Algebro-geometric solutions for the two-component Camassa–Holm Dym hierarchy

• Two-component Camassa–Holm Dym hierarchy is constructed by polynomial recursive formalism.
• The associated hyperelliptic curve for the Camassa–Holm Dym hierarchy is given.
• Fritz’s method is used to construct algebro-geometric solutions of the Camassa–Holm Dym hierarchy.

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa–Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker–Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire CHD2 hierarchy.