Nonlinear Surface Waves on Water of Finite Depth

[9971764]有限深度水面上的非线性表面波孙树明本项目研究有限深度水面上的非线性毛细重力波。所谓“水”，是指一种不可压缩的、密度不变的无粘流体，在具有表面张力的重力作用下在自由表面上运动，流动是无旋的。本文主要研究以刚性水平底和自由表面为界的水中非线性毛细重力波的数学理论。它由两部分组成。第一部分给出了二维大表面张力孤立波分叉的三维传播波存在的数学证明。第二部分讨论了二维矩形区域内毛细重力波在自由水面上的运动。水波理论在流体力学的发展中起着至关重要的作用。除了在工程和应用科学的各个分支中具有重要意义外，许多水波现象在日常生活中也会发生。船只在河流或水库中产生的波浪以及风或地震在海洋中产生的波浪可能是最熟悉的例子。本工作着重研究海洋或水库自由水面上的某些波浪运动，试图给出产生某些波的条件，并获得这些波的大时间特性。这项研究的结果可以解释一些自然现象，如地震在海洋中产生的波浪传播，并提供一些可以观察到的水库波浪运动的描述。这种对海洋或水库中波浪运动的了解可以用来预测某些自然现象，以防止由它们引起的人类灾难，并从中受益，例如发电。

9971764NONLINEAR SURFACE WAVES ON WATER OF FINITE DEPTHShu-Ming Sun The project concerns nonlinear capillary-gravity waves on water of finite depth. By ``water'', it means that an incompressible, inviscid fluid with constant density is moving under the influence of the gravity with surface tension on the free surface and the flow is irrotational. The proposed research focuses on the mathematical theory of nonlinear capillary-gravity waves on water bounded by a rigid horizontal bottom and a free surface. It consists of two parts. The first part intends to give a mathematical proof of the existence of three-dimensional propagating waves bifurcating from a two-dimensional solitary wave with large surface tension. The second part deals with the motion of capillary-gravity waves on the free surface of water in a two-dimensional rectangular region. The theory of water waves plays a crucial role in the development of fluid mechanics. Apart from being important in various branches of engineering and applied sciences, many water-wave phenomena happen in every-day experience. Waves generated by ships in rivers or reservoirs and waves generated by wind or earthquakes in oceans are probably the most familiar examples. This work focuses on certain wave motions on the free surface of water in oceans or reservoirs and intends to give some conditions for generating certain kinds of waves and obtain the large-time behavior of these waves. The results obtained from this research may present explanations of some natural phenomena, such as wave propagation in oceans generated by earthquakes, and provide descriptions of some wave motions in reservoirs that can be observed. Such understanding of those wave motions in oceans or reservoirs can be used to predict certain natural phenomena for preventing human disasters caused by them as well as benefiting from them, such as generating more electricity.