Multi-Hamiltonian Structure for the Finite Defocusing Ablowitz-Ladik Equation

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 Caratheodory 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. (C) 2008 Wiley Periodicals, Inc.