Wave Propagation in Heterogeneous Nonlinear Dispersive Systems

该项目旨在开发数学工具，用于研究空间不均匀的物理系统中的波传播，主要是（a）海底地形周期性变化的水道中的表面水波和（B）成分周期性或随机变化的固体材料中的振动通道。例如，在设计用于非破坏性测试和减震的材料时会出现这种问题。在通道底部平坦或固体均匀的情况下，有成熟的定量和定性理论，这些理论在数学上都是严格的，并且在应用中常用。然而，异质性的存在（由于通道底部地形或材料中的夹杂物/缺陷）会导致意想不到的物理现象（例如，出现“锯齿状”的脉冲），本研究将有助于阐明和预测。调查的关键数学工具是以严格的方式适应Korteweg-de弗里斯（KdV）近似，使其适用于异构问题。这样做需要应用椭圆均匀化理论的方法来研究非线性色散系统。这种近似表明，非均匀系统存在解，粗略地说，这些解是孤立波。然而，典型的KdV近似结果仅在时间间隔上有效，该时间间隔虽然长，但具有有限的持续时间。这样的问题是否近似系统承认一个真正的全球时间对应孤立波是悬而未决的。的KdV近似将作为一个出发点的调查非常困难的问题的存在“广义行波”的异质系统。在存在空间周期系数的情况下，在移动参考系中静止的经典行波极不可能存在。相反，人们期望的是移位周期解，即在适当的移动标架中是时间周期的解。另一个复杂的是空间异质性，意外地进入问题作为一个奇异扰动。这样做的主要后果是，波不会在空间无穷远处收敛到零，而是接近极小振幅的空间周期解。这样的波具有无穷大的总能量，从而极大地复杂化了长时间渐近性的结构。特别是，这表明，虽然真正本地化的有限能量行波可能不存在，它们的亚稳态类似物可能。

This project is aimed at developing mathematical tools for the study of wave propagation in physical systems that are not spatially uniform, chiefly (a) surface water waves in a channel whose bottom topography varies periodically and (b) the passage of vibrations through a solid material whose composition varies either periodically or randomly. Problems of this kind arise, for example, in the design of materials for use in non-destructive testing and shock absorption. In the settings where the channel's bottom is flat or the solid is homogeneous, there are well-developed quantitative and qualitative theories which are both mathematically rigorous and are commonly used in applications. However, the presence of heterogeneity (due to the channel bottom topography or inclusions/defects in the material) leads to unexpected physical phenomena (for example, pulses that appear "jagged") that this research will help to elucidate and predict. The key mathematical tool for the investigation is to adapt the Korteweg-de Vries (KdV) approximation in a rigorous way so that it applies to heterogeneous problems. Doing so requires applying methods from elliptic homogenization theory to the study of nonlinear dispersive systems. This sort of approximation suggests that there are solutions to the heterogeneous systems which are, roughly speaking, solitary waves. However, a typical KdV approximation result is only valid on a time interval which, while long, is of finite duration. As such the question of whether or not the approximated system admits a genuine global in time counterpart to the solitary wave is left unresolved. The KdV approximation will serve as a point of departure for investigations into the very difficult question of the existence of "generalized traveling waves" for heterogeneous systems. In the presence of spatially periodic coefficients, classical traveling waves, which are static in a moving reference frame, are highly unlikely to exist. What one expects instead are shift-periodic solutions, i.e. solutions which, in an appropriate moving frame, are time periodic. An additional complication is the spatial heterogeneity that unexpectedly enters the problem as a singular perturbation. The main consequence of this is that the waves are not expected to converge to zero at spatial infinity but instead approach extremely small amplitude spatially periodic solutions. Such waves have infinite total energy, thus complicating the structure of long time asymptotics enormously. In particular this indicates that, while truly localized finite energy traveling waves may not exist, their metastable analogs may.