The algebro-geometric solutions for Degasperis-Procesi hierarchy

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 $\mathcal{K}_{r-2}$ 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.