Wave Propagation in Heterogeneous Nonlinear Dispersive Systems

Drexel University, Philadelphia PA

Investigators

Abstract

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.