2025/05/08 by James Woodfield, Woodfield, James · 1 voice
Computer Science · Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS) #nlin.PS
paper · pdf · doi:10.48550/arxiv.2505.05167
openalex publication_date 2025/05/08 · arxiv published 2025/05/08 · arxiv updated 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper numerically investigates Euler-Poincaré equations arising from a self-semidirect product group structure. Nonlinearly coupled systems of equations emerge from the semidirect product action where one set of dynamics can be considered in the frame of another. A monolithic energy-preserving continuous Galerkin finite element method is used to study geodesic equations associated with the semidirect product of the diffeomorphism group on a circle with itself. Theoretically predicted peakon solutions are observed as an emergent behaviour. In addition, complicated nonlinear transfers of energy are associated with the semidirect product coupling, where amongst various nonlinear interactions, we observe coupled peakon behaviour. A mimetic (C-grid) finite difference method is used to study the geodesic flow of the semidirect product of the volume preserving diffeomorphism group with itself, where similar coupling behaviour is observed in the vorticity variables. We also investigate coadjoint and Lie-Poisson structures in the context of geodesic equations on semidirect product groups, where the underlying group is first extended by central extension or semidirect product.