Journal
ANNALES DE L INSTITUT FOURIER
Volume 67, Issue 3, Pages 1185-1230Publisher
ANNALES INST FOURIER
Keywords
Two-component Camassa-Holm hierarchy; real-valued algebro-geometric solutions; isospectral tori; self-adjoint Hamiltonian systems; Weyl-Titchmarsh theory
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Funding
- Research Council of Norway
- Austrian Science Fund (FWF) [J3455, P26060]
- Austrian Science Fund (FWF) [J3455, P26060] Funding Source: Austrian Science Fund (FWF)
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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 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.
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