Three-Dimensional Nonlinear Gravity-Capillary Water Waves

本计画研究流体中之三维非线性重力-毛细波理论。 研究的重点是不可压缩的无粘流体的恒定密度，移动的重力和表面张力的影响下的自由表面，无旋流。 这项工作集中在三个问题领域。 第一个将建立精确的双孤子解的存在性，相互作用相同的二维孤立波。 在第二个领域的工作旨在证明存在的三维传播波衰减到零的传播方向和横向在水中具有大的表面张力。 第三个问题领域涉及的三维传播波分叉的存在，从一个二维的广义孤立波，孤立波的波纹在无穷远，在水中具有小的表面张力。 在这项工作中，精确的完全非线性控制方程，而不是近似的模型方程，研究三维传播表面波在水中。 理论流体动力学和应用分析的相互作用对该项目至关重要。水波理论对于理解和控制许多重要的自然现象是必不可少的，例如风或地震产生的海浪，以及船舶产生的波浪。 该项目侧重于三维水波的数学理论。 该研究将有助于设计具有显著降低的波浪阻力的船舶，以及对实验中观察到的水波现象的理解。 低波浪阻力的船舶设计对于消除快速渡轮产生的巨浪尤为重要，这些巨浪威胁着海岸线，并被归咎于许多船只事故。

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.