A Multiscale Camassa–Holm equation

Darryl D. Holm and Maneesh Kumar Singh

Chaos, Solitons & Fractals, 2026, vol. 202, issue P1

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 S1 are given to demonstrate its compound peakon propagation and collision dynamics.

Keywords: Geometric mechanics; Energy cascade; Coherent structures (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925014225

DOI: 10.1016/j.chaos.2025.117409

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