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

Nicholas M. Ercolani, Guadalupe I. Lozano

Research output: Contribution to journalArticle

10 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)105-121
Number of pages17
JournalPhysica D: Nonlinear Phenomena
Volume218
Issue number2
DOIs
StatePublished - Jun 15 2006

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Keywords

  • Bi-Hamiltonian structures
  • Discrete integrable equations
  • Inverse scattering
  • Lattice dynamics
  • Poisson geometry

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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