The periodic two-dimensional $\mu$-$b$-equation as an EPDiff equation

TL;DR

This paper introduces new periodic two-dimensional $$-equations as geodesic flows on diffeomorphism groups, providing derivations, geometric insights, well-posedness results, and curvature computations within the framework of Arnold's theory.

Contribution

It presents the formulation and analysis of novel 2D $$-equations as geodesic flows, extending Arnold's geometric approach to these equations.

Findings

Derivation of 2D $$-equations within Arnold's framework

Well-posedness results for the introduced equations

Explicit curvature computations related to the equations

Abstract

We introduce a periodic two-dimensional $\mu$-$b$-equation and a periodic two-dimensional two-component $(\mu)$-Camassa-Holm equation which we study as geodesic flows on the diffeomorphism group of the torus and a semidirect product respectively. The paper explains the derivation of these equations within V.I. Arnold's (1966) general framework, some analogies to recently discussed related equations and gives a self-contained presentation of the geometric aspects. As an application, we obtain well-posedness results and some explicit curvature computations.