A bi-Hamiltonian structure for the integrable, discrete non-linear Schrödinger system

This paper shows that the AL (Ablowitz-Ladik) hierarchy of (integrable) equations can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J, and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R = K J^{- 1}. In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations.