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

TL;DR

This paper constructs algebro-geometric solutions for the two-component Camassa-Holm hierarchy using a new zero-curvature approach, characterizing the isospectral set as a real n-dimensional torus and employing spectral theory techniques.

Contribution

It introduces a novel zero-curvature formalism for the CH-2 hierarchy and describes the isospectral set of solutions as a real torus, advancing spectral analysis methods.

Findings

Characterization of the isospectral set as a real torus

Construction of solutions via zero-curvature formalism

Application of Weyl-Titchmarsh theory to singular Hamiltonian systems

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 $n$th equation of the stationary CH-2 hierarchy as the real $n$-dimensional torus $\mathbb{T}^n$. 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.