an:04182456
Zbl 0717.58051
Adams, M. R.; Harnad, J.; Hurtubise, J.
Isospectral Hamiltonian flows in finite and infinite dimensions. II: Integration of flows
EN
Commun. Math. Phys. 134, No. 3, 555-585 (1990).
00156265
1990
j
37C10 37J35 37K10 35Q55
isospectral flows; complete integrability; coupled non-linear Schr??dinger equation
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 Schr??dinger equation (CNLS). The authors illustrate their general constructions by explicitly solving CNLS as well as the Rosochatius equation.
W.J.Satzer jun
Zbl 0659.58022