Rays, waves and asymptotics

Rays , Waves and Asymptotics

Exponential Asymptotics and Capillary Waves

Recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow (for example, stagnation points and corners) meet it. The ampli...

Semiclassical Asymptotics for Weakly Nonlinear Bloch Waves

We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schrödinger equations with a fast periodic potential and a slowly varying confinement potential. A rigorous two-scale WKB–analysis, locally in time, is performed. The main nonlinear phenomenon is a modification of the Berry phase.

Asymptotics of gravitational collapse of scalar waves

Through a combination of analytic and numerical techniques, the formation of a black hole by a self-gravitating, spherically symmetric, massless scalar field is investigated. The evolution algorithm incorporates a Penrose compactification so that the Bondi mass, the news function, and other radiation zone limits can be obtained numerically. The late time behavior recently established by Christo...

Clusters of Galaxies: Shock Waves and Cosmic Rays

Recent observations of galaxy clusters in radio and X-ray indicate that cosmic rays and magnetic fields may be energetically important in the intracluster medium. According to the estimates based on theses observational studies, the combined pressure of these two components of the intracluster medium may range between 10% ∼ 100% of gas pressure, although their total energy is probably time depe...