A LONG WAVE APPROXIMATION FOR CAPILLARY-GRAVITY WAVES AND THE KAWAHARA EQUATION
A LONG WAVE APPROXIMATION FOR CAPILLARY-GRAVITY WAVES AND THE KAWAHARA EQUATION
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发表时间：
2013
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通讯作者：
T. Iguchi
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作者：
T. Iguchi
Kawahara 方程是一个高阶 Korteweg-de Vries 方程，带有附加的五阶导数项。它是由桥本推导出来的，作为当邦德数接近三分之一时，在长波状态下平底无限长运河中的毛细重力波模型。在本文中，我们对该模型给出了严格的数学证明，并表明 Kawahara 方程的解在适当的意义上在很长的时间间隔内近似于毛细管重力波的完整问题。我们还考虑了底部不平坦的情况，并推导出耦合的 Kawahara 型方程，其解近似于该情况下整个问题的解。
The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justification of this modeling and show that the solution of the Kawahara equation approximates that of the full problem of capillary-gravity waves in an appropriate sense for a long time interval. We also consider the case where the bottom is not flat and derive coupled Kawahara type equations whose solution approximates that of the full problem in that case.