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.
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Fritz Gesztesy, Helge Holden, Algebro-Geometric Solutions of the Camassa–Holm hierarchy. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 73–142DOI 10.4171/RMI/339