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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 | Affiliation: | Dept. of Mathematics, University of Arizona, Tucson, AZ, USA), Guadalupe I. Lozano (Dept. of Mathematics, University of Michigan, Ann Arbor, MI, USA | Abstract: | This 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 | Services: | Forum | Review | PDF | Favorites |
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