Algebro-geometric Solutions for the Degasperis-Procesi Hierarchy
Algebro-geometric Solutions for the Degasperis-Procesi Hierarchy
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DOI：
10.1137/12089689x
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发表时间：
2012-04
期刊：
影响因子：
--
通讯作者：
Yuanyuan Hou;Peng Zhao;E. Fan;Z. Qiao
中科院分区：
文献类型：
--
作者：
Yuanyuan Hou;Peng Zhao;E. Fan;Z. Qiao
Though the 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 $\mathcal{K}_{r-2}$ with genus $r-2$, 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-geom...
