Abstract:
We study first order phase transitions in the gravitational collapse of spherically symmetric skyrmions. Static sphaleron solutions are shown to play the role of critical solutions separating black-hole spacetimes from no-black-hole spacetimes. In particular, we find a new type of first order phase transition where subcritical data do not disperse but evolve towards a static regular stable solution. We also demonstrate explicitly that the near-critical solutions depart from the intermediate asymptotic regime along the unstable manifold of the critical solution.

Abstract:
We consider the problem of asymptotic stability of a self-similar attractor for a simple semilinear radial wave equation which arises in the study of the Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In the first step we determine the spectrum of linearized perturbations about the attractor using a method of continued fractions. In the second step we demonstrate numerically that the resulting eigensystem provides an accurate description of the dynamics of convergence towards the attractor.

Abstract:
We present the numerical evidence for fractal threshold behavior in the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi type-IX ansatz. In other words, we show that a flip of the wings of a butterfly may influence the process of the black hole formation.

Abstract:
We consider the long-time behavior of small amplitude solutions of the semilinear wave equation $\Box \phi =\phi^p$ in odd $d\geq 5$ spatial dimensions. We show that for the quadratic nonlinearity ($p=2$) the tail has an anomalously small amplitude and fast decay. The extension of the results to more general nonlinearities involving first derivatives is also discussed.

Abstract:
We present results from a numerical study of spherically-symmetric collapse of a self-gravitating, SU(2) gauge field. Two distinct critical solutions are observed at the threshold of black hole formation. In one case the critical solution is discretely self-similar and black holes of arbitrarily small mass can form. However, in the other instance the critical solution is the n=1 static Bartnik-Mckinnon sphaleron, and black hole formation turns on at finite mass. The transition between these two scenarios is characterized by the superposition of both types of critical behaviour.

Abstract:
We continue our study of the gravitational collapse of spherically symmetric skyrmions. For certain families of initial data, we find the discretely self-similar Type II critical transition characterized by the mass scaling exponent $\gamma \approx 0.20$ and the echoing period $\Delta \approx 0.74$. We argue that the coincidence of these critical exponents with those found previously in the Einstein-Yang-Mills model is not accidental but, in fact, the two models belong to the same universality class.

Abstract:
We study the asymptotic behavior of spherically symmetric solutions in the Skyrme model. We show that the relaxation to the degree-one soliton (called the Skyrmion) has a universal form of a superposition of two effects: exponentially damped oscillations (the quasinormal ringing) and a power law decay (the tail). The quasinormal ringing, which dominates the dynamics for intermediate times, is a linear resonance effect. In contrast, the polynomial tail, which becomes uncovered at late times, is shown to be a \emph{nonlinear} phenomenon.

Abstract:
We discuss the nonlinear origin of the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in even-dimensional spacetimes. Using third-order perturbation method, we derive explicit expressions for the tail (the decay rate and the amplitude) for solutions starting from small initial data and we verify this prediction via numerical integration of the Einstein-scalar field equations in four and six dimensions. Our results show that the coincidence of decay rates of linear and nonlinear tails in four dimensions (which has misguided some tail hunters in the past) is in a sense accidental and does not hold in higher dimensions.

Abstract:
We consider the late-time tails of spherical waves propagating on even-dimensional Minkowski spacetime under the influence of a long range radial potential. We show that in six and higher even dimensions there exist exceptional potentials for which the tail has an anomalously small amplitude and fast decay. Along the way we clarify and amend some confounding arguments and statements in the literature of the subject.

Abstract:
This article investigates the interaction of a spherically symmetric massless scalar field with a strong gravitational field. It focuses on the propagation of waves in regions outside any horizons. The two factors acting on the waves can be identified as a redshift and a backscattering. The influence of backscattering on the intensity of the outgoing radiation is studied and rigorous quantitative upper bounds obtained. These show that the total flux may be decreased if the sources are placed in a region adjoining an apparent horizon. Backscattering can be neglected in the case $2m_0 /R<< 1$, that is when the emitter is located at a distance from a black hole much larger than the Schwarzschild radius. This backscattering may have noticeable astrophysical consequences.