55 research outputs found
The algebro-geometric solutions for Degasperis-Procesi hierarchy
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 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.Comment: 65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004
by other author
Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy
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 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 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
Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model
This paper devotes to present analysiswork on the fifth order Camassa-Holm (FOCH) modelwhich recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data
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