On the higher-order b-family equation and Euler equations on the circle

Min Zhu

doi:10.3934/dcds.2014.34.3013

Article Contents

2014, Volume 34, Issue 7: 3013-3024. Doi: 10.3934/dcds.2014.34.3013

This issue Previous Article The Aubry-Mather theorem for driven generalized elastic chains Next Article

On the higher-order b-family equation and Euler equations on the circle

Min Zhu^1,

1.
Department of Mathematics, Nanjing Forestry University, Nanjing 210037

Received: May 31, 2013

Revised: August 31, 2013

Published: June 2014

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Abstract

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 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\ge 1. $

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Mathematics Subject Classification: Primary: 35Q35, 58D05.

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