Singular solutions of the r-Camassa–Holm equation

Abstract

Abstract 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 W 1 , r metric. The conserved energy for the problem is given by the full W 1 , 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.