Axially-symmetric Solutions Research Articles

Axisymmetric solutions to Einstein field equations via integral transforms

In this paper, we present new axisymmetric and reflection symmetric vacuum solutions to the Einstein field equations. They are obtained using the Hankel integral transform method and all three solutions exhibit naked singularities. Our results further reinforce the importance and special character of axisymmetric solutions in general relativity and highlight the role of integral transforms methods in solving complex problems in this field. We compare our results to already existing solutions which exhibit the same type of singularities. In this context we notice that most known axial-symmetric solutions possess naked singularities. A discussion of characteristic features of the newly found metrics, e.g., blueshift and the geometry of the singularities, is given.

Global Regular Axially-Symmetric Solutions to the Navier–Stokes Equations with Small Swirl

Axially symmetric solutions to the Navier–Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and the angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.

Towards a full general relativistic approach to galaxies

We analyse the dynamics of a single disk galaxy from a general relativistic viewpoint. We investigate dark matter (DM) effects in terms of a known family of stationary axially-symmetric solutions of Einstein equations coupled to a rotating dust. These effects are generated by the non-Newtonian features of such solutions and are ascribed to the essential role of frame dragging. Indeed, in such models, the off-diagonal elements of the metric are, in general, of the same order of magnitude of the diagonal ones. We generalize the results of Balasin and Grumiller (BG) to the physical case of differentially rotating dust. In particular, we find that for differential rotation the amount of energy density required to account for the flat rotation curves of disk galaxies is reduced with respect to the BG rigid rotation case. This stresses the discrepancy between Newtonian gravity and general relativity (GR), even at low velocities and low energy densities.

On the Local Smoothness of Some Class of Axially-Symmetric Solutions to the MHD Equations

A special class of weak axially-symmetric solutions to the MHD equations for which the velocity field has only poloidal component and the magnetic field is toroidal is considered. For such solutions a local regularity is proved. The global strong solvability of the initial boundary-value problem for the corresponding system in a cylindrical domain with non-slip boundary conditions for the velocity on the cylindrical surface is established as well.

Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field

Incompressible viscous axially-symmetric magnetohydrodynamics is considered in a bounded axially-symmetric cylinder. Vanishing of the normal components, azimuthal components and also azimuthal components of rotation of the velocity and the magnetic field is assumed on the boundary. First, global existence of regular special solutions, such that the velocity is without the swirl but the magnetic field has only the swirl component, is proved. Next, the existence of global regular axially-symmetric solutions which are initially close to the special solutions and remain close to them for all time is proved. The result is shown under sufficiently small differences of the external forces. All considerations are performed step by step in time and are made by the energy method. In view of complicated calculations estimates are only derived so existence should follow from the Faedo-Galerkin method.

Addendum to: The Liouville Theorem for the Steady-State Navier–Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl

We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in \({\mathbb{R}^{3}}\). We prove that there are no nontrivial solution of these equations defined in the whole space \({\mathbb{R}^{3}}\) for axially symmetric case with no swirl (the Liouville theorem). Also we prove the conditional Liouville type theorem for axial symmetric solutions to the Euler system.

Charged axially-symmetric solution in f(T) gravity theory

In present article, an axially symmetric tetrad field has been applied to the charged field equations of f(T) gravity theory. Some constraints have been imposed to solve the resulting non-linear partial differential equations. An exact non-vacuum charged solution with three constants of integration is derived. The solution does not have non-trivial scalar torsion, $${\mathcal {T}}={T^i}_{jk}{S_i}^{jk}$$ . Total conserved charges, using Poincare gauge version, are calculated to understand the physical meaning of the three constants of integration. It has been shown that these constants are gravitational mass, angular momentum of the rotating source and charge parameter.

Charged static axial symmetric solutions and scalar structures

This paper is devoted to the study of static axi- ally symmetric spacetime with anisotropic fluid by means of structure scalars in the presence of electromagnetic field. The structure scalars in terms of physical variables are eval- uated through the Einstein-Maxwell field equations and the inhomogeneity factors are identified. We also explore ana- lytic solutions for isotropic as well as anisotropic fluids. It is found that isotropic solution turns out to be a charged so- lution which has no correspondence with the Weyl metrics while the anisotropic solution has the correspondence with the Weyl metrics.

Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space

We consider positive solutions of $$\varDelta u=0$$ in $$\mathbf{R}_+^n$$ , $$\partial _{\nu }u=u^q$$ on $$\partial \mathbf{R}_+^n$$ , where $$n\ge 3$$ and $$q>n/(n-2)$$ . We investigate the qualitative property of positive $$x_n$$ -axial symmetric solutions. In particular, we are concerned with the asymptotic expansion and the intersection property of positive $$x_n$$ -axial symmetric solutions.

Rocket Noise Sources Localization Through a Tailored Beam-Forming Technique

A source localization technique is applied to analyze wall pressure measurements undertaken on the surface of a space launcher mock-up. The technique is based on the combined use of a standard beam-forming technique and elementary acoustic solutions tailored to the space launcher geometry. Two classes of acoustic elementary solutions are considered: plane waves impinging on the surface of an in nite cylinder computed analytically trough a literature formula, and plane waves impinging on the real space launcher computed numerically through a computational acoustic technique. In both cases, the 3D acoustic eld resulting from the impingement of an arbitrarily oriented plane wave is obtained by a truncated series summation of elementary axial-symmetric solutions. These two classes of basis functions are proven to provide consistent source localization results in the addressed frequency regime. However, although the analytical functional basis enables a faster localization of the noise sources that take place during the rocket ring, the numerical functional basis is expected to enable, at an acceptable computational cost, a more reliable reconstruction of the acoustic loads on the space launcher surface in a higher frequency regime. The proposed CAA-based beam-forming and source reconstruction technique is therefore a useful tool for the vibro-acoustic design of future launch vehicles.

Hopfion canonical quantization

We study the effect of the canonical quantization of the rotational mode of the charge Q=1 and Q=2 spinning Hopfions. The axially-symmetric solutions are constructed numerically, it is shown the quantum corrections to the mass of the configurations are relatively large.

Deformed BPS monopole in Ω-background

We study the BPS condition in the Ω-deformed N=2 super Yang–Mills theory when one of the ϵ-parameters of the background is zero. We obtain the deformed BPS equation for dyons and the formula for their central charge. In particular, we find that the deformed BPS monopole equation has axially-symmetric solution and is equivalent to the Ernst equation. The monopole charge is shown to be undeformed. We construct one-monopole solution explicitly and examine its profile.

On the use of tailored functional bases for space launcher noise sources localization and reconstruction

A source localization and inverse reconstruction methodology is applied to analyze wall pressure signals measured on the surface of a space launcher mock-up. The methodology is based on the use of elementary acoustic solutions tailored to the space launcher geometry as functional basis. Two classes of functional bases are considered: plane waves impinging on the surface of an infinite cylinder computed analytically through a literature formula, and plane waves impinging on the real space launcher computed numerically through a FEM computational acoustic technique. In both cases, the 3D acoustic field resulting from the impingement of an arbitrarily oriented plane wave is obtained by a truncated series summation of elementary axial-symmetric solutions. These two classes of functional bases are proven to provide consistent results in the addressed frequency range. However, although the analytical basis enables a faster localization of the noise sources that take place during the rocket firing, the numerical basis is expected to enable, at an acceptable computational cost, a more reliable reconstruction of the acoustic loads on the space launcher surface in a higher frequency range. The proposed FEM-based beam-forming and source reconstruction technique is therefore a useful tool for the vibro-acoustic design of future launch vehicles.

TWISTED SEMILOCAL STRINGS WITH TWO HIGGS-DOUBLETS

The standard electroweak model is extended by means of a second Brout–Englert–Higgs–doublet. The symmetry breaking potential is chosen in such a way that (i) the Lagrangian possesses a custodial symmetry, (ii) a stationary, axially symmetric ansatz of the bosonic fields consistently reduces the Euler–Lagrange equations to a set of differential equations. The potential involves, in particular, a direct interaction between the two doublets. Magnetic, stationary, axially-symmetric solutions of the classical equations are constructed. Some of them can be assimilated to embedded Nielsen–Olesen strings. From these solutions there are bifurcations and new solutions appear which exhibit the characteristics of the recently constructed twisted semilocal strings. A special emphasis is set on "doubly-twisted" solutions for which the two doublets present different time-dependent phase factors. They are regular and have a finite energy which can be lower than the energy of the embedded twisted solution. Electric-type solutions, such that the fields oscillate asymptotically far from the symmetry-axis, are also reported.

Some exact solutions of geophysical fluid‐dynamics equations for testing models in spherical and plane geometry

AbstractA methodology is presented for constructing a family of exact axially‐symmetric solutions to various geophysical fluid‐dynamics equation sets, with the aim of facilitating the development and testing of numerical models. The construction is done first for the shallow‐water equations in spherical geometry, and then used to construct solutions for the shallow‐water equations in Cartesian geometry, the vertical‐slice Euler equations on an f–F plane, and the three‐dimensional deep‐ and shallow‐atmosphere Euler equations in spherical geometry. The solutions are steady, vortical, axially symmetric, and non‐divergent. Illustrative solutions are presented for five simple model problems: an equatorial jet, twin mid‐latitude jets, an isolated polar vortex, multiple Jovian jets, and an exponentially‐decaying vortex. It is also shown how to make a non‐hydrostatic vertical‐slice Euler‐equations model on an f–F plane emulate a shallow‐water model on an f plane by suitably setting parameter values and initial conditions. Crown Copyright 2007. Reproduced with the permission of the Controller of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.

General Axisymmetric Solutions and Self-Tuning in 6D Chiral Gauged Supergravity

We re-examine the properties of the axially-symmetric solutions to chiral gauged 6D supergravity, recently found in refs. hep-th/0307238 and hep-th/0308064. Ref. hep-th/0307238 finds the most general solutions having two singularities which are maximally-symmetric in the large 4 dimensions and which are axially-symmetric in the internal dimensions. We show that not all of these solutions have purely conical singularities at the brane positions, and that not all singularities can be interpreted as being the bulk geometry sourced by neutral 3-branes. The subset of solutions for which the metric singularities are conical precisely agree with the solutions of ref. hep-th/0308064. Establishing this connection between the solutions of these two references resolves a minor conflict concerning whether or not the tensions of the resulting branes must be negative. The tensions can be both negative and positive depending on the choice of parameters. We discuss the physical interpretation of the non-conical solutions, including their significance for the proposal for using 6-dimensional self-tuning to understand the small size of the observed vacuum energy. In passing we briefly comment on a recent paper by Garriga and Porrati which criticizes the realization of self-tuning in 6D supergravity.

A source of a quasi-spherical space–time: The case for the M–Q solution

We present a physically reasonable source for an static, axially-symmetric solution to the Einstein equations. Arguments are provided, supporting our belief that the exterior space-time produced by such source, describing a quadrupole correction to the Schwarzschild metric, is particularly suitable (among known solutions of the Weyl family) for discussing the properties of quasi-spherical gravitational fields.

Integral Equation Approach for the Propagation of TE-Waves in a Nonlinear Dielectric Cylindrical Waveguide

We consider the propagation of TE-polarized electromagnetic waves in cylindrical dielectric waveguides of circular cross section filled with lossless, nonmagnetic, and isotropic medium exhibiting a local Kerr-type dielectric nonlinearity. We look for axially-symmetric solutions and reduce the problem to the analysis of the associated cubic-nonlinear equation. We show that the solution in the form of a TE-polarized electromagnetic wave exists and can be obtained by iterating a cubic-nonlinear integral equation. We derive the associated dispersion equation and prove that it has a root that determines this solution.

Noncoaxial multivortices in the complex sine-Gordon theory on the plane

We construct explicit multivortex solutions for the complex sine-Gordon equation (the Lund–Regge model) in two Euclidean dimensions. Unlike the previously found (coaxial) multivortices, the new solutions comprise n single vortices placed at arbitrary positions (but confined within a finite part of the plane.) All multivortices, including the single vortex, have an infinite number of parameters. We also show that, in contrast to the coaxial complex sine-Gordon multivortices, the axially-symmetric solutions of the Ginzburg–Landau model (the stationary Gross–Pitaevskii equation) do not belong to a broader family of noncoaxial multivortex configurations.

Bound states of the elliptic sine-Gordon equation

We give a detailed study of finite-energy solutions to elliptic sine-Gordon (SG) equation in the plane with point-like singularities. These bound-state solutions (in a sense of scalar field theory) with only one singularity at the origin demonstrate a target-like annular soliton pattern at large distance from the origin. An effective radius of this pattern is calculated both analytically and numerically for the case of axial symmetric solutions. The analytic study is based on an isomonodromic deformation method for the third Painlevé equation, which distinguishes bound-state solutions as separatrices in a manifold of general (infinite-energy) solutions. Exact analytic estimates give us a tool to study bounded-state solutions to the non-integrable SG equation with forcing. Namely, for large intensity at the singularity we derive a critical value of forcing, which governs the existence and stability of the bound-state solutions. This plays a crucial role for two concrete physical applications dealing with large area Josephson junctions and nematic liquid crystals in a rotating magnetic field. For both examples we compute critical values of field and driving forces which enables the formation of modes with finite energy. These numerical computed critical values correlate well with computer simulations and experimental data.