Flat Galaxies with Dark Matter Halos—Existence and Stability

We consider a model for a flat, disk-like galaxy surrounded by a halo of dark matter, namely a Vlasov-Poisson type system with two particle species, the stars which are restricted to the galactic plane and the dark matter particles. These constituents interact only through the gravitational potential which stars and dark matter create collectively. Using a variational approach we prove the existence of steady state solutions and their nonlinear stability under suitably restricted perturbations.     

Rotating black branes in the presence of nonlinear electromagnetic field

In this paper, we consider a class of gravity whose action represents itself as a sum of the usual Einstein–Hilbert action with cosmological constant and an U(1) gauge field for which the action is given by a power of the Maxwell invariant. We present a class of rotating black branes with Ricci flat horizon and show that the presented solutions may be interpreted as black brane solutions with two event horizons, extreme black hole and naked singularity provided the parameters of the solutions are chosen suitably. We investigate the properties of the solutions and find that for the special values of the nonlinear parameter, the solutions are not asymptotically anti-de Sitter. Finally, we obtain the conserved quantities of the rotating black branes and find that the nonlinear source has effects on the electric field, the behavior of spacetime, the type of singularity and other quantities.     

Nonlinear r-modes in rapidly rotating relativistic stars

The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications, provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3D general-relativistic hydrodynamical evolutions. We find that (1) on dynamical time scales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one-the maximum r-mode amplitude is of order unity. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes. (3) The kinematical drift associated with r-modes appears to be present in our simulations, but confirmation requires more precise initial data.     

Magnetic Dilaton Rotating Strings in the Presence of Exponential Nonlinear Electrodynamics

In this paper, we construct a new class of four-dimensional spinning magnetic dilaton string solutions which produces a longitudinal nonlinear electromagnetic field. The Lagrangian of the matter field has the exponential form. We study the physical properties of the solution in ample details. Geometrical, causal and geodisical structures of the solutions are investigated, separately. We confirm that the spacetime is both null and geodesically complete. We find that these solutions have no curvature singularity and no horizon, but have a conic geometry. We investigate the effects of variation of charge and the intensity of the dilaton field, on the deficit angle. Due to the presence of the dilaton field, the asymptotic behavior of the solutions are neither flat nor (anti-) de Sitter [(A)dS]. Furthermore, we extend our study to the higher dimensions and obtain the (n+1)-dimensional magnetic rotating dilaton strings with k≤[n/2] rotation parameters and calculate conserved quantities of the solutions. Although these solutions are not asymptotically (A)dS, we use counterterm method to calculate conserved quantities. We also calculate electric charge and show that the net electric charge of the spinning string is proportional to the rotating parameter and the electric field only exists when the rotation parameter does not vanish.     

Differentially Rotating Disks of Dust

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer—Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Bäcklund transformations to appropriate seed solutions of the Weyl class.     

Mass and stability of rigidly rotating relativistic polytropes by energy method

The energy of a rigidly rotating star is written in the first and second post-Newtonian approximation. Conditions for equilibrium and stability are derived, as well as evolutionary paths for stars shedding angular momentum or mass. For polytropic index n > 1, these analytic results agree with exact numerical results to within a few percent as long as the general relativity index (P/c²)_c < 0.1. The energy method is applied to low mass white dwarfs, to semirelativistic neutron stars and to supermassive stars.     

Rotating Kaluza-Klein monopoles and dyons

Conformastationary solutions of the five-dimensional vacuum Einstein equations, depending on one or two harmonic potentials, are constructed. We thus obtain solutions describing systems of rotating electric or magnetic monopoles, as well as rotating dyon solutions, satisfying the principle of equivalence.     

Traveling-waves and solitons in a generalized time-variable coefficients nonlinear Schrödinger equation with higher-order terms

In this work, we study exact solutions of a generalized nonautonomous cubic–quintic nonlinear Schrödinger equation with higher-order terms, and the dispersion and the nonlinear coefficients engendering temporal dependency. Similarity transformations are used to convert the nonautonomous equation into autonomous one and then we present solutions in a general way. These solutions are obtained for the first class by using the F-expansion method and for the second class constituted by most general bright, dark and front by a direct substitution. We also generalize the external potential which traps the system and the nonlinearities. Finally, the stability of the soliton solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed analytically and numerically. The results reveal that solitons can propagate in a stable way under slight disturbance of the constraint conditions and initial perturbation of a 10% white noise.     

The stability of relativistic stars and the role of the adiabatic index

We study the stability of three analytical solutions of the Einstein’s field equations for spheres of fluid. These solutions are suitable to describe compact objects including white dwarfs, neutron stars and supermassive stars and they have been extensively employed in the literature. We re-examine the range of stability of the Tolman VII solution, we focus on the stability of the Buchdahl solution which is under contradiction in the literature and we examine the stability of the Nariai IV solution. We found that all the mentioned solutions are stable in an extensive range of the compactness parameter. We also concentrate on the effect of the adiabatic index on the instability condition. We found that the critical adiabatic index, depends linearly on the ratio of central pressure over central energy density \(P_c/{\mathcal{E}}_c\), up to high values of the compactness. Finally, we examine the possibility to impose constraints, via the adiabatic index, on realistic equations of state in order to ensure stable configurations of compact objects.     

Heisenberg Model in a Rotating Magnetic Field

We study the Heisenberg model under the influence of a rotating magnetic field. By using a time-dependent unitary transformation, the time evolution operator for the Schrödinger equation is obtained, which involves no chronological product. The spin vectors (mean values of the spin operators) are obtained as explicit functions of time in the most general case. A series of cyclic solutions are presented. The nonadiabatic geometric phases of these cyclic solutions are calculated, and are expressed in terms of the solid angle subtended by the closed trace of the total spin vector, as well as in terms of those of the individual spins.     

Analytical periodic motions in a parametrically excited, nonlinear rotating blade

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated.     

An improved symplectic precise integration method for analysis of the rotating rigid–flexible coupled system

This paper presents an improved symplectic precise integration method (PIM) to increase the accuracy and keep the stability of the computation of the rotating rigid–flexible coupled system. Firstly, the generalized Hamilton's principle is used to establish a coupled model for the rotating system, which is discretized and transferred into Hamiltonian systems subsequently. Secondly, a suitable symplectic geometric algorithm is proposed to keep the computational stability of the rotating rigid–flexible coupled system. Thirdly, the idea of PIM is introduced into the symplectic geometric algorithm to establish a symplectic PIM, which combines the advantages of the accuracy of the PIM and the stability of the symplectic geometric algorithm. In some sense, the results obtained by this method are analytical solutions in computer for a long span of time, so the time-step can be enlarged to speed up the computation. Finally, three numerical examples show the stability of computation, the accuracy of solving stiff equations and the capability of solving nonlinear equations, respectively. All these examples prove the symplectic PIM is a promising method for the rotating rigid–flexible coupled systems.     

Stable Directions for Small Nonlinear Dirac Standing Waves

We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schrödinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation     

Frame Dragging Effect on Properties of Rotating Neutron Stars with Strong Magnetic Field

The general relativistic frame dragging effect on the properties, such as the moments of inertia and the radii of gyration of fast rotating neutron stars with a uniform strong magnetic field, is calculated accurate to the first order in the uniform angular velocity. The results show that compared with the corresponding non-rotating static spherical symmetric neutron star with a weaker magnetic field, a fast rotating neutron star (millisecond pulsar) with a stronger magnetic field has a relative smaller moment of inertia and radius of gyration.     

Nonlinear torsional oscillations in rotating systems

Torsional oscillations, temporal variations in the differential rotation in rotating systems, are a well-established phenomenon in the fluid regions of convective stars and planets. We show for the first time that nonlinear torsional oscillations can be generated and maintained by convective instabilities in rotating systems.     

Isotropic Steady States in Galactic Dynamics

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov--Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against {general}, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy. Received: 24 October 2000 / Accepted: 7 January 2001     

Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.     

Brane-world stars from minimal geometric deformation, and black holes

Using the effective four-dimensional Einstein field equations, we build analytical models of spherically symmetric stars in the brane-world, in which the external space-time contains both an ADM mass and a tidal charge. In order to determine the interior geometry, we apply the principle of minimal geometric deformation, which allows one to map general relativistic solutions to solutions of the effective four-dimensional brane-world equations. We further restrict our analysis to stars with a radius linearly related to the total general relativistic mass, and obtain a general relation between the latter, the brane-world ADM mass and the tidal charge. In these models, the value of the star’s radius can then be taken to zero smoothly, thus obtaining brane-world black hole metrics with a tidal charge solely determined by the mass of the source and the brane tension. We find configurations which entail a partial screening of the gravitational mass, and general conclusions regarding the minimum mass for semiclassical black holes are also drawn.     

Asymptotic Analysis of Conserved Densities Evaluated on Invariant Solutions Associated with Large Scale Nonlinear Zonal Flows Around the Rotating Sphere

We study the asymptotic behavior of the conserved densities deduced form the Lagrangian corresponding to the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the equations in question can be used to model zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. The conserved densities were analyzed and visualized by using the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.     

Rotating hairy black holes

We construct stationary black-hole solutions in SU(2) Einstein-Yang-Mills theory which carry angular momentum and electric charge. Possessing nontrivial non-Abelian magnetic fields outside their regular event horizon, they represent nonperturbative rotating hairy black holes.