Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy

Details Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy

2003-07-18

arxiv.org/abs/nlin/0307030v8

J. Math. Sci. Univ. Tokyo 11 (2004), 91-131

Nonlinear Sciences

Exactly Solvable and Integrable Systems

latex2e, 36 pp, no figure; (v2) minor changes, mostly typos; (v3) Title changed, text fully revised with new results; (v4) ser

Scientific paper

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 $A_n(z)$, $n = 1,2,...$, 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.

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