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


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 ∞.


Spectral Data Integration Method Matrix Representation Quantum Computing Spectral Parameter 
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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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