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 𝕋n
. 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.
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.