Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy

Authors

Yu Hou
Peng Zhao
Engui Fan
Zhijun Qiao, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

5-7-2013

Abstract

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-geometric solutions are obtained for all equations in the whole DP hierarchy.

Comments

© 2013, Society for Industrial and Applied Mathematics
Read More: https://locus.siam.org/doi/abs/10.1137/12089689X?journalCode=sjmaah

Recommended Citation

Hou, Yu, Peng Zhao, Engui Fan, and Zhijun Qiao. 2013. “Algebro-Geometric Solutions for the Degasperis--Procesi Hierarchy.” SIAM Journal on Mathematical Analysis 45 (3): 1216–66. https://doi.org/10.1137/12089689X.

Publication Title

SIAM Journal on Mathematical Analysis

DOI

10.1137/12089689X