EVOLUTION OF PACKETS OF WATER-WAVES
EVOLUTION OF PACKETS OF WATER-WAVES
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DOI：
10.1017/s0022112079000835
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发表时间：
1979-01-01
影响因子：
3.7
通讯作者：
SEGUR, H
中科院分区：
文献类型：
--
作者：
ABLOWITZ, MJ;SEGUR, H
我们认为，主要是在一个方向上旅行的水波包的演变，但在两个水平方向上的波幅调制缓慢。两个单独的模型进行了讨论，这取决于是否长的波与流体深度相比。这些模型是Korteweg-de弗里斯方程（长波）和立方非线性薛定谔方程（短波）的二维推广。在这两种情况下，我们发现，二维的波包的演变从根本上取决于无量纲的表面张力和流体深度。特别是，对于长波，一维（KdV）孤子变得不稳定，甚至更长的横向扰动时，表面张力参数变得足够大，即在非常薄的水片。二维长波（“块”）在所有水平方向上代数地衰减，并像孤子一样相互作用，只有当一维孤子不稳定时才存在。然而，表面张力和深度的最戏剧性后果发生在足够深的水中的毛细型波。在这里，一个到处都很小（但不是无穷小）并且在两个水平维度上都被调制的波包可以在有限的时间内“聚焦”，产生一个波振幅有限的区域。这种非线性不稳定性应该比迄今为止所研究的线性不稳定性更强、更明显，应该很容易观察到。二维短波包演化的另一个特征是，所有一维孤子对于长横向扰动都是不稳定的。最后，我们确定了一些精确的相似解的发展方程。
We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrödinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.