Stability of Solitary Waves on Water of Finite Depth

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

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.