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.

本研究项目涉及的稳定性的二维和三维局部表面波（也称为孤立波）的水界定下由一个刚性的水平底部和自由表面以上。 波浪在重力和自由表面上的表面张力的影响下运动。 近似模型方程和精确的欧拉方程控制的流动将被用来研究这些波的稳定性。 该项目包括三个问题。 第一部分研究了广义Boussinesq系统孤立波的稳定性，证明了广义Boussinesq系统孤立波是稳定的。 第二个问题是研究孤立波的谱稳定性，建立在孤立波解附近线性化的方程的不稳定特征值存在或不存在的判据。 第三个问题是在初始波接近孤立波的假设下，在任意有限时间内存在三维孤立波的条件稳定性。 在这里，流体动力学和应用分析理论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。 这类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，孤立波沿沿着传播的渠道或开放的海洋有一个显着的性质的持久性。 然而，这种波的稳定性仍然是一个数学上未解决的问题。 本研究项目以孤立波的稳定性分析为中心，对地震引起的海啸波在海洋中的传播、高速渡轮产生的威胁海岸线的巨浪等涉及流体界面、波的传播和相互作用的数学、科学和工程学的许多领域都具有潜在的影响，以及风暴或飓风引起的波浪，这些波浪会对海上石油钻井平台造成巨大破坏。