General Initial Conditions Research Articles

Excited hairy black holes: Dynamical construction and level transitions

We study the dynamics of unstable Reissner-Nordstr\"om anti-de Sitter black holes under charged scalar field perturbations in spherical symmetry. We unravel their general behavior and approach to the final equlibrium state. In the first part of this work, we present a numerical analysis of massive charged scalar field quasinormal modes. We identify the known mode families - superradiant modes, zero-damped modes, AdS modes, and the near-horizon mode - and we track their migration under variation of the black hole and field parameters. We show that the zero-damped modes become superradiantly unstable for large RNAdS with large gauge coupling; the leading unstable mode is identified with the near-horizon condensation instability. In the second part, we present results of numerical simulations of perturbed large RNAdS, showing the nonlinear development of these unstable modes. For generic initial conditions, charge and mass are ransferred from the black hole to the scalar field, until an equilibrium solution with a scalar condensate is reached. We use results from the linear analysis, however, to select special initial data corresponding to an unstable overtone mode. We find that these data evolve to produce a new equilibrium state - an excited hairy black hole with the scalar condensate in an overtone configuration. This state is, however, unstable, and the black hole eventually decays to the generic end state. Nevertheless, this demonstrates the potential relevance of overtone modes as transients in black hole dynamics.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

Abstract Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

On the Metastability of the 1-d Allen\u2013Cahn Equation

We apply an energy method for metastability, developed in an earlier work with Otto, to the Allen–Cahn equation on the line and a broad class of initial data. In the earlier work, we for simplicity considered the equation on (0,L)times (0,infty ) with periodic boundary conditions and an initial condition with two simple zeros. In this paper we explain the implications of the metastability framework (slightly modified as in previous joint work with Scholtes) for the equation on {mathbb {R}}times (0,infty ) and a more general initial condition. Our goal is to make clear the strength of the metastability framework and to highlight the difference in the analysis between the second-order Allen–Cahn equation and the fourth-order Cahn–Hilliard equation.

Dynamics of closed singularities

Les flots géométriques paraboliques ont la propriété de régulariser en temps court, néanmoins, en temps long, l’apparition de singularités est inévitable et elle peuvent être compliquées et impossibles à classifier. L’idée qui sous-tend cet article est que l’utilisation des propriétés dynamiques du flot permet d’obtenir une régularisation en temps long aussi, pour des conditions initiales génériques. Combiné avec un article récent, il nous permet de montrer, dans un cas particulier important, que les singularités sont les plus simples possibles.

Fluorescence recovery after photobleaching: direct measurement of diffusion anisotropy.

Fluorescence recovery after photobleaching (FRAP) is a widely used technique for studying diffusion in biological tissues. Most of the existing approaches for the analysis of FRAP experiments assume isotropic diffusion, while only a few account for anisotropic diffusion. In fibrous tissues, such as articular cartilage, tendons and ligaments, diffusion, the main mechanism for molecular transport, is anisotropic and depends on the fibre alignment. In this work, we solve the general diffusion equation governing a FRAP test, assuming an anisotropic diffusivity tensor and using a general initial condition for the case of an elliptical (thereby including the case of a circular) bleaching profile. We introduce a closed-form solution in the spatial coordinates, which can be applied directly to FRAP tests to extract the diffusivity tensor. We validate the approach by measuring the diffusivity tensor of [Formula: see text] FITC-Dextran in porcine medial collateral ligaments. The measured diffusion anisotropy was [Formula: see text] (SE), which is in agreement with that reported in the literature. The limitations of the approach, such as the size of the bleached region and the intensity of the bleaching, are studied using COMSOL simulations.

Interplay Between Homophily-Based Appraisal Dynamics and Influence-Based Opinion Dynamics: Modeling and Analysis

In social systems, the evolution of interpersonal appraisals and individual opinions are not independent processes but intertwine with each other. Despite extensive studies on both opinion dynamics and appraisal dynamics separately, no previous work has ever combined these two processes together. In this letter, we propose a novel and intuitive model on the interplay between homophily-based appraisal dynamics and influence-based opinion dynamics. We assume that individuals' opinions are updated via the influence network constructed from their interpersonal appraisals, which are in turn updated based on the individual opinions via the homophily mechanism. By theoretical analysis, we characterize the set of equilibria and some transient behavior of our model. Moreover, we establish the equivalence among the convergence of the appraisal network to social balance, the modulus consensus of individual opinions, and the non-vanishing appraisals. Monte Carlo validations further show that the non-vanishing appraisals condition holds for generic initial conditions. Compared with previous works that explain the emergence of social balance via person-to-person homophily mechanism, our model provides an alternative explanation in terms of the person-to-entity homophily mechanism. In addition, our model also describes how individuals' opinions on multiple irrelevant issues become correlated and converge to modulus consensus over time-varying influence networks.

Asymptotic behavior of solutions of nonautonomous neutral dynamical systems

This paper studies the dynamics of families of monotone nonautonomous neutral functional differential equations with nonautonomous operator, of great importance for their applications to the study of the long-term behavior of the trajectories of problems described by this kind of equations, such us compartmental systems and neural networks among many others. Precisely, more general admissible initial conditions are included in the study to show that the solutions are asymptotically of the same type as the coefficients of the neutral and non-neutral part.

Renormalizing the Kardar\u2013Parisi\u2013Zhang Equation in dge 3 in Weak Disorder

We study Kardar–Parisi–Zhang equation in spatial dimension 3 or larger driven by a Gaussian space–time white noise with a small convolution in space. When the noise intensity is small, it is known that the solutions converge to a random limit as the smoothing parameter is turned off. We identify this limit, in the case of general initial conditions ranging from flat to droplet. We provide strong approximations of the solution which obey exactly the limit law. We prove that this limit has sub-Gaussian lower tails, implying existence of all negative (and positive) moments.

Integrodifferential evolution systems with nonlocal initial conditions

The paper deals with systems of abstract integrodifferential equations subject to general nonlocal initial conditions. In order to allow the nonlinear terms of the equations to behave independently as much as possible, we use a vector approach based on matrices, vector-valued norms and a vector version of Krasnoselskii's fixed point theorem for a sum of two operators. The assumptions take into account the support of the nonlocal initial conditions and the hybrid character of the system. Two examples are given to illustrate the theory.

Cosmic censorship and the evolution of d -dimensional charged evaporating black holes

The cosmic censorship conjecture essentially states that naked singularities should not form from generic initial conditions. Since black hole parameters can change their values under Hawking evaporation, one has to ask whether it is possible to reach extremality by simply waiting for the black hole to evaporate. If so a slight perturbation would likely render the singularity naked. Fortunately, at least for the case of asymptotically flat 4-dimensional Reissner-Nordstr\"om black hole, Hiscock and Weems showed that it can never reach extremality despite the fact that for a sufficiently massive black hole, its charge-to-mass ratio can increase during Hawking evaporation. Hence cosmic censorship is never violated by Hawking emission. However, we know that under some processes, it is easier to violate cosmic censorship in higher dimensions, therefore it is crucial to generalize Hiscock and Weems model to dimensions above four to check cosmic censorship. We found that Hawking evaporation cannot lead to violation of cosmic censorship even in higher dimensional Reissner-Nordstr\"om spacetimes. Morerover, it seems to be more difficult to reach extremality as number of dimension increases.

Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula.

We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.

Generalized hydrodynamic approach to charge and energy currents in the one-dimensional Hubbard model

We have studied nonequilibrium dynamics of the one-dimensional Hubbard model using the generalized hydrodynamic theory. We mainly investigated the spatiotemporal profile of particle density (equivalent to charge density), energy density, and their currents using the partitioning protocol; the initial state consists of two semi-infinite different thermal equilibrium states joined at the origin. In this protocol, a transient region where currents flow appears around the origin, and this region expands its size linearly with time. We examined how density and current profiles depend on initial conditions. Inside the transient region, we have found a clogged region where charge current is zero but nonvanishing energy current flows. This phenomenon is similar to spin-charge separation in Tomonaga-Luttinger liquids, in which spin and charge excitations propagate with different velocities. This region appears when one of the initial states has half-filled electron density, and it is located adjacent to this initial state. We have proved analytically the existence of the clogged region in the infinite temperature case of the half-filled initial state. The existence is confirmed also for finite temperatures by numerical calculations of generalized hydrodynamics. A similar analytical proof is also given for a clogged region of spin current when magnetic field is applied to one and only one of the two initial states. A universal proportionality of charge and spin currents is also proved for a special region, for general initial conditions of electron density and magnetic field. To examine the clogged region of charge current, we have calculated the current contributions of different types of quasiparticles in the Hubbard model. It is found that the charge current component carried by scattering states (i.e., Fermi liquid type) is canceled completely in the clogged region by the counter flows carried by bound-state quasiparticles (called $k\text{\ensuremath{-}}\mathrm{\ensuremath{\Lambda}}$ strings). Their contributions are not canceled for energy current, which is also proved analytically for special cases, and these different behaviors are the origin of the clogged region. Except for the clogged region, charge and energy densities are in good proportion to each other, and their ratio depends on the initial conditions. This proportionality, in nonequilibrium dynamics, is reminiscent of Wiedemann-Franz law in thermal equilibrium, which states current proportionality determined by temperature. The long-time stationary values of charge and energy currents were also studied with varying initial conditions. We have compared the results with the values of noninteracting systems and discussed the effects of electron correlations. The ratio of these stationary currents is also analyzed. We have found that the temperature dependence of the ratio is strongly suppressed by electron correlations and even reversed.

РОЗВ’ЯЗУВАНIСТЬ ОСЕРЕДНЕНИХ ЗАДАЧ У ЗГОРТКАХ ДЛЯ СЛАБО ПОРИСТИХ СЕРЕДОВИЩ

Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.

On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions

In weak turbulence theory, the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number $|p|=\infty$ is $0$, as time evolves, the energy is gradually accumulated at $\{|p|=\infty\}$. Finally, all the energy of the system is concentrated at $\{|p|=\infty\}$ and the energy function becomes a Dirac function at infinity $E\delta_{\{|p|=\infty\}}$, where $E$ is the total energy. The existence of this class of solutions is, in some sense, {the first complete rigorous} mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well.

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics

The incompressible 2D Euler equations on a sphere constitute a fundamental model in hydrodynamics. The long-time behaviour of solutions is largely unknown; statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness. Such numerical results were obtained using artificial hyperviscosity to account for the cascade of enstrophy into smaller scales. Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour. Here we develop an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs. Long-time simulations on a non-rotating sphere then reveal three possible outcomes for generic initial conditions: the formation of either 2, 3, or 4 coherent vortex structures. These numerical results contradict the statistical mechanics theory and show that previous numerical results, suggesting 4 coherent vortex structures as the generic behaviour, display only a special case. Through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describes the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral $\gamma$ (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: $\gamma$ small, intermediate, and large yields most likely 4, 3, or 2 coherent vortex formations. Our findings thus suggest that the likely long-time behaviour can be predicted from the first integral $\gamma$.

Scaling behavior of the stationary states arising from dissipation at continuous quantum transitions

We study the critical behavior of the nonequilibrium dynamics and of the steady states emerging from the competition between coherent and dissipative dynamics close to quantum phase transitions. The latter is induced by the coupling of the system with a Markovian bath, such that the evolution of the system's density matrix can be effectively described by a Lindblad master equation. We devise general scaling behaviors for the out-of-equilibrium evolution and the stationary states emerging in the large-time limit for generic initial conditions in terms of the parameters of the Hamiltonian providing the coherent driving and those associated with the dissipative interactions with the environment. Our framework is supported by numerical results for the dynamics of a one-dimensional lattice fermion gas undergoing a quantum Ising transition in the presence of dissipative mechanisms which include local pumping and decay of particles.

Modulus of continuity of polymer weight profiles in Brownian last passage percolation

In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0)\in \mathbb{R}^{2}$, and the other is varied horizontally, over $(z,1)$, $z\in \mathbb{R}$, so that the polymer weight profile is a function of $z\in \mathbb{R}$. This profile is known to manifest a one-half power law, having $1/2$-Hölder continuity. The polymer weight profile may be defined beginning from a much more general initial condition. In this article, we present a more general assertion of this one-half power law, as well as a bound on the polylogarithmic correction. The polymer weight profile admits a modulus of continuity of order $x^{1/2}(\log x^{-1})^{2/3}$, with a high degree of uniformity in the scaling parameter and over a very broad class of initial data.

Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions

We investigate the Vlasov-Poisson equations perturbed by a strong external uniform magnetic field. We study the asymptotic behavior of the solutions, based on averaging techniques. We analyze the case of general initial conditions. By filtering out the oscillations, we are led to a profile. We prove strong convergence results and establish second order estimates.

Dynamic social balance and convergent appraisals via homophily and influence mechanisms

Social balance theory describes allowable and forbidden configurations of the topologies of signed directed social appraisal networks. In this paper, we propose two discrete-time dynamical systems that explain how an appraisal network converges to social balance from an initially unbalanced configuration. These two models are based on two different socio-psychological mechanisms, respectively: the homophily mechanism and the influence mechanism. Our main theoretical contribution is a comprehensive analysis for both models in three steps. First, we establish the well-posedness and bounded evolution of the interpersonal appraisals. Second, we fully characterize the set of equilibrium points; for both models, each equilibrium network is composed of an arbitrary number of complete subgraphs satisfying structural balance. Third, we establish the equivalence among three distinct properties: non-vanishing appraisals, convergence to all-to-all appraisal networks, and finite-time achievement of social balance. In addition to theoretical analysis, Monte Carlo validations illustrate how the non-vanishing appraisal condition holds for generic initial conditions in both models. Moreover, a numerical comparison between the two models indicates that the homophily-based model might be a more universal explanation for the emergence of social balance. Finally, adopting the homophily-based model, we present numerical results on the mediation and globalization of local conflicts, the competition for allies, and the asymptotic formation of a single versus two factions.

On the Boundary Value Problem for Functional Differential Inclusion of Fractional Order with General Initial Condition in a Banach Space

We consider the problem for a functional differential inclusion of fractional order with a general initial condition expressed in the form of an operator inclusion in a Banach space. At the beginning of the paper, an introduction is presented in which the relevance of the study is substantiated, then preliminary information from fractional analysis, the theory of measures of noncompactness and condensing mappings, as well as some information from multivalued analysis are given. In the second subsection, we state the problem and its solution on the basis of the theory of condensing multivalued mappings. In the last subsection, we give an example of a particular case of the solved problem, in the case of an antiperiodic boundary condition.