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

By Adrian Constantin, David Lannes. 2007

1,080 citations recorded by the source corpus.

In recent years two nonlinear dispersive partial differential equations have attracted much 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 accommodate wave breaking phenomena.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b2a1ef43b6f75012f69fe4bf52e5b425588f6e0e

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Citations in the corpus

The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation (2010, methodology): “[10] A. Constantin, D. Lannes, The hydro-dynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch.”
Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces (2020, neutral): “Another well-known integrable equation admitting peakons is the Degasperis-Procesi (DP) equation [15, 20] { ∂tu+ u∂xu = − 3 2 ∂x(1 − ∂ 2 x) u(2), (x, t) ∈ R× R, u(x, t = 0) = u0, x ∈ R.”
On the global existence and wave-breaking criteria for the two-component Camassa-Holm system (2010, background): “1), one obtains the classical Camassa–Holm model [5], whose relevance for water waves was established in [10,27].”
Inverse scattering transform for the Degasperis–Procesi equation (2010, background): “However, among the various equations that arise in this way, there are only two which admit a bi-Hamiltonian structure (see [10, 35]): the Camassa-Holm (CH) equation [4] and the DP equation (DP).”
Well-posedness and Continuity Properties of the Fornberg-Whitham Equation in Besov Spaces (2016, methodology): “Concerning it’s physical relevance, we refer the reader to the works by Johnson [23], [24] and Constantine and Lannes [6].”
On the Cauchy problem for the two-component Camassa–Holm system (2011, background): “The first one is the equation possesses the solutions in the form of peaked solitons or ’peakons’ [5,15], The peakon u(t, x) = ce−|x−ct |, c = 0 is smooth except at its crest and the tallest among all waves of fixed energy.”
Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces (2012, background): “Its hydrodynamical relevance has been pointed out in [4], [8].”
Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons (2010, background): “5) It is regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the Camassa–Holm shallow water equation, and it can also be obtained from the governing equations for water waves [21].”
Infinite propagation speed for a two component Camassa-Holm equation (2009, background): “Firstly, it has been shown to model a variety of phenomena, including the propagation of unidirectional shallow water waves over a flat bed [6, 20 , 26, 36, 37] (also allowing for an”
Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “A similar approach was followed by Constantin and Lannes in [5] where it is shown that CH is a generalization of the BBM equations with appropriate scaling and choice of the parameters involved.”
Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations (2012, background): “[6] Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch.”