Isospectral Hamiltonian flows in finite and infinite dimensions |
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Authors: | M R Adams J Harnad J Hurtubise |
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Institution: | (1) Department of Mathematics, University of Georgia, 30602 Athens, GA, USA;(2) Centre de Recherches Mathématiques, Université de Montréal, C. P. 6128-A, H3C 3J7 Montréal, Qué, Canada;(3) Department of Mathematics, Concordia University, Montréal, Qué, Canada;(4) Department of Mathematics, McGill University, H3A 2K6 Montréal, Qué, Canada |
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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
of the positive part of the loop algebra
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 bundleT P
1(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map
from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles
over a suitably desingularized curve
may be used to determine both the flow of matricial polynomialsL( ) and the Hamiltonian flow in the spaceM
N,r×MN,r in terms of -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
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.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army grant DAA L03-87-K-0110 |
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