Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy (Q1687892)

The purpose of this paper is to study the two-component Camassa-Holm (CH-2) hierarchy. First, the basic polynomial recursion formalism using a new zero-curvature matrix approach is recalled. This polynomial defines the hyperelliptic curve \(\mathcal K_n\) underlying the stationary CH-2 hierarchy. Then \(\mathcal K_n\) in polynomial form and an associated meromorphic function are analyzed. It is shown that the zeros of a certain polynomial satisfies a first-order system of differential equations, and a connection to the stationary Dubrovin equation is found. The Weyl-Titschmarsh theory for singular Hamiltonian systems is summarized, and finally, the real-valued algebro-geometric solutions of the CH-2 hierarchy associated with \(\mathcal K_n\) are studied by using a Nevanlinna-Herglotz function and some Lagrange-type interpolation formulas. By the polynomial definition as multiple sum, this theory has certain similarities to a Sato theory.