Elliptic spectral parameter and infinite dimensional Grassmann variety

Details Elliptic spectral parameter and infinite dimensional Grassmann variety Elliptic spectral parameter and infinite dimensional Grassmann variety

2003-12-08

arxiv.org/abs/nlin/0312016v3

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Contribution to Faro conference "Infinite dimensional algebras and quantum integrable systems", latex2e, usepackage amssymb, 3

Scientific paper

Recent results on the Grassmannian perspective of soliton equations with an elliptic spectral parameter are presented along with a detailed review of the classical case with a rational spectral parameter. The nonlinear Schr\"odinger hierarchy is picked out for illustration of the classical case. This system is formulated as a dynamical system on a Lie group of Laurent series with factorization structure. The factorization structure induces a mapping to an infinite dimensional Grassmann variety. The dynamical system on the Lie group is thereby mapped to a simple dynamical system on a subset of the Grassmann variety. Upon suitable modification, almost the same procedure turns out to work for soliton equations with an elliptic spectral parameters. A clue is the geometry of holomorphic vector bundles over the elliptic curve hidden (or manifest) in the zero-curvature representation.

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