Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 2, Page 91--131.

Takasaki, Kanehisa
Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy
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Two ``elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition to the usual fields $u,v$, these elliptic analogues have new dynamical variables called ``Tyurin parameters,'' which are connected with a family of vector bundles over the elliptic curve in consideration. The zero-curvature equations of these systems are formulated by a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,\ldots$, of elliptic functions. In addition to a fixed pole at $z = 0$, these matrices have several extra poles. Tyurin parameters consist of the coordinates of those poles and some additional parameters that describe the structure of $A_n(z)$'s. Two distinct solutions of the auxiliary linear equations are constructed, and shown to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert pair is used to define a mapping to an infinite dimensional Grassmann variety. The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety.

Keywords: soliton equation, elliptic curve, holomorphic bundle, Grassmann variety

Mathematics Subject Classification (2000): 35Q58, 37K10, 58F07.
Mathematical Reviews Number: MR2081422

Received: 2003-12-18