Stability of Solitary Waves on Water of Finite Depth

这项研究项目涉及二维和三维局部面波(也称为孤立波)在以下由刚性水平底部和以上由自由表面包围的水中的稳定性。波在自由面上的重力和表面张力的影响下运动。用近似模型方程和精确的欧拉方程来研究这些波的稳定性。该项目由三个问题组成。第一个是研究广义Boussinesq系统孤立波的稳定性，证明了这类系统的孤立波是稳定的。第二个问题旨在研究孤立波谱稳定性，并建立在孤立波解附近线性化的方程的不稳定本征值存在或不存在的判据。第三个问题研究了三维孤立波的条件稳定性，假设最初靠近孤立波的波在任意有限时间内存在。在这里，流体力学理论和应用分析的相互作用是必不可少的。水波理论已经发展了150多年，人们从实验、数值和数学上研究了许多现实世界的现象，如湖泊中的船只或海洋中的船只所产生的波浪。这些海浪的稳定性研究是该领域的重点和难点课题之一。特别是，实验和观测表明，沿航道或公海传播的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。这项研究集中于分析孤立波的稳定性，在涉及流体界面和波传播与相互作用的许多数学、科学和工程领域具有潜在的影响，例如地震引起的海啸波在海洋中的传播，由快速渡轮产生的威胁海岸线的巨浪被指责为许多船只事故的罪魁祸首，以及风暴或飓风引起的对近海石油钻井平台造成巨大破坏的海浪。

This research project concerns stability of two- and three-dimensional localized surface waves (also called solitary waves) on water bounded below by a rigid horizontal bottom and above by a free surface. The waves move under the influence of gravity and surface tension on the free surface. Approximate model equations and exact Euler equations governing the flow will be used to study the stability of these waves. The project consists of three problems. The first one is to study the stability of solitary waves for generalized Boussinesq systems and show that the solitary waves for such a system are stable. The second problem intends to study the spectral stability of solitary waves and establish the criteria for the existence or nonexistence of unstable eigenvalues for the equations linearized around the solitary-wave solution. The third problem deals with the conditional stability of three-dimensional solitary waves under the assumption that waves initially close to the solitary wave exist for any finite time. Here, interplay of the theories in fluid dynamics and applied analysis is essential. The theory of water waves has developed for more than 150 years, and numerous real-world phenomena, such as waves generated by boats in lakes or ships in oceans, have been studied experimentally, numerically, and mathematically. Research on the stability of these waves is one of the important and difficult subjects in this area. In particular, experiments and observations have shown that the 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. This research project, which is focused on analysis of the stability of solitary waves, has potential impact in many areas of mathematics, science, and engineering that involve fluid interfaces and wave propagation and interactions, such as the propagation of tsunami waves in oceans caused by earthquakes, the giant waves generated from fast ferries that threaten coastlines and have been blamed for many boat accidents, and the waves induced by storms or hurricanes that cause tremendous damage to offshore oil rigs.