The algebro-geometric solutions for Degasperis-Procesi hierarchy
Abstract
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.
Keywords
Cite
@article{arxiv.1204.2140,
title = {The algebro-geometric solutions for Degasperis-Procesi hierarchy},
author = {Yu Hou and Peng Zhao and Engui Fan and Zhijun Qiao},
journal= {arXiv preprint arXiv:1204.2140},
year = {2012}
}
Comments
65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004 by other authors