Three-Dimensional Nonlinear Gravity-Capillary Water Waves

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This project studies the theory of three-dimensional nonlinear gravity-capillary waves in fluid bounded below by a rigid horizontal bottom and above by a free surface. The research is focused on incompressible inviscid fluids of constant density, moving under the influence of gravity and surface tension on the free surface, with irrotational flow. The work centers on three problem areas. The first will establish the existence of exact twin-soliton solutions, obliquely-interacting identical two-dimensional solitary waves. Work in the second area aims to prove the existence of three-dimensional propagating waves that decay to zero in both propagating and transverse directions in water with large surface tension. The third problem area concerns the existence of three-dimensional propagating waves bifurcating from a two-dimensional generalized solitary wave, a solitary wave with ripples at infinity, in water with small surface tension. In this work, the exact fully nonlinear governing equations, rather than approximate model equations, are employed to study three-dimensional propagating surface waves in water. Interplay of theoretical fluid dynamics and applied analysis is essential to the project. The theory of water waves is essential for understanding and control of many important natural phenomena, such as ocean waves generated by wind or earthquakes, and waves generated by ships. This project focuses on the mathematical theory of three-dimensional water waves. The research will contribute to the design of ships with significantly reduced wave resistance, as well as understanding of water-wave phenomena observed in experiments. Design of ships with low wave resistance is especially important for eliminating giant waves generated by fast ferries, which threaten coastlines and have been blamed for many boat accidents.