The periodic $\mu$-$b$-equation and Euler equations on the circle

TL;DR

This paper investigates the geometric structure of the periodic $ ext{b}$-equation and its $ ext{μ}$-variant, revealing that only for $b=2$ does it correspond to a metric Euler equation on the diffeomorphism group of the circle, with implications for understanding related integrable equations.

Contribution

It characterizes when the $ ext{μ}$-$b$-equation can be realized as a metric Euler equation on the circle's diffeomorphism group, specifically for $b=2$, and shows that for $b=3$ it cannot, extending previous results.

Findings

The $ ext{μ}$-$b$-equation is a metric Euler equation only for $b=2$.

The inertia operator for $b=2$ is explicitly given by $L=μ-rac{d^2}{dx^2}$.

The $ ext{μ}$-Degasperis-Procesi equation ($b=3$) is not a metric Euler equation for any regular inertia operator.

Abstract

In this paper, we study the $\mu$-variant of the periodic $b$-equation and show that this equation can be realized as a metric Euler equation on the Lie group $\Diff^{\infty}(\S)$ if and only if $b=2$ (for which it becomes the $\mu$-Camassa-Holm equation). In this case, the inertia operator generating the metric on $\Diff^{\infty}(\S)$ is given by $L=\mu-\partial_x^2$. In contrast, the $\mu$-Degasperis-Procesi equation (obtained for $b=3$) is not a metric Euler equation on $\Diff^{\infty}(\S)$ for any regular inertia operator $A\in\mathcal L_{\text{is}}^{\text{sym}}(C^{\infty}(\S))$. The paper generalizes some recent results of [J. Escher and B. Kolev, DOI 10.1007/s00209-010-0778-2], [J. Escher and J. Seiler, J. Math. Phys. 51 (2010), 053101.1-053101.6] and [B. Kolev, Wave Motion 46 (2009), 412-419].