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Multi‐Hamiltonian structure for the finite defocusing Ablowitz‐Ladik equation
Author(s) -
Gekhtman Michael,
Nenciu Irina
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20255
Subject(s) - mathematics , poisson bracket , symplectic geometry , algebraic structure , poisson distribution , hamiltonian (control theory) , poisson manifold , algebraic number , pure mathematics , mathematical analysis , integrable system , lie algebra , mathematical optimization , statistics
We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc.