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On the higher-order b-family equation and Euler equations on the circle. (English) Zbl 1292.35249

Summary: Considered herein is a geometric investigation on the higher-order b-family equation describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. It is shown that the higher-order \(b\)-family equation can only be realized as an Euler equation on the Lie group \(\mathrm{Diff}(\mathbb{S}^1)\) of all smooth and orientation preserving diffeomorphisms on the circle if the parameter \(b=2\) which corresponds to the higher-order Camassa-Holm equation with the metric \(H^k\), \(k \geq 1\).

MSC:

35Q35 PDEs in connection with fluid mechanics
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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