Stability of Solitary Waves on Water of Finite Depth

有限深度水面上孤立波的稳定性

• 批准号: 0807597
• 负责人: Shu-Ming Sun
• 金额: $ 11.85万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807597&HistoricalAwards=false
• 关键词: Stability; Solitary; Waves; Water; Finite

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.

该研究项目涉及二维局部表面波（也称为孤立波）的稳定性，在下面的水上由刚性水平底部及以上的自由表面界定。 在重力和表面张力对自由表面的影响下，波浪移动。 近似模型方程和控制流量的精确欧拉方程将用于研究这些波的稳定性。 该项目包括三个问题。 第一个是研究广义Boussinesq系统的孤立波的稳定性，并表明该系统的孤立波是稳定的。 第二个问题旨在研究孤立波的光谱稳定性，并确定对围绕孤立波解决方案线性线性的方程的存在或不存在的不稳定特征值的标准。 第三个问题涉及三维孤立波的条件稳定性，即在任何有限的时间内都存在最初接近孤立波的波的假设。 在这里，流体动力学和应用分析中的理论相互作用至关重要。 水浪理论已经发展了150多年，并且在实验，数值和数学上研究了许多现实现象，例如湖泊或海洋中的船只或海洋船只产生的波浪。 研究这些波的稳定性是该领域的重要且困难的主题之一。 特别是，实验和观察结果表明，沿着通道或开海传播的孤立波具有显着的永久特性。 然而，这种波浪的稳定性在数学上仍然是一个未解决的问题。 该研究项目集中在分析孤立波的稳定性上，在许多数学，科学和工程学领域都具有潜在的影响飓风会对海上石油钻机造成巨大破坏。