Quasi-linear hyperbolic and surface waves

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This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of physical systems. It focuses on nondispersive waves, especially surface waves that propagate on interfaces such as discontinuities in vorticity, vortex sheets, material boundaries, water waves, and shock waves. Many of the wave motions considered in the proposal have constant, nonzero frequency in the linearized limit. These waves form a comparatively little studied class of nondispersive waves, and the proposed research aims to develop an understanding of their nonlinear dynamics, which is qualitatively different from that of dispersive waves or nondispersive hyperbolic waves. The proposed research will derive and study asymptotic descriptions of these waves and will also develop normal form transformations for quasi-linear wave equations. Hamiltonian dynamics provides unifying framework for most of the nonlinear wave motions to be studied in the proposed research. For small-amplitude waves, this description is more easily carried out in spectral form, which is particularly appropriate for the surface waves considered in the proposed research because of the spatial nonlocality of their interactions. The issue of understanding the relationship between the spectral and spatial descriptions of the resulting nonlinear dynamics is a fundamental one and one that is relevant to many other problems. A further topic of the proposed research is a study of the glancing Mach reflection of shock waves. Shock reflection is one of the most important multi-dimensional problems for hyperbolic conservation laws, leading to remarkably interesting and complex phenomena.These results should also shed light on related problems in transonic aerodynamics.Surface waves are waves that propagate along a boundary or interface. The most familiar example consists of the water waves on the surface of a body of water, like an ocean. Another type of surface wave consists of the Rayleigh waves on a solid interface. These waves are generated by earthquakes, and they are also used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones. A further example consists of the electromagnetic surface waves, or surface plasmons, on the interface between a metal and an insulator, which find applications in photonics. Small-amplitude waves are well-described by linear equations, but at larger amplitudes nonlinear effects become important. These effects lead to qualitatively new phenomena such as wave-breaking, the formation of shock waves or other singularities, and the generation of new waves by nonlinear wave-interactions. Nonlinearity, and the possibility of a free surface that moves with the wave, 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 surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.