The algebro-geometric solutions for Degasperis-Procesi hierarchy
Yu Hou, Peng Zhao, Engui Fan, Zhijun Qiao

TL;DR
This paper develops algebro-geometric solutions for the entire Degasperis-Procesi hierarchy using algebraic curves and Baker-Akhiezer functions, overcoming previous difficulties related to non-hyper-elliptic curves.
Contribution
It introduces a third order algebraic curve and explicit theta function representations to construct solutions for all DP hierarchy equations.
Findings
Explicit algebro-geometric solutions for the DP hierarchy derived.
Construction of Baker-Akhiezer functions and meromorphic functions.
Application of algebraic curve theory to integrable systems.
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…
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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic structures and combinatorial models · Nonlinear Photonic Systems
