Work

A Multiscale Camassa–Holm equation

Year: 2025

Type: article

Abstract: 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.

Source: Chaos Solitons & Fractals

DOI: 10.1016/j.chaos.2025.117409

Authors: Darryl D. Holm, Maneesh Kumar Singh

Institution: Imperial College London

Language: English

Topic: Nonlinear Waves and Solitons

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,970

Funders: European Research Council, Office of Naval Research