Revista Matemática Iberoamericana

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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) Department of Mathematics, Baylor University, Sid Richardson 305I, One Bear Place #97328, TX 76798-7328, Waco, USA
(2) Department of Mathematical Sciences, University of Trondheim, Alfred Getz vei 1, 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 Fritz, Holden Helge: Algebro-Geometric Solutions of the Camassa–Holm hierarchy. Rev. Mat. Iberoamericana 19 (2003), 73-142. doi: 10.4171/RMI/339