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Volume 18, Issue 1, March 2011, Pages 1 - 8
The Periodic μ-b-Equation and Euler Equations on the Circle
Authors
Martin Kohlmann
Institute for Applied Mathematics, University of Hannover, D-30167 Hannover, Germany,kohlmann@ifam.uni-hannover.de
Received 9 October 2010, Accepted 21 November 2010, Available Online 7 January 2021.
- DOI
- 10.1142/S1402925111001155How to use a DOI?
- Keywords
- μ-b-equation; diffeomorphism group of the circle; metric and non-metric Euler equations
- Abstract
In this paper, we study the μ-variant of the periodic b-equation and show that this equation can be realized as a metric Euler equation on the Lie group
Diff ∞(𝕊) if and only if b = 2 (for which it becomes the μ-Camassa–Holm equation). In this case, the inertia operator generating the metric onDiff ∞(𝕊) is given by L = μ − ∂x2. In contrast, the μ-Degasperis–Procesi equation (obtained for b = 3) is not a metric Euler equation onDiff ∞(𝕊) for any regular inertia operator A ∈ ℒissym. The paper generalizes some recent results of [13, 16, 24].- Copyright
- © 2011 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
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Cite this article
TY - JOUR AU - Martin Kohlmann PY - 2021 DA - 2021/01/07 TI - The Periodic μ-b-Equation and Euler Equations on the Circle JO - Journal of Nonlinear Mathematical Physics SP - 1 EP - 8 VL - 18 IS - 1 SN - 1776-0852 UR - https://doi.org/10.1142/S1402925111001155 DO - 10.1142/S1402925111001155 ID - Kohlmann2021 ER -