Algebro-geometric Solutions for the Derivative Burgers Hierarchy

Abstract

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi(DP)equationarecastinthesamepeakonfamily,theypossessthe 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-hyperelliptic curves lead to great difficulty in the construction of algebro- geometric solutions of the DP equation. In this paper, we study algebro-geometric solutions for the derivative Burgers (DB) equation, which is derived by Qiao and Li (2004) as a short wave model of the DP equation with the help of functional gradient and a pair of Lenard operators. Based on the characteristic polynomial of a Lax matrix for the DB equation, we introduce a third order algebraic curve Kr −1 with genus r − 1, from which the associated Baker-Akhiezer functions, meromorphic function, and Dubrovin-type equations are constructed. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker- Akhiezerfunctionsandthemeromorphicfunction.Inparticular,thealgebro-geometric solutions are obtained for all equations in the whole DB hierarchy.