18,492 research outputs found
Elliptic Algebra and Integrable Models for Solitons on Noncummutative Torus
We study the algebra and the basis of the Hilbert space in terms of the functions of the positions of solitons. Then
we embed the Heisenberg group as the quantum operator factors in the
representation of the transfer matrice of various integrable models. Finally we
generalize our result to the generic case.Comment: Talk given by Bo-Yu Hou at the Joint APCTP-Nankai Symposium. Tianjin
(PRC), Oct. 2001. To appear in the proceedings, to be published by Int. J.
Mod. Phys. B. 7 pages, latex, no figure
The algebro-geometric solutions for Degasperis-Procesi hierarchy
Though completely integrable Camassa-Holm (CH) equation and
Degasperis-Procesi (DP) equation are cast in the same peakon family, they
possess the second- and third-order Lax operators, respectively. From the
viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic
and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great
difficulty in the construction of algebro-geometric solutions of the DP
equation. In this paper, we derive the DP hierarchy with the help of Lenard
recursion operators. Based on the characteristic polynomial of a Lax matrix for
the DP hierarchy, we introduce a third order algebraic curve
with genus , from which the associated Baker-Akhiezer
functions, meromorphic function and Dubrovin-type equations are established.
Furthermore, the theory of algebraic curve is applied to derive explicit
representations of the theta function for the Baker-Akhiezer functions and the
meromorphic function. In particular, the algebro-geometric solutions are
obtained for all equations in the whole DP hierarchy.Comment: 65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004
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