Nonlinear hyperbolic waves and interfaces

University Of California-Davis, Davis CA

Investigators

Abstract

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.