Communications in Mathematical Physics

, Volume 134, Issue 3, pp 555–585 | Cite as

Isospectral Hamiltonian flows in finite and infinite dimensions

II. Integration of flows
  • M. R. Adams
  • J. Harnad
  • J. Hurtubise
Article

Abstract

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad*-invariant Poisson submanifolds of the dual\((\widetilde{gl}(r)^ + )*\) of the positive part of the loop algebra\(\widetilde{gl}(r)\) is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleTP1(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map\(\tilde J_r :M_{N,r} \times M_{N,r} \to (\widetilde{gl}(r)^ + )*\) from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles\(E_t \to \tilde S\) over a suitably desingularized curve\(\tilde S\) may be used to determine both the flow of matricial polynomialsL(λ) and the Hamiltonian flow in the spaceMN,r×MN,r in terms of θ-functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles\(E_t \to \tilde S\) under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrödinger) equation.

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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • M. R. Adams
    • 1
  • J. Harnad
    • 2
    • 3
  • J. Hurtubise
    • 4
  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada
  3. 3.Department of MathematicsConcordia UniversityMontréalCanada
  4. 4.Department of MathematicsMcGill UniversityMontréalCanada

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