A note on multi-dimensional Camassa–Holm-type systems on the torus

We present a 2n-component nonlinear evolutionary PDE which includes the n-dimensional versions of the Camassa–Holm and the Hunter–Saxton systems as well as their partially averaged variations. Our goal is to apply Arnold's geometric formalism (Arnold 1966 Ann. Inst. Fourier (Grenoble) 16 319–61; Ebin and Marsden 1970 Ann. Math. 92 102–63) to this general equation in order to obtain results on well-posedness, conservation laws or stability of its solutions. Following the line of arguments of Kohlmann (2011 J. Phys. A: Math. Theor. 44 465205), we present geometric aspects of a two-dimensional periodic μ-b-equation on the diffeomorphism group of the torus in this context. (paper)