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Isospectral Hamiltonian flows in finite and infinite dimensions. II: Integration of flows. (English) Zbl 0717.58051
In Part I of this paper [ibid. 117, No.3, 451-500 (1988; Zbl 0659.58022)], the authors showed how isospectral Hamiltonian flows in the space of rank r perturbations, \({\mathcal M}_ A\), of an \(n\times n\) matrix A can be derived from the Adler-Kostant-Symes theorem. These flows arise through the use of a moment map from \({\mathcal M}_ A\) into the dual, (\(gl(r)^+)^*\), of the positive part of the loop algebra \(gl(r)\). Such systems were shown to be completely integrable under special assumptions on the spectrum of A and the resulting matrix polynomial L(\(\lambda\))\(\in (gl(r)^+)^*.\)
The purpose of this part II is to provide a more unified, streamlined formulation which allows A and L(\(\lambda\)) to have more general spectra. Such a generalization is necessary to be able to treat important examples of integrable systems such as the coupled non-linear Schrödinger equation (CNLS). The authors illustrate their general constructions by explicitly solving CNLS as well as the Rosochatius equation.
Reviewer: W.J.Satzer jun

37C10 Dynamics induced by flows and semiflows
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI
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