Camassa-Holm, Korteweg-de Vries and related models for water waves. (English) Zbl 1037.76006

This paper studies Korteweg-de Vries (KdV) equation, shallow-water equation, regularized long-wave equation, Camassa-Holm (CH) equation, and Green-Naghdi equation. The author describes the current methods for obtaining the CH equation in the context of water wave theory, presents the corresponding higher-order KdV results that are, in a sense, an analogue of the CH equation,and show that the CH equation does indeed arise in the water-wave problem, but in a careful limiting process. Moreover, some properties of this equation, and how it relates to the description of surface waves, are discussed. Finally, a possibility of extending the calculations to different scenarios is addressed, and, as an example, a two-dimensional CH equation is derived for water waves.


76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
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