Article

**Ann. Inst. Fourier. 67, 1185-1230 (2017)**[DOI: 10.5802/aif.3107]

## Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa-Holm Hierarchy

### Jonathan Eckhardt, Fritz Gesztesy, Helge Holden, Aleksey Kostenko, and Gerald Teschl

We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy
employing a new zero-curvature formalism and identify and
describe in detail the isospectral set associated to all real-valued, smooth, and bounded
algebro-geometric solutions of the

*n*th equation of the stationary CH-2 hierarchy as the real*n*-dimensional torus*𝕋*. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint singular Hamiltonian systems. In particular, we employ Weyl-Titchmarsh theory for singular (canonical) Hamiltonian systems.^{n}While we focus primarily on the case of stationary algebro-geometric CH-2 solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.

** MSC91:** Primary 35Q51, 35Q53, 37K15; Secondary 37K10, 37K20.

**Keywords:** *Two-component Camassa-Holm hierarchy, real-valued algebro-geometric solutions, isospectral tori, self-adjoint Hamiltonian systems, Weyl-Titchmarsh theory.*

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