Collaborative Research: Nonlinear Dispersive Waves With Weak Dissipation

许多物理系统都允许非线性色散波存在，其传播速度取决于波的频率和振幅。这些系统通常是耗散的，但耗散可能很弱。本文主要研究存在弱耗散的非线性色散波，特别强调了表面水波。这一提议的pi和其他人最近的工作表明，在某些情况下，即使是微弱的耗散也会深刻地影响非线性色散波的稳定性。本提案的总体主题是研究弱耗散存在下的非线性色散波。我们将使用最先进的实验室设备在深水中进行波浪实验。这些实验将指导数学模型的发展并检验其有效性。我们将探讨的问题包括(a)弱耗散如何影响比普通平面波更复杂的表面波的稳定性和性质？(b)在水波和光学实验中观察到的非线性色散波的窄带谱峰的降移原因是什么？(c)这项理论和实验室工作是否适用于实际的海浪？(d)是什么导致了表面水波的耗散？这项研究的具体目标包括获得大振幅海浪动力学的更好的预测模型，以及发生在海洋-大气边界的交换过程。这些交换过程的精确模型对于预测全球变暖和相关的全球现象至关重要。此外，精确的波浪动力学模型可能有助于理解极端波浪，如异常浪。许多要进行的工作在本质上是基础的，许多物理系统都使用数学，因此对水波获得的结果可能会影响我们对许多物理系统的理解。这些系统包括光纤中的光波，磁性薄膜中的自旋波等。

Many physical systems admit nonlinear, dispersive waves, for which the speed of propagation depends on both the frequency and amplitude of the wave. These systems are usually dissipative, but the dissipation might be weak. This proposal focusses on nonlinear dispersive waves in the presence of weak dissipation, with special emphasis on surface water waves. Recent work by the PIs of this proposal and others has shown that even weak dissipation can profoundly affect the stability of nonlinear dispersive waves in certain situations. The overall theme of this proposal is to study nonlinear dispersive waves in the presence of weak dissipation. We will use a state-of-the-art laboratory facility to conduct experiments on waves on deep water. These experiments will guide the development of mathematical models and test their validity. Questions we will explore include (a) How does weak dissipation affect the stability and properties of surface wave patterns that are more complicated than ordinary plane waves? (b) What causes the downshifting of the peak of a narrow-banded spectrum of nonlinear, dispersive waves, as observed in experiments on water waves and in optics? (c) Does this theoretical and laboratory work apply to actual ocean waves? (d) What causes the dissipation in surface water waves?Particular goals of this research include obtaining better predictive models of the dynamics of large-amplitude ocean waves, and of exchange processes that occur at the ocean-atmosphere boundary. Accurate models of these exchange processes will be essential in predicting global warming and related global phenomena. In addition, accurate models of wave dynamics might prove useful in understanding extreme waves such as rogue waves. Much of the work to be conducted is fundamental in nature and the mathematics is shared by many physical systems, so the results obtained for water waves could affect our understanding of many physical systems. These other systems include light waves in an optical fiber, spin waves in a thin magnetic film and others.