The effect of the induced mean flow on solitary waves in deep water
The effect of the induced mean flow on solitary waves in deep water
复制标题
诱发平均流对深水中孤立波的影响
DOI：
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发表时间：
1998
影响因子：
3.7
通讯作者：
R. Grimshaw
中科院分区：
文献类型：
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作者：
T. Akylas;F. Dias;R. Grimshaw
重力-毛细孤立波的两个分支是由一列无穷小周期波在相速度的最小值处分叉而成的。一般来说，这些孤立波的特征是振幅呈指数衰减的振荡尾，在小振幅极限下，它们可以被解释为非线性薛定谔方程的包络孤子解，使得包络以与载波振荡相同的速度传播。然而，在无限深的水中，根据Hogan（1985）导出的四阶包络方程，由于NLS方程中没有考虑的诱导平均流，这些重力-毛细孤立波的轮廓实际上在无穷远处代数衰减（如1/x2）。基于全水波方程的数值计算证实了深水中孤立波尾的代数衰减。此外，在Benjamin（1992）提出的双流体系统界面波模型方程的孤立波解的尾部也发现了同样的行为。
Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.