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

Abstract

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.