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Communications in Mathematical Physics

, Volume 155, Issue 2, pp 385–413 | Cite as

Darboux coordinates and Liouville-Arnold integration in loop algebras

  • M. R. Adams
  • J. Harnad
  • J. Hurtubise
Article

Abstract

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part\(\tilde{\mathfrak{g}}^+\) of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.

Keywords

Spectral Data Integration Method Matrix Representation Quantum Computing Spectral Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • M. R. Adams
    • 1
  • J. Harnad
    • 2
    • 3
  • J. Hurtubise
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of GeorgiaAthensGeorgia
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  3. 3.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada
  4. 4.Department of MathematicsMcGill UniversityMontréalCanada

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