The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
Kouranbaeva, Shinar
arXiv, 9807021 / Harvested from arXiv
Text on arXiv
Misiolek has shown that the Camassa-Holm (CH) equation is a geodesic flow on the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm equation for the case $\kappa =0$ is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the $H^1$ inner product over the entire group. This paper uses the right-trivialisation technique to rigorously verify that the Euler-Poincar\'{e} theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds (see Refs. \cite{HMR} and \cite{SH}).
Publié le : 1998-07-21
Classification:  Mathematical Physics,  Mathematics - Group Theory,  58B99,  57R57 
@article{9807021,
     author = {Kouranbaeva, Shinar},
     title = {The Camassa-Holm equation as a geodesic flow on the diffeomorphism group},
     journal = {arXiv},
     volume = {1998},
     number = {0},
     year = {1998},
     language = {en},
     url = {http://dml.mathdoc.fr/item/9807021}
}

Kouranbaeva, Shinar. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. arXiv, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/9807021/