A note on multi-dimensional Camassa-Holm type systems on the torus
| Autor: | Kohlmann, Martin |
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| Rok vydání: | 2011 |
| Předmět: | |
| Zdroj: | J. Phys. A: Math. Theor. 45 (2012) 125205 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8113/45/12/125205 |
| Popis: | We present a $2n$-component nonlinear evolutionary PDE which includes the $n$-dimensional versions of the Camassa-Holm and the Hunter-Saxton systems as well as their partially averaged variations. Our goal is to apply 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], [D.G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92(2) (1970) 102-163] geometric formalism to this general equation in order to obtain results on well-posedness, conservation laws or stability of its solutions. Following the line of arguments of the paper [M. Kohlmann, The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus. J. Phys. A.: Math. Theor. 44 (2011) 465205 (17 pp.)] we present geometric aspects of a two-dimensional periodic $\mu$-$b$-equation on the diffeomorphism group of the torus in this context. Comment: 14 pages |
| Databáze: | arXiv |
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