Wave Propagation in Heterogeneous Nonlinear Dispersive Systems

该项目的目的是开发数学工具，用于研究空间上不均匀的物理系统中的波传播，主要是(A)底部地形周期性变化的渠道中的表面水波，以及(B)振动通过成分周期性或随机变化的固体材料。例如，在无损检测和减震材料的设计中就会出现这类问题。在沟槽底部平坦或固体均匀的环境中，有成熟的定量和定性理论，这些理论在数学上都是严格的，在应用中也是常用的。然而，异质性的存在(由于沟道底部的地形或材料中的夹杂物/缺陷)导致了意想不到的物理现象(例如，脉冲看起来“参差不齐”)，这项研究将有助于阐明和预测。研究的关键数学工具是以严格的方式调整Korteweg-de Vries(KdV)近似，使其适用于异质问题。要做到这一点，需要将椭圆齐次化理论的方法应用到非线性色散系统的研究中。这种近似表明，非均匀系统存在解，粗略地说，这些系统是孤立波。然而，典型的KdV近似结果仅在时间间隔上有效，虽然时间间隔很长，但具有有限的持续时间。因此，近似的系统是否在时间上接纳了孤立波的真正的全球对应物的问题尚未解决。KdV近似将作为研究非均质系统存在“广义行波”这一非常困难的问题的起点。在空间周期系数的存在下，在运动坐标系中静止的经典行波是极不可能存在的。相反，人们期望的是移位周期解，即在适当的移动帧中的解是时间周期的。另一个复杂的问题是空间异质性，它出人意料地以奇异扰动的形式进入这个问题。这样做的主要结果是，预计波不会在空间无穷远处收敛到零，而是接近极小幅度的空间周期解。这样的波具有无限的总能量，因此极大地使长时间渐近的结构复杂化。特别是，这表明，虽然真正局域的有限能量行波可能不存在，但它们的亚稳态类似物可能存在。

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.