The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

2007/09/06 by Adrian Constantin, David Lannes · 1,023 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Burgers' equation #Camassa–Holm equation #Classical mechanics #Dispersionless equation #Integrable system #Kadomtsev–Petviashvili equation #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Partial differential equation #Physics #Quantum mechanics #Relevance (law) #Shallow water equations #Waves and shallow water #math.AP #physics.ao-ph

paper · pdf · doi:10.1007/s00205-008-0128-2

published in Archive for Rational Mechanics and Analysis 192(1), 165-186 (Springer Science+Business Media)

arxiv created 2007/09/06 · openalex publication_date 2008/05/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In recent years two nonlinear dispersive partial differential equations have attracted a lot of attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accomodate wave breaking phenomena.

Citations

• Global Conservative Solutions of the Camassa–Holm Equation
• Wave breaking for nonlinear nonlocal shallow water equations

Cited by

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• Non-uniform dependence for the Novikov equation in Besov spaces
• Stability of peakons for the generalized modified Camassa-Holm equation
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• On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations
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• Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function
• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
• Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations
• The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces
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• On sharp global well-posedness and Ill-posedness for a fifth-order\n KdV-BBM type equation
• On the Cauchy problem for a two-component Degasperis-Procesi system
• Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude
• Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data
• On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations
• Dynamical behavior near self-similar blowup waves for the generalized b-equation
• Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space
• On Isospectral flows related to (dual) cubic strings: Novikov, DP peakons and B, C-Toda lattices
• The Cauchy problem for the shallow water typ equations in low regularity spaces on the circle
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• An integrable semi-discrete Degasperis-Procesi equation
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• The algebro-geometric solutions for the modified Camassa-Holm hierarchy
• The Burgers equation and the Korteweg-de Vries equation with quadratic nonlinearity
• Singularity formation for the Serre-Green-Naghdi equations and applications to abcd-Boussinesq systems
• Well-Posedness of a kind of the free surface equation of shallow water wave
• On uniqueness of dissipative solutions of the Camassa-Holm equation
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• The Camassa-Holm equation as an incompressible Euler equation: a geometric point of view
• Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities
• Shock creation and Painlevé property of colliding peakons in the Degasperis-Procesi Equation
• Eigenvalues, bifurcation and one-sign solutions for the periodic p-Laplacian
• Orbital and asymptotic stability of a train of peakons for the Novikov equation
• The continuous dependence and non-uniform dependence of the rotation Camassa-Holm equation in Besov spaces
• The algebro-geometric solutions for Degasperis-Procesi hierarchy
• Ill-posedness for the higher dimensional Camassa-Holm equations in Besov spaces
• A Hamiltonian approach to the CH-KP equation
• Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation,
• On permanent and breaking waves in hyperelastic rods and rings
• Stability of stationary solutions for nonintegrable peakon equations
• On a Lagrangian Reduction and a Deformation of Completely Integrable Systems
• Traveling surface waves of moderate amplitude in shallow water
• Non-uniform Dependence on Initial Data for the Generalized Camassa–Holm–Novikov Equation in Besov Space
• A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System
• Hamiltonian formulation of the extended Green–Naghdi equations
• DECOUPLED AND UNIDIRECTIONAL ASYMPTOTIC MODELS FOR THE PROPAGATION OF INTERNAL WAVES
• Shallow water asymptotic models for the propagation of internal waves
• Hamiltonian structure for two-dimensional extended Green–Naghdi equations
• On the Cauchy problem for a new integrable two-component system with peakon and weak kink solutions
• The Degasperis–Procesi equation as a non-metric Euler equation
• Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation
• The 2-component dispersionless Burgers equation arising in the modelling of blood flow
• An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation
• Camassa–Holm equation [wikipedia]
• Recent advances on the global regularity for irrotational water waves. [europepmc]
• Generalizations of the short pulse equation. [europepmc]
• Stability analysis and novel solutions to the generalized Degasperis Procesi equation: An application to plasma physics. [europepmc]

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