, Volume 134, Issue 3, pp 555-585

Isospectral Hamiltonian flows in finite and infinite dimensions

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 bundleTP 1(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 spaceM N,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.

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
Communicated by A. Jaffe