This paper introduces the r-Camassa–Holm (r-CH) equation, which describes
a geodesic flow on the manifold of diffeomorphisms acting on the real line
induced by the W1,r metric. The conserved energy for the problem is given
by the full W1,r
norm. For r = 2, we recover the Camassa–Holm equation.
We compute the Lie symmetries for r-CH and study various symmetry reductions. We introduce singular weak solutions of the r-CH equation for r ⩾ 2
and demonstrates their robustness in numerical simulations of their nonlinear
interactions in both overtaking and head-on collisions. Several open questions
are formulated about the unexplored properties of the r-CH weak singular
solutions, including the question of whether they would emerge from smooth
initial conditions.