Recursion operators and bi-Hamiltonian structures in multidimensions. I
type: Item
instance of: scholarly article
title: Recursion operators and bi-Hamiltonian structures in multidimensions. I
zbMATH Open document ID: 0674.35074
DOI: 10.1007/BF01218017
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: In a previous paper [Stud. Appl. Math. 75, 179-185 (1986; Zbl 0613.35073)] the authors constructed surprisingly a recursion operator for a famous \(2+1\) (i.e., 2 spatially and one temporally) dimensional hierarchy of nonlinear evolution equations. In the present paper under review the authors develop a general theory for the algebraic structure of recursion operators in \(2+1\) dimensions. A number of important notions, especially the hereditary symmetry and the Hamiltonian pair, that were developed thus far for \(1+1\) dimensions, are successfully extended to the \(2+1\) dimensions. As illustrative examples the authors study the hierarchies related to the following two linear eigenvalue problems: \[ w_{xx}+q(x,y)w+\alpha w_ y=0, \] where \(\alpha\) is a constant, and \(W_ x=JW_ x+QW\) where \(J=diag(\alpha,-\alpha),\) Q is a \(2\times 2\) off-diagonal matrix containing the potentials \(q_ 1(x,y)\) and \(q_ 2(x,y)\).
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 4103620
zbMATH Keywords: recursion operator
zbMATH Keywords: hierarchy of nonlinear evolution equations
zbMATH Keywords: algebraic structure
zbMATH Keywords: hereditary symmetry
zbMATH Keywords: Hamiltonian pair
author: P. M. Santini
author: A. S. Fokas
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01218017
OpenAlex ID: W4214735065
description: scientific article; zbMATH DE number 4103620