Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity

2007/01/01 by Delia Ionescu-Kruse · 13 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Numerical methods for differential equations #Vorticity #Conservative vector field #Zero (linguistics) #Potential vorticity #Waves and shallow water #Camassa–Holm equation #Shallow water equations #Flow (mathematics) #Vorticity equation #Mathematical analysis #Physics #Mathematics #Classical mechanics #Mechanics #Vortex #Integrable system #Compressibility #Thermodynamics

paper · pdf · doi:10.3934/dcds.2007.19.531

published in Discrete and Continuous Dynamical Systems 17(3), 531-543 (American Institute of Mathematical Sciences)

openalex publication_date 2007/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/19

Abstract

We describe the physical hypotheses underlying thederivation of an approximate model of water waves. For unidirectional surface shallow water waves moving over an irrotational flow as well as over a non-zero vorticity flow, we derive the Camassa-Holm equation by an interplay of variational methods and small-parameterexpansions.

Citations

• On the Inverse Spectral Problem for the Camassa–Holm Equation
• The Camassa–Holm equation for water waves moving over a shear flow
• Equations of Camassa‐Holm type and Jacobi ellipsoidal coordinates
• On the scattering problem for the Camassa-Holm equation
• The Lie Algebra Structure of Nonlinear Evolution Equations Admitting Infinite Dimensional Abelian Symmetry Groups
• A shallow water equation as a geodesic flow on the Bott-Virasoro group
• Wave breaking for nonlinear nonlocal shallow water equations
• Existence of permanent and breaking waves for a shallow water equation: a geometric approach
• Geodesic flow on the diffeomorphism group of the circle
• Stability of the Camassa-Holm solitons
• On solutions of the Camassa-Holm equation
• Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation

Cited by

• Camassa–Holm equation [wikipedia]