A connection between the shallow-water equations and the Euler-Poincaré equations

2014/04/18 by Roberto Camassa, Camassa, Roberto, Long Lee +1
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1404.4900

arxiv created 2014/04/18 · openalex publication_date 2014/04/18 · arxiv updated 2014/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Euler-Poincaré differential (EPDiff) equations and the shallow water (SW) equations share similar wave characteristics. Using the Hamiltonian structure of the SW equations with flat bottom topography, we establish a connection between the EPDiff equations and the SW equations in one and multi-dimensions. Additionally, we show that the EPDiff equations can be recast in a curl formulation.

Citations

• Solitary waves and N-particle algorithms for a class of Euler-Poincaré equations
• Interaction Dynamics of Singular Wave Fronts

Related

• Extended shallow water wave equations
• Conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Lagrange's variables
• Construction of Hamiltonian and Nambu forms for the shallow water equations
• Rigorous derivation from the water waves equations of some full\n dispersion shallow water models
• Free-surface Euler equations with density variations, and shallow-water limit