Journal of Nonlinear Science

, Volume 25, Issue 1, pp 1–35 | Cite as

Algebro-geometric Solutions for the Derivative Burgers Hierarchy

Article
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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-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 \(\mathcal {K}_{r-1}\) 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–Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DB hierarchy.

Keywords

Algebro-geometric solutions The derivative Burgers hierarchy The third order algebraic curve Baker–Akhiezer functions 

Mathematics Subject Classification

37K10 37K15 70H16 14F43 
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Notes

Acknowledgments

The authors would like to express their sincerest thanks to referees and editors for constructive suggestions and helpful comments. This work was partially supported by grants from the National Science Foundation of China (Project Nos. 10971031; 11271079; 11075055; 11171295), Doctoral Programs Foundation of the Ministry of Education of China, and the Shanghai Shuguang Tracking Project (Project 08GG01). Hou and Qiao was partially supported by the Norman Hackerman Advanced Research Program under Grant Number 003599-0001-2009.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical Science, Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear ScienceFudan UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of Texas-Pan AmericanEdinburgUSA
  3. 3.School of Mathematical SciencesZhaoqing UniversityZhaoqingPeople’s Republic of China

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