A long wave approximation for capillary-gravity waves and an effect of the bottom

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18 Citations (Scopus)

Abstract

The Korteweg-de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the 2-dimensional water wave problem, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. In this article, we consider the validity of this approximation in the case of the presence of the surface tension. Moreover, we consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation. We derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval. We also prove that if the initial data and the bottom decay at infinity in a suitable sense, then the KdV equation takes the place of the coupled equations.

Original languageEnglish
Pages (from-to)37-85
Number of pages49
JournalCommunications in Partial Differential Equations
Volume32
Issue number1
DOIs
Publication statusPublished - 2007 Jan 1

Keywords

  • Capillary-gravity waves
  • KdV equation
  • Long wave approximation
  • Water waves

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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