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This paper introduces the r-Camassa-Holm (r-CH) equation, which describes ageodesic flow on the manifold of diffeomorphisms acting on the real lineinduced by the W1,r metric. The conserved energy is for the problem is given bythe full W1,r norm and the for r = 2, we recover the Camassa-Holm equation. Wecompute the Lie symmetries for r-CH and study various symmetry reductions. Weintroduce singular weak solutions of the r-CH equation for r >= 2 anddemonstrates their robustness in numerical simulations of their nonlinearinteractions in both overtaking and head-on collisions. Several open questionsare formulated about the unexplored properties of the r-CH weak singularsolutions, including the question of whether they would emerge from smoothinitial conditions.

The r-Camassa-Holm equation: smooth and singular solutions