Camassa–Holm, Korteweg–de Vries and related models for water waves

By R. Johnson. Journal of Fluid Mechanics, 2002

842 citations recorded by the source corpus.

Subfields: Oceanography, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e430e3f9254eb525ce55098d0dd2773dca579a33

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In Oceanography

In Statistical and Nonlinear Physics

Citations in the corpus

Blow-up of solutions to a nonlinear dispersive rod equation (2006, methodology): “Recently, the alternative derivation of the Camassa-Holm equation as a model for water waves was presented by Johnson [ 15 ].”
Unsteady undular bores in fully nonlinear shallow-water theory (2005, positive): “For uni-directional propagation, this leads to the KdV equation for ζ = η − η0, ∂tζ + ∂xζ + 3 2 ζ∂xζ + 1 6 ∂3xxxζ = 0 (3) (see Choi &Camassa 1999, Johnson 2002).”
Non-uniform dependence on initial data for the CH equation on the line (2009, methodology): “ water wave equation by Camassa and Holm [CH], who showed that CH is biHamiltonian and studied its \peakon" solutions. Since then CH has been rederived in various ways by Misio lek [Mi], Johnson [J], Constantin and Lannes [CL], and Ionescu-Kruse [I]. Well-posedness on the line was rst establish”
A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation (2006, background): “…(1.5) appears as a water-wave equation at quadratic order in an asymptotic expansion for unidirectional shallow water waves modeled by the incompressible Euler equations, whereas the KdV and BBM equations appear at first order in this asymptotic expansion (Camassa and Holm, 1993; Johnson, 2002).”
Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves (2003, methodology): “In Johnson (2002) the 6fth-order derivative, which is an essential part of our derivation, was omitted. In Section 4 we shall show that the CH equation is asymptotically equivalent to the KdV5 equation, which involves the 6fth-order derivative. In obtaining (3.16), the free parameters in the Kodama ”
Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation (2009, background): “the main interest of this model is that in the case of flat bottoms (b = 0), its unidirectional limit gives the Cammassa-Holm equation ([33, 34, 35, 36]).”