Some Mathematical Problems on Exact Solitary Water Waves

本研究考虑了有限深度水中在重力和小（或零）表面张力作用下二维和三维行波存在和稳定性的数学理论。将使用精确的控制流体流动的全非线性方程。该项目包括三个问题。第一部分是基于最近在线性渐近稳定性方面的突破，研究了零表面张力水面上孤立波的非线性稳定性。第二个问题是对具有小表面张力的水的多驼峰（或多孤立）波的存在给出严格的证明，因为这种波已经在实验中观察到并使用模型方程推导出来。第三个问题是在小表面张力的水中存在二维孤立波分岔的三维波（也称为破维分岔），这可能证实了许多大规模实验观察到的现象。在这里，主要的重点是使用精确方程，而不是近似方程，来研究有限深度水中的波浪。流体动力学理论和应用分析的相互作用对这项研究至关重要。水体自由表面上波浪运动的数学理论是一门迷人的学科，在应用数学和纯数学研究方面有着悠久的历史，并且与人类与河流和海洋有关的事业不断相关。现实世界中的许多观察，如湖泊中的船只或海洋中的船只产生的波浪，已经通过实验、数值和数学进行了研究。这些表面波的存在性和稳定性的数学研究是该领域的重要和困难的研究课题之一。特别是，实验和观察已经证实，单驼峰波（称为孤波）沿海峡或公海传播具有显著的持久性。然而，这种波的稳定性在数学上仍然是一个未解决的问题。虽然在许多实验中已经观察到多驼峰波或由二维波分叉的三维波，但对这些波的数学研究还比较落后。所提出的研究目标是在这些问题上取得一些重大的数学进展。该研究可能对涉及流体界面和波的传播与相互作用的许多数学、物理和工程科学研究产生潜在影响。例如，该理论可以对地震引起的海啸波在海洋中的传播提供有益的预报和有用的信息，防止由快速渡轮产生的巨浪，这种巨浪被指责为许多船只事故，以及海洋中风暴或飓风引起的波浪的性质，这些波浪可能对近海石油钻井平台或海洋结构造成巨大破坏。

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.