12,011 research outputs found
Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy
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 , 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 matrices , , of elliptic
functions. In addition to a fixed pole at , these matrices have several
extra poles. Tyurin parameters consist of the coordinates of those poles and
some additional parameters that describe the structure of '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.Comment: 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
KP and Toda tau functions in Bethe ansatz
Recent work of Foda and his group on a connection between classical
integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum
integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin
1/2, the phase model on a finite chain, etc.) is reviewed. Some additional
information on this issue is also presented.Comment: latex2e, using ws-procs9x6 package, 19 pages, contribution to the
festschrift volume for the 60th anniversary of Tetsuji Miw
Area-Preserving Diffeomorphisms and Nonlinear Integrable Systems
Present state of the study of nonlinear ``integrable" systems related to the
group of area-preserving diffeomorphisms on various surfaces is overviewed.
Roles of area-preserving diffeomorphisms in 4-d self-dual gravity are reviewed.
Recent progress in new members of this family, the SDiff(2) KP and Toda
hierarchies, is reported. The group of area-preserving diffeomorphisms on a
cylinder plays a key role just as the infinite matrix group GL() does
in the ordinary KP and Toda lattice hierarchies. The notion of tau functions is
also shown to persist in these hierarchies, and gives rise to a central
extension of the corresponding Lie algebra.Comment: 16 page
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