### 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 language | English |
---|---|

Pages (from-to) | 37-85 |

Number of pages | 49 |

Journal | Communications in Partial Differential Equations |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jan |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**A long wave approximation for capillary-gravity waves and an effect of the bottom.** / Iguchi, Tatsuo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A long wave approximation for capillary-gravity waves and an effect of the bottom

AU - Iguchi, Tatsuo

PY - 2007/1

Y1 - 2007/1

N2 - 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.

AB - 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.

KW - Capillary-gravity waves

KW - KdV equation

KW - Long wave approximation

KW - Water waves

UR - http://www.scopus.com/inward/record.url?scp=33847163682&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847163682&partnerID=8YFLogxK

U2 - 10.1080/03605300601088708

DO - 10.1080/03605300601088708

M3 - Article

AN - SCOPUS:33847163682

VL - 32

SP - 37

EP - 85

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 1

ER -