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

Hou, Yaxin; Fan, Engui

doi:10.1016/j.chaos.2014.06.009

Cited by 4 publications

(10 citation statements)

References 24 publications

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“…Various multicomponent extensions of the Camassa-Holm equation and its generalizations are discussed in, e.g., [6], [7], [19], [38], [40], [46], [48], [52]- [54]. Closest to the investigations in this paper is the derivation of the CH-2 hierarchy and its algebro-geometric solutions in [35].…”

Section: Introductionmentioning

confidence: 94%

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

Eckhardt¹,

Gesztesy²,

Holden³

et al. 2017

Ann. inst. Fourier

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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 Dubrovintype 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. Contents 1. Introduction 1 2. The CH-2 Hierarchy, Recursion Relations, and Hyperelliptic Curves 4 3. The Stationary Algebro-Geometric CH-2 Formalism 8 4. Basic Facts on Self-Adjoint Hamiltonian Systems 16 5. Real-Valued Algebro-Geometric CH-2 Solutions and the Associated Isospectral Torus 19 References 32

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Section: Introductionmentioning

confidence: 94%

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

Eckhardt¹,

Gesztesy²,

Holden³

et al. 2017

Ann. inst. Fourier

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“…The representation (4.35) for u on Ω follows from trace formula (3.46) and the expression (F.88 [15]) for n j=0 µ j . To prove the constraint (4.36), one can refs Theorem 4.5 in our latest paper [18]. Equations (4.37) is clear from (4.15).…”

Section: Stationary Algebro-geometric Solutions Of Hs Hierarchymentioning

confidence: 99%

“…[12]- [15]. Recently, we investigated algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy, the Degasperis-Procesi hierarchy and the modified Camassa-Holm hierarchy [16]- [18].…”

Section: Introductionmentioning

confidence: 99%

Algebro-geometric solutions for the Hunter–Saxton hierarchy

Hou

Fan

Zhao

2013

Z. Angew. Math. Phys.

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This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the Hunter-Saxton (HS) hierarchy through studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the HS hierarchy, 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 Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire HS hierarchy.

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“…2,5,6 A systematic approach, proposed by Gesztesy and Holden 7-10 to construct algebrogeometric solutions for integrable equations, has been extended to the whole (1+1) dimensional integrable hierarchy, such as the AKNS hierarchy, the Camassa-Holm hierarchy, and so on. Recently, Fan and his co-authors [11][12][13][14][15][16] investigated the Hunter-Saxton hierarchy, the modified Camassa-Holm hierarchy, the Degasperis-Procesi hierarchy, etc., and obtained their algebro-geometric solutions.…”

Section: Introductionmentioning

confidence: 99%

Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy

Yue

Xia

2014

Journal of Mathematical Physics

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This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the complex Sharma-Tasso-Olver (CSTO) hierarchy. Our main tools include the polynomial recursive formalism to derive the CSTO hierarchy, the hyper-elliptic 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. By use of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire CSTO hierarchy. C 2014 AIP Publishing LLC. [http://dx.

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

Cited by 4 publications

References 24 publications

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy

Algebro-geometric solutions for the Hunter–Saxton hierarchy

Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy

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