TY - JOUR

T1 - A bi-Hamiltonian structure for the integrable, discrete non-linear Schrödinger system

AU - Ercolani, Nicholas M.

AU - Lozano, Guadalupe I.

N1 - Funding Information:
G.I. Lozano would like to thank Hermann Flaschka for useful discussions and feedback. N.M. Ercolani and G.I. Lozano were supported in part by NSF grant no. 0073087.

PY - 2006/6/15

Y1 - 2006/6/15

N2 - 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.

AB - 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.

KW - Bi-Hamiltonian structures

KW - Discrete integrable equations

KW - Inverse scattering

KW - Lattice dynamics

KW - Poisson geometry

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U2 - 10.1016/j.physd.2006.04.014

DO - 10.1016/j.physd.2006.04.014

M3 - Article

AN - SCOPUS:33745433249

VL - 218

SP - 105

EP - 121

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 2

ER -