Stability of Solitary Waves on Water of Finite Depth

本研究项目关注的是二维和三维局部表面波（也称为孤立波）在水面上的稳定性，水面的底部是刚性的水平底部，上面是自由的表面。波浪在自由表面的重力和表面张力的影响下运动。将采用近似模型方程和精确欧拉方程来研究这些波的稳定性。这个项目包括三个问题。第一部分研究了广义Boussinesq系统的孤立波的稳定性，并证明了这种系统的孤立波是稳定的。第二个问题是研究孤立波的谱稳定性，建立围绕孤立波解线性化的方程存在或不存在不稳定特征值的判据。第三个问题是在假定初始波在任意有限时间内接近孤立波的前提下，研究三维孤立波的条件稳定性问题。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，许多现实世界的现象，如湖泊中的船只或海洋中的船只产生的波浪，已经被实验，数值和数学研究过。这些波的稳定性研究是这一领域的重点和难点之一。特别是，实验和观测表明，沿海峡或公海传播的孤立波具有显著的持久性。然而，这种波的稳定性在数学上仍然是一个未解决的问题。这个研究项目的重点是分析孤立波的稳定性，它对涉及流体界面和波的传播和相互作用的数学、科学和工程的许多领域都有潜在的影响，例如地震引起的海啸波在海洋中的传播，快速渡轮产生的巨浪威胁着海岸线，并被归咎于许多船只事故。风暴或飓风引起的海浪对海上石油钻井平台造成了巨大的破坏。

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.