Some Mathematical Problems on Exact Solitary Water Waves

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The proposed research considers mathematical theory on the existence and stability of two- and three-dimensional traveling waves on water of finite depth moving under the influence of gravity and small (or zero) surface tension. The exact fully nonlinear equations governing the fluid flows will be used. The project includes three problems.The first one is to study the nonlinear stability of solitary waves on water with zero surface tension, based upon recent breakthrough on the linear asymptotic stability. The second problem intends to give a rigorous proof on the existence of multi-hump (or multi-solitary) waves for water with small surface tension, since such waves have been observed in experiments and derived using model equations. The third problem deals with the existence of three-dimensional waves bifurcating from two-dimensional solitary waves (also called dimension-breaking bifurcation) on water with small surface tension, which may confirm the phenomena observed from many large-scale experiments. Here, the main thrust is to use the exact equations, rather than approximate equations,to study the waves on water of finite depth. An interplay of the theories in fluid dynamics and applied analysis will be essential to this research.The mathematical theory of wave motions on a free surface over a body of water is a fascinating subject, with a long history in both applied and pure mathematical research, and with a continuing relevance to the enterprises of mankind having to do with rivers and oceans. Numerous observations in the real world, such as waves generated by boats in lakes or ships in oceans, have been studied experimentally, numerically,and mathematically. The mathematical study on the existence and stability of these surface waves is one of the important and difficult research subjects in this area. In particular, experiments and observations have confirmed that single-hump waves (called 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. Although multi-hump waves or three-dimensional waves bifurcating from two dimensional waves have been observed in many experiments, mathematical research on these waves is still lagging behind. The goal of proposed research is to make some significant mathematical advances on these problems. The research may have potential impact on many scientific research in mathematics, physics and engineering that involve fluid interfaces and wave propagation and interactions. For example, the theory may provide helpful forecasts and useful information on the propagation of tsunami waves in oceans caused by earthquakes, the prevention of giant waves generated from fast ferries that have been blamed for many boat accidents, and the properties of waves induced by storms or hurricanes in oceans which may cause tremendous damage to offshore oil rigs or marine structures.