Abstract The higher-dimensional b -equation is a family of PDEs, introduced by Holm and Staley (SIAM J Appl Dyn Syst 2 (3): 323–380, 2003), that describe the motion of shallow water waves in n -dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid, where the constant b can be thought of as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of Rⁿ R n and show that this connection is Levi-Civita with respect to a right-invariant Riemannian metric only when b=2 b = 2. Using this framework and the methods of Ebin and Marsden (Ann Math (2): 92: 102–163, 1970), we show local well-posedness of the b -equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the b -equation as a smooth ODE on a Hilbert manifold, applying Picard–Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution.
Justin Valletta (2026) studied this question.