Elliptic spectral parameter and infinite dimensional Grassmann variety

Kanehisa Takasaki

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arXiv · nlin/0312016

Elliptic spectral parameter and infinite dimensional Grassmann variety

Kanehisa Takasaki

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Abstract

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 Schrödinger 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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Kanehisa Takasaki. 2004-08-18. Elliptic spectral parameter and infinite dimensional Grassmann variety. https://arxiv.org/abs/nlin/0312016

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