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

Abstract

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 nth equation of the stationary CH-2 hierarchy as the real n-dimensional torus T n. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for selfadjoint 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.