Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

Chiral CP^2 skyrmions in three-band superconductors

It is shown that under certain conditions, three-component superconductors (and in particular three-band systems) allow stable topological defects different from vortices. We demonstrate the existence of these excitations, characterized by a $CP^2$ topological invariant, in models for three-component superconductors with broken time reversal symmetry. We term these topological defects "chiral $GL^{(3)}$ skyrmions", where "chiral" refers to the fact that due to broken time reversal symmetry, these defects come in inequivalent left- and right-handed versions. In certain cases these objects are energetically cheaper than vortices and should be induced by an applied magnetic field. In other situations these skyrmions are metastable states, which can be produced by a quench. Observation of these defects can signal broken time reversal symmetry in three-band superconductors or in Josephson-coupled bilayers of $s_\pm$ and s-wave superconductors.Comment: minor presentation changes; replaced journal version; 30 pages, 21 figure

Continuum Limit of 2D Fractional Nonlinear Schr\"odinger Equation

We prove that the solutions to the discrete Nonlinear Schr\"odinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schr\"odinger Equation (FNLSE), as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $\alpha \in (1,2)$ approaches the boundaries.Comment: Revised Articl

Weighted inequalities in Fluid mechanics and general relativity: Carleman estimates and cusped travelling waves

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-07-2019Esta tesis tiene embargado el acceso al texto completo hasta el 16-01-202

Super-Eddington Atmospheres that Don't Blow Away

We show that magnetized, radiation dominated atmospheres can support steady state patterns of density inhomogeneity that enable them to radiate at far above the Eddington limit, without suffering mass loss. The inhomogeneities consist of periodic shock fronts bounding narrow, high-density regions, interspersed with much broader regions of low density. The radiation flux avoids the regions of high density, which are therefore weighed down by gravity, while gas in the low-density regions is slammed upward into the shock fronts by radiation force. As the wave pattern moves through the atmosphere, each parcel of matter alternately experiences upward and downward forces, which balance on average. Magnetic tension shares the competing forces between regions of different densities, preventing the atmosphere from blowing apart. We calculate the density structure and phase speed of the wave pattern, and relate these to the wavelength, the density contrast, and the factor by which the net radiation flux exceeds the Eddington limit. In principle, this factor can be as large as the ratio of magnetic pressure to mean gas pressure, or the ratio of radiation pressure to gas pressure, whichever is smaller. Although the magnetic pressure must be large compared to the mean gas pressure in order to support a large density contrast, it need not be large compared to the radiation pressure. These highly inhomogeneous flows could represent the nonlinear development of the "photon bubble" instability discovered by Gammie. We briefly discuss the applicability of these solutions to astrophysical systems.Comment: 11 pages, 1 figure, accepted for publication in The Astrophysical Journa

NEUTROSOPHIC LOGIC, WAVE MECHANICS, AND OTHER STORIES

There is beginning for anything; we used to hear that phrase. The same wisdom word applies to the authors too. What began in 2005 as a short email on some ideas related to interpretation of the Wave Mechanics results in a number of papers and books up to now. Some of these papers can be found in Progress in Physics or elsewhere. It is often recognized that when a mathematician meets a physics-inclined mind then the result is either a series of endless debates or publication. In this story, authors preferred to publish rather than perish. Therefore, the purpose with this book is to present a selection of published papers in a compilation which enable the readers to find some coherent ideas which appear in those articles. For this reason, the ordering of the papers here is based on categories of ideas

Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black spacetime. A prime application of characteristic evolution is to compute waveforms via Cauchy-characteristic matching, which is also reviewed.Comment: Published version http://www.livingreviews.org/lrr-2005-1

Soliton interactions and bound states

The research presented in this thesis is concerned with soliton interactions and bound states. We consider a on-topological soliton in (1 + 1) dimensions and topological models in (2 + 1) and (3 + 1) dimensions. In chapter 2 we consider Qballs, which are non-topological solitons, in (1 + 1) dimensions. Here we note the semi-integrable behaviour of small-charge Qballs. This leads us to propose a possible mechanism to explain the two distinct oscillatory modes of a Qball breather. In chapter 3 we are interested in the (2+1)-dimensional baby-skyrme model, which is a lower-dimensional analogue of the Skyrme theory. We discover new chain-like bound-state minimum-energy solutions. We then analyse whether these solutions are the minimum-energy solutions on a cylinder, and then finally on the torus. In chapter 4 we discuss a new (2 + 1)-dimensional model containing a baby skyrmion coupled to a vector meson. This is an analogue of the (3 + 1)-dimensional Skyrme theory containing a vector meson. We use this lower-dimensional analogue to numerically justify the use of a rational map ansatz in the analysis of the (3 + 1)-dimensional skyrmion. Also we analytically prove why the baby-skyrme model, and the model containing a baby skyrmion stabilised by a vector meson, have very similar solutions. Chapter 5 discusses Hopf solitons. Instead of being lumps, Hopf solitons actually resemble loops of string. Their charge is related to the string's knotting and twisting. In this chapter we include an extra mass term in the Skyrme-Faddeev theory; this gives solitons which are exponentially localised. We then explore the infinite-coupling case, which gives compact Hopons. This chapter is part of an ongoing investigation. All of the original research results presented are my own results

Numerical methods for finding stationary gravitational solutions

The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS. We also include several tools and tricks that have been useful throughout the literature