2003/07/31 by Kanehisa Takasaki
Physics and Astronomy · Mathematics · #nlin.SI #hep-th #math-ph #math.MP #math.QA
published as J. Math. Sci. Univ. Tokyo 11 (2004), 91-131 · latex2e, 36 pp, no figure; (v2) minor changes, mostly typos; (v3) Title changed, text fully revised with new results; (v4) serious errors in section 5 corrected; (v5) proof of main results is improved; (v6) minor change in proof of Lemma 10 etc; (v7) final version for publication; (v8) typos corrected. Journal of Mathematical Sciences, University of Tokyo (to appear)
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 × 2 matrices An(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 An(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.