The project addresses the mathematical modeling and analysis of nonlinear, nondispersive wave propagation in continuum mechanics. It focuses on waves modeled by nonlinear hyperbolic PDEs, and related equations, that propagate along boundaries or interfaces (such as discontinuities in vorticity, vortex sheets, material boundaries, and shock waves). These surface waves often display a complex nonlocal, nonlinear behavior which is not well-understood. The principal investigator will derive and study reduced asymptotic equations that describe these waves in a range of physical applications. A typical feature of the resulting nonlocal quasilinear equations is that they are Hamiltonian, and they may be expressed in both spectral and spatial forms, leading to connections with multilinear harmonic analysis. Fundamental questions concerning these equations include the life-span of smooth solutions, the formation and physical interpretation of singularities, and the global existence of weak solutions.
Surface waves are waves that propagate along a boundary or interface. Since they are guided along an interface, they decay more slowly than bulk waves, which explains why the surface seismic waves generated by an earthquake are the most destructive far from their source. Surface waves are widely used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones or nanophotonic surface plasmon devices, because they are directly accessible to detection and manipulation. Small-amplitude waves are well-described by linear equations, but nonlinear effects become important at larger amplitudes and lead to qualitatively new phenomena such as the formation of singularities (for example, shock waves in a compressible fluid). Nonlinearity makes the mathematical analysis of these problems very challenging. An additional feature of surface waves is that the effects of nonlinearity may be nonlocal because what happens at one point on the surface can influence what happens elsewhere on the surface through the bulk medium. The principal investigator plans to study the fundamental qualitative properties of such nonlinear, nonlocal surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, including transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.
Waves of all kinds occur throughout nature and technology. For example, we use light waves to see, sound waves to hear, and radio waves to communicate. The physical examples of wave motions are diverse, but the underlying mathematics is the same. Most models of waves consist of partial differential equations, and for a large class of nondispersive wave motions -- including most of the waves studied in this project -- these equations are of hyperbolic type. At low intensities, waves behave linearly: the combined effect of many waves is simply the sum of their individual effects. At high intensities, waves behave nonlinearly, leading to new phenomena such as shock waves in a sonic boom. Nonlinear waves are much harder to analyze than linear waves because they cannot be decomposed into simpler continuant parts. This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of physical systems; the goal is to uncover new nonlinear phenomena in the context of specific applications. The project focuses especially on waves that propagate on surfaces or interfaces. The most familiar type of surface wave consists of the waves on a body of fluid, ranging from ripples in a coffee cup to tsunamis on the pacific ocean, but there are many other kinds of surface waves. One type of wave considered in the project consists of the Rayleigh waves that propagate on the surface of an elastic solid. These waves are generated by earthquakes (where they are often the most destructive seismic waves because they are guided along the surface of the earth) and on a much smaller scale they are used in analog filters for cell phones and other electronic devices. Another, closely related, type of magnetohydrodynamic surface wave propagates along tangential discontinuities in a plasma. Such discontinuities provide a model for the magnetopause or heliopause in the magnetic field of the earth or sun. The project also analyzes a relatively little studied class of nondispersive waves whose frequency is independent of their wavelength. By contrast, the much better studied class of nondispersive, hyperbolic waves consists of waves whose speed is independent of their wavelength. Physical examples of such constant-frequency waves include inertial oscillations in a rotating shallow fluid, which provides a model of geophysical flows in the ocean or atmosphere, and waves on a vorticity discontinuity in an inviscid, incompressible fluid flow. Specific outcomes of this project include: A study of surface waves on a tangential discontinuity in magnetohydrodynamics near the onset of Kelvin-Helmholtz instability, showing that nonlinearity causes an earlier onset of the instability than the one predicted by linearized theory. A study of two-dimensional shock diffraction, showing that nonlinearity enhances the diffraction. This problem is part of a long-term research program to better understand the propagation of shock waves in several space dimensions, as well as related problems in transonic flow, which has important applications in aerodynamics. The derivation of a new quasi-linear Schrodinger equation that describes the effect of weak pressure gradients on large amplitude inertial oscillations in a rotating shallow fluid. The development of novel normal form methods that apply to quasi-linear wave equations. In particular, they give a proof that constant-frequency oscillations delay steepening and singularity-formation in a Burgers-Hilbert equation that describes surface waves on a vorticity discontinuity. Two graduate students were supported by this award and received their Ph.D.'s during the award period. One is now working as a mathematician for the U.S. Navy, and the other has a postdoctoral position.
