Isospectral hamiltonian flows in finite and infinite dimensions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Isospectral hamiltonian flows in finite and infinite dimensions

Authors:	M R Adams J Harnad E Previato

Institution:	(1) Department of Mathematics, University of Georgia, 30602 Athens, GA, USA;(2) Département de Mathémathiques Appliquées, Ecole Polytechnique, Succ. A, C.P. 6079, H3C 3A7 Montréal, Qué, Canada;(3) Department of Mathematics, Boston University, 02215 Boston, MA, USA

Abstract:	A moment map $\tilde J_r :M_A \to (\widetilde{gl(r)}^ + )^$ is constructed from the Poisson manifold _A of rank-r perturbations of a fixedN×N matrixA to the dual $(\widetilde{gl(r)}^ + )^$ of the positive part of the formal loop algebra $\widetilde{gl(r)}$ =gl(r), ^–1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on $(\widetilde{gl(r)}^ + )^$ . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in _A. The latter may be identified with flows on finite dimensional coadjoint orbits in $(\widetilde{gl(r)}^ + )^$ and linearized on the Jacobi variety of an invariant spectral curveX _r which, generically, is anr-sheeted Riemann surface. Reductions of _A are derived, corresponding to subalgebras ofgl(r, ) andsl(r, ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of $\widetilde{sl(r,\mathbb{C}})$ . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.This research was partially supported by NSF grants MCS-8108814 (A03), DMS-8604189, and DMS-8601995

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