Mathematical general relativity: a sampler

We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.Comment: Some updates. A shortened version, to appear in the Bulletin of the AMS, is available online at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2010-01304-

Numerical computation of travelling breathers in Klein-Gordon chains

We numerically study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by $p$ sites along the chain after a fixed propagation time $T$ (these solutions generalize the concept of solitary waves for which $p=1$). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently (G. James and Y. Sire, to appear in {\sl Comm. Math. Phys.}, 2004), the tail being exponentially small with respect to the central oscillation size. In this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein-Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition we numerically observe the existence of these solutions in the case of non even potentials

Is general relativity `essentially understood' ?

The content of Einstein's theory of gravitation is encoded in the properties of the solutions to his field equations. There has been obtained a wealth of information about these solutions in the ninety years the theory has been around. It led to the prediction and the observation of physical phenomena which confirm the important role of general relativity in physics. The understanding of the domain of highly dynamical, strong field configurations is, however, still quite limited. The gravitational wave experiments are likely to provide soon observational data on phenomena which are not accessible by other means. Further theoretical progress will require, however, new methods for the analysis and the numerical calculation of the solutions to Einstein's field equations on large scales and under general assumptions. We discuss some of the problems involved, describe the status of the field and recent results, and point out some open problems.Comment: Extended version of a talk which was to be delivered at the DPG Fruehjahrstagung in Berlin, 5 March 200

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Cumulative reports and publications through December 31, 1988

This document contains a complete list of ICASE Reports. Since ICASE Reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms