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

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


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