FRG: Collaborative Research: Fully Nonlinear, Three-Dimensional, Surface Water Waves in Arbitrary Depth

这项工作的总体目标是全面了解有限振幅的三维水波，并最终建立一个实用的模型来有效地描述这些波。一个既准确又计算效率高的模型可以有许多实际应用。需要解决的具体问题是：(1)在整个深度范围内三维、双周期、行水波模式的存在性和稳定性；(2)海浪中普遍存在六角形、矩形或新月形波（或其他多重周期波型）；(3)水波的长波和调制描述，以及在这种情况下可行的后续稳定性分析；(4)设计和实现算法，使浅水和深水中渐近模型的精确解得到实际应用；(5)三维非线性波浪的详细动力学与一些常用的海浪输运模式的关系；(6)在大振幅非线性波存在或非线性波聚焦条件下，非线性波动力学的详细局部描述对这些输运模型的影响。这些问题将使用分析、计算、渐近和代数几何来研究，涉及完整方程和近似模型，所有这些都与最先进的物理实验相结合。大浪的破坏力是众所周知的。大尺度的海浪对远洋船舶、近海石油平台和沿海环境中其他结构的设计有重大影响。这些波浪还影响航运模式的调度和路线，并强烈影响海空运输过程。然而，目前使用的大多数海浪理论模型都是基于小振幅的海浪。在这项研究中，我们的重点是发展对大振幅波的透彻理解。最终目标是开发一个实用的数学模型，可以在上面列出的应用程序中进行操作。特别是，研究人员计划以他们最近的工作为基础，在这项工作中，他们已经观察到某些大振幅波的相干模式。他们在实验室实验中观察到这些模式，作为众所周知的水波方程的解，以及作为其他（更）近似水波模型的方程的解。他们的工作涉及各种数学和计算工具以及最先进的实验室实验。在目前的工作中，研究人员将结合他们所有的工具来理解和描述这些连贯的模式，并将它们作为实际海浪模型的基石。

The overall objectives of this work are to develop a thorough understanding of three-dimensional water waves of finite amplitude, and ultimately to develop a practical model to describe these waves efficiently. A model that is both accurate and computationally efficient could have many practical applications. Specific problems to be addressed are: (1) the existence and stability of three-dimensional, doubly-periodic, traveling water-wave patterns, through the full range of depths; (2) the prevalence of hexagonal, rectangular or crescent-shaped waves (or other multiply periodic wave patterns) among ocean waves; (3) the long-wave and modulational descriptions of water waves, and the subsequent stability analyses that are feasible in these cases; (4) the design and implementation of algorithms to make practical use of exact solutions of asymptotic models in shallow and deep water; (5) the relation between the detailed dynamics of three-dimensional, nonlinear waves and some commonly used ocean-wave transport models; and (6) the impact of a detailed local description of nonlinear wave dynamics on these transport models, in the presence of large amplitude nonlinear waves or under conditions of nonlinear wave focusing. These problems will be studied using analysis, computation, asymptotics, and algebraic geometry, involving the full equations and approximate models, all in conjunction with state-of-the-art physical experiments.The destructive force of large-amplitude ocean waves is well known. Large-scale ocean waves have a major impact on the design of ocean-going ships, of off-shore oil platforms, and of other structures in a coastal environment. These waves also impact the scheduling and routing of shipping patterns, and they strongly affect air-sea transport processes. Yet most theoretical models of ocean waves now in use are based on waves of small amplitude. In this investigation we focus on developing a thorough understanding of large-amplitude waves. The ultimate goal is to develop a practical, mathematical model that may be used operationally in the applications listed above. In particular, the investigators plan to build on their recent work in which they have observed certain coherent patterns of large-amplitude waves. They have observed these patterns in laboratory experiments, as solutions to the well-known equations of water waves, and as solutions to other equations that are (more) approximate models of water waves. Their work involves a variety of mathematical and computational tools as well as state-of-the-art laboratory experiments. In the present work the investigators will combine all of their tools to understand and describe these coherent patterns and to use them as the building blocks for a practical model of ocean waves.