Nonlinear Surface Waves on Water of Finite Depth

9971764有限深度水上的非线性表面波该项目研究有限水深上的非线性毛细管重力波。所谓“水”，是指一种密度恒定的不可压缩、无粘性的流体，在重力的作用下在自由面上以表面张力运动，流动是无旋的。本文的研究重点是刚性水底和自由水面上的非线性毛细重力波的数学理论。它由两部分组成。第一部分给出了具有大表面张力的二维孤立波分叉出的三维传播波存在的数学证明。第二部分研究了二维矩形区域内自由水面上毛细重力波的运动。水波理论在流体力学的发展中起着至关重要的作用。除了在工程学和应用科学的各个分支中都很重要外，许多水波现象在日常生活中也会发生。最熟悉的例子可能是河流或水库中的船只产生的海浪和海洋中的风或地震产生的海浪。这项工作主要研究海洋或水库中自由水面上的某些波动，旨在给出某些波动产生的条件，并得到这些波动的大时间行为。这项研究的结果可以解释一些自然现象，如地震引起的波浪在海洋中的传播，并对一些可以观测到的水库中的波动进行描述。对海洋或水库中波动的这种了解可以用来预测某些自然现象，以防止由它们造成的人类灾难，并从它们中受益，例如产生更多的电力。

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.