1. P. M. Santini

    Present address: Dipartimento di Fisica, Università di Roma, La Sapienza, I-00185, Roma, Italy

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  1. Department of Mathematics and Computer Science and Institute for Nonlinear Studies, Clarkson University, 13676, Potsdam, NY, USA

    P. M. Santini & A. S. Fokas

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  1. P. M. Santini
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Communicated by A. Jaffe

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Santini, P.M., Fokas, A.S. Recursion operators and bi-Hamiltonian structures in multidimensions. I. Commun.Math. Phys. 115, 375–419 (1988). https://doi.org/10.1007/BF01218017

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