Geometric finite difference schemes for the generalized hyperelastic-rod wave equation

Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa–Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.