no. 3
Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy
[Solutions algebro-géométriques à valeurs réelles de la hierarchie de Camassa–Holm à deux composantes]
Eckhardt, Jonathan1 ; Gesztesy, Fritz2 ; Holden, Helge3 ; Kostenko, Aleksey1 ; Teschl, Gerald1, 4
1 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Wien (Austria)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematical Sciences Norwegian University of Science and Technology NO–7491 Trondheim (Norway)
4 International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien (Austria)
Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1185-1230.

Nous présentons une construction de la hiérarchie de l’équation de Camassa–Holm à deux composantes (CH-2) en utilisant un nouveau formalisme de courbure nulle. Nous décrivons en détail et identifions l’ensemble isospectral associé à toutes les solutions algébro-géométriques à valeur réelle, réguliéres et bornées de la n-ème équation de l’équation stationnaire de la hiérarchie CH-2 au tore 𝕋 n de dimension n. Nous utilisons des équations de type Dubrovin pour les diviseurs auxiliaires et certains aspects de la théorie spectrale et d’inversion spectrale pour les systèmes Hamiltoniens singuliers auto-adjoints. En particulier, nous utilisons la théorie de Weyl–Titchmarsh pour les systèmes (canoniques) Hamiltoniens singuliers.

Bien que nous nous concentrons principalement sur le cas des solutions algébro-géométriques stationnaires pour CH-2, nous remarquons que le cas de la solution évolutive qui dépend du temps est subordonné au cas stationnaire en ce qui concernent les questions isospectrales liées au tore.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3107
Classification : 35Q51, 35Q53, 37K15, 37K10, 37K20
Keywords: Two-component Camassa–Holm hierarchy, real-valued algebro-geometric solutions, isospectral tori, self-adjoint Hamiltonian systems, Weyl–Titchmarsh theory
Mot clés : Hiérarchie de Camassa–Holm à deux composantes, solutions algebro-géométriques à valeurs réelles, théorie isospectrale, systèmes hamiltoniens auto-adjoints, théorie de Weyl–Titchmarsh
Eckhardt, Jonathan 1 ; Gesztesy, Fritz 2 ; Holden, Helge 3 ; Kostenko, Aleksey 1 ; Teschl, Gerald 1, 4

1 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Wien (Austria)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematical Sciences Norwegian University of Science and Technology NO–7491 Trondheim (Norway)
4 International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien (Austria)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de l'Institut Fourier},
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Eckhardt, Jonathan; Gesztesy, Fritz; Holden, Helge; Kostenko, Aleksey; Teschl, Gerald. Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy. Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1185-1230. doi : 10.5802/aif.3107. https://aif.centre-mersenne.org/articles/10.5802/aif.3107/

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