Revista Matemática Iberoamericana


Full-Text PDF (479 KB) | Table of Contents | RMI summary
Volume 19, Issue 1, 2003, pp. 73–142
DOI: 10.4171/RMI/339

Algebro-Geometric Solutions of the Camassa–Holm hierarchy

Fritz Gesztesy[1] and Helge Holden[2]

(1) Mathematics Department, University of Missouri, 202 Mathematical Sciences Bldg, MO 65211, COLUMBIA, UNITED STATES
(2) Department of Mathematical Sciences, University of Trondheim, Alfred Getz vei 1, NO-7491, TRONDHEIM, NORWAY

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial value problem, and derive the theta function representations of algebro-geometric solutions of the CH hierarchy.

Keywords: Camassa–Holm hierarchy, algebro-geometric solutions, Dubrovin equations, trace formulas

Gesztesy F, Holden H. Algebro-Geometric Solutions of the Camassa–Holm hierarchy. Rev. Mat. Iberoamericana 19 (2003), 73-142. doi: 10.4171/RMI/339