The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus

Details The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus

2011-07-20

arxiv.org/abs/1107.3998v2

J. Phys. A: Math. Theor. 44 (2011) 465205

Mathematics

Analysis of PDEs

21 pages. arXiv admin note: text overlap with arXiv:1107.5404

Scientific paper

10.1088/1751-8113/44/46/465205

In this paper, the two-dimensional periodic $b$-equation is discussed under geometric aspects, i.e., as a geodesic flow on the diffeomorphism group of the torus $\T=S^1\times S^1$. In the framework of Arnold's [V.I. Arnold, Sur la g\'eom\'etrie diff\'erentielle des groupes de Lie de dimension infinie et ses applications \`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361] famous approach, we achieve some well-posedness results for the $b$-equation and we perform explicit curvature computations for the 2D Camassa-Holm equation, which is obtained for $b=2$. Finally, we explain the special role of the choice $b=2$ by giving a rigorous proof that $b=2$ is the only case in which the associated geodesic flow is weakly Riemannian.

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