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 ]
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 $n$th 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 : 2016-03-01
Accepté le : 2016-08-11
Publié le : 2017-05-30
DOI : https://doi.org/10.5802/aif.3107
Classification : 35Q51,  35Q53,  37K15,  37K10,  37K20
Mots clés : Hiérarchie de Camassa–Holm à deux composantes, solutions algebro-géométriques à valeurs réelles, théorie isospectrale, systèmes hamiltoniens auto-adjoints, théorie de Weyl–Titchmarsh
@article{AIF_2017__67_3_1185_0,
author = {Eckhardt, Jonathan and Gesztesy, Fritz and Holden, Helge and Kostenko, Aleksey and Teschl, Gerald},
title = {Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa--Holm Hierarchy},
journal = {Annales de l'Institut Fourier},
pages = {1185--1230},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {3},
year = {2017},
doi = {10.5802/aif.3107},
language = {en},
url = {archive.numdam.org/item/AIF_2017__67_3_1185_0/}
}
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. http://archive.numdam.org/item/AIF_2017__67_3_1185_0/

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