Some Mathematical Problems on Exact Solitary Water Waves

所提出的研究考虑了在重力和小(或零)表面张力的影响下，有限水深运动的二维和三维行波的存在和稳定性的数学理论。将使用精确的、完全非线性的控制流体流动的方程。该项目包括三个问题。第一个问题是在线性渐近稳定性的最新突破的基础上，研究零表面张力水中孤立波的非线性稳定性。第二个问题是关于小表面张力水中存在多峰(或多孤立)波的严格证明，因为这种波是在实验中观察到的，并用模型方程推导出来的。第三个问题是关于在小表面张力的水中存在由二维孤立波分叉而来的三维波(也称为破维分叉)，这可能证实了从许多大型实验中观察到的现象。这里的主要目的是用精确方程而不是近似方程来研究有限水深的水波。流体力学和应用分析理论的相互作用对这项研究将是必不可少的。水体上自由表面波动的数学理论是一门令人着迷的学科，在应用和纯数学研究方面都有很长的历史，并与人类与河流和海洋有关的事业持续相关。现实世界中的大量观测，如湖泊中的船只或海洋中的船只产生的波浪，已经被实验、数值和数学研究。这些面波的存在性和稳定性的数学研究是这一领域的重点和难点研究课题之一。特别是，实验和观测已经证实，沿海峡或公海传播的单峰波(称为孤立波)具有显著的持久性。然而，这种波浪的稳定性在数学上仍然是一个悬而未决的问题。虽然在许多实验中观察到了从二维波浪中分叉出来的多峰波或三维波，但对这些波的数学研究还相对滞后。拟议研究的目标是在这些问题上取得一些重大的数学进展。这项研究可能会对数学、物理和工程中的许多涉及流体界面和波的传播和相互作用的科学研究产生潜在的影响。例如，该理论可以为地震引起的海啸波在海洋中的传播提供有用的预测和有用的信息，防止由快速渡轮产生的巨浪，这些巨浪被指责为许多船只事故的罪魁祸首，以及可能对近海石油钻井平台或海洋结构物造成巨大破坏的风暴或飓风在海洋中引发的海浪的性质。

The proposed research considers mathematical theory on the existence and stability of two- and three-dimensional traveling waves on water of finite depth moving under the influence of gravity and small (or zero) surface tension. The exact fully nonlinear equations governing the fluid flows will be used. The project includes three problems.The first one is to study the nonlinear stability of solitary waves on water with zero surface tension, based upon recent breakthrough on the linear asymptotic stability. The second problem intends to give a rigorous proof on the existence of multi-hump (or multi-solitary) waves for water with small surface tension, since such waves have been observed in experiments and derived using model equations. The third problem deals with the existence of three-dimensional waves bifurcating from two-dimensional solitary waves (also called dimension-breaking bifurcation) on water with small surface tension, which may confirm the phenomena observed from many large-scale experiments. Here, the main thrust is to use the exact equations, rather than approximate equations,to study the waves on water of finite depth. An interplay of the theories in fluid dynamics and applied analysis will be essential to this research.The mathematical theory of wave motions on a free surface over a body of water is a fascinating subject, with a long history in both applied and pure mathematical research, and with a continuing relevance to the enterprises of mankind having to do with rivers and oceans. Numerous observations in the real world, such as waves generated by boats in lakes or ships in oceans, have been studied experimentally, numerically,and mathematically. The mathematical study on the existence and stability of these surface waves is one of the important and difficult research subjects in this area. In particular, experiments and observations have confirmed that single-hump waves (called solitary waves) propagating along a channel or an open sea have a remarkable property of permanence. Yet, the stability of such waves remains an unsolved problem mathematically. Although multi-hump waves or three-dimensional waves bifurcating from two dimensional waves have been observed in many experiments, mathematical research on these waves is still lagging behind. The goal of proposed research is to make some significant mathematical advances on these problems. The research may have potential impact on many scientific research in mathematics, physics and engineering that involve fluid interfaces and wave propagation and interactions. For example, the theory may provide helpful forecasts and useful information on the propagation of tsunami waves in oceans caused by earthquakes, the prevention of giant waves generated from fast ferries that have been blamed for many boat accidents, and the properties of waves induced by storms or hurricanes in oceans which may cause tremendous damage to offshore oil rigs or marine structures.