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29 March 2024
 
  » arxiv » math.SG/0504348

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A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System
Nicholas M. Ercolani ; Guadalupe I. Lozano ;
Date 17 Apr 2005
Subject Symplectic Geometry; Dynamical Systems MSC-class: 37K10, 37K60, 37K25, 53D17 | math.SG math.DS
AffiliationDept. of Mathematics, University of Arizona, Tucson, AZ, USA), Guadalupe I. Lozano (Dept. of Mathematics, University of Michigan, Ann Arbor, MI, USA
AbstractThis paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) 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 (obtained by composing K and the inverse of J.) 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.
Source arXiv, math.SG/0504348
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