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 $n$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.

35 pages. arXiv admin note: substantial text overlap with arXiv:nlin/0208021

References in corpus (5)

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• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
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Cited by in corpus (4)

• Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow
• Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches
• The Classical Moment Problem and Generalized Indefinite Strings
• The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation