Journal of Nonlinear Mathematical Physics

Volume 18, Issue 1, March 2013, 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
https://doi.org/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 on Diff(𝕊) is given by L = μ − ∂x2. In contrast, the μ-Degasperis–Procesi equation (obtained for b = 3) is not a metric Euler equation on Diff(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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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
18 - 1
Pages
1 - 8
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1142/S1402925111001155How to use a DOI?
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/).

Cite this article

TY  - JOUR
AU  - Martin Kohlmann
PY  - 2021
DA  - 2021/01
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  - https://doi.org/10.1142/S1402925111001155
ID  - Kohlmann2021
ER  -