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Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy | | Kanehisa Takasaki
; | | Date: |
18 Jul 2003 | | Journal: | J. Math. Sci. Univ. Tokyo 11 (2004), 91-131 | | Subject: | Exactly Solvable and Integrable Systems; Quantum Algebra; Mathematical Physics | nlin.SI hep-th math-ph math.MP math.QA | | Abstract: | Two "elliptic analogues’’ of the nonlinear Schrödinger 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 $2 imes 2$ 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 Schrödinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety. | | Source: | arXiv, nlin.SI/0307030 | | Services: | Forum | Review | PDF | Favorites | |
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