A Multiscale Camassa–Holm equation

A system of equations is introduced for Multiscale Camassa–Holm (MCH) dynamics on the manifold of smooth invertible maps (diffeomorphisms). The emergent multiscale singular solutions of MCH generalise the peakon solutions of the CH equation. The MCH paradigm is inspired by L. F. Richardson’s rhyme in 1922 of whirls within whirls within whirls to illustrate the cascade of energy in fluid turbulence. The MCH dynamics illustrates the cascade of energy in collisions of peakons within peakons within peakons. Numerical simulations of MCH on S 1 are given to demonstrate its compound peakon propagation and collision dynamics. • We derive the Multiscale Camassa–Holm (MCH) system and numerically simulate its solutions. • L.F. Richardson’s rhyme (1921) represents turbulence as whirls within whirls within whirls. • MCH represents multiscale compound dynamics of peakons within peakons within peakons. • The MCH system governs creation, propagation and collisions of multiscale compound peakons.