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

TL;DR

This paper introduces the r-Camassa-Holm equation, exploring its geometric structure, symmetries, and singular solutions, and demonstrates their robustness through numerical simulations, raising questions about their origins from smooth initial data.

Contribution

It presents the r-Camassa-Holm equation, analyzes its symmetries, introduces singular weak solutions, and studies their nonlinear interactions numerically.

Findings

Singular weak solutions are robust in simulations.

The r-CH equation generalizes the Camassa-Holm equation for r ≥ 2.

Symmetry reductions reveal underlying geometric structures.

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 W1,r metric. The conserved energy is for the problem is given by the full W1,r norm and the 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.