Nonstationary Solutions Research Articles

Convoluted solutions in supergravity

Inspired by the convoluted solutions for two intersecting M2 branes in elevendimensional supergravity, in which one brane in the system is completely localized along the overall and relative transverse coordinates while the other brane in the system is localized only along the overall transverse coordinates, we construct two classes of exact solutions to Einstein-Maxwell theory in six and higher dimensions. We show that the membrane configuration preserves four supersymmetries and upon dimensional reduction, the solutions provide intersecting configurations of three D-branes in type IIA supergravity. Moreover, we show that the metric functions in six and higher dimensions can be written as convolution-like integrals of two special functions. The solutions are regular everywhere and show a bolt structure on a single point in any dimensionality. Also, we find the exact nonstationary solutions to the Einstein-Maxwell theory with positive cosmological constant. We show that the cosmological solutions are expanding patches in asymptotically de Sitter spacetime.

Surface waves in a square container due to its resonant horizontal elliptic motion

Surface waves in a square container due to its resonant horizontal elliptic or linear motion are investigated theoretically. The motion of the container is characterized by the ratio, expressed as , of the length of the minor axis to the length of the major axis of its elliptic orbit, and by the angle θ between the directions of the major axis and one of its sidewalls. Using the reductive perturbation method, non-linear time evolution equations for the complex amplitudes of two degenerate modes excited by this motion are derived with the inclusion of linear damping. When is small, for any θ these equations have two kinds of stable stationary solutions corresponding to regular co-rotating waves whose direction of rotation is the same as that of the container and regular counter-rotating waves of the opposite direction of rotation. As increases to one, the region of forcing frequency in which stable regular counter-rotating waves are observed shrinks and then disappears for any θ. Solutions with chaotic or periodic slow variations in amplitude and phase of excited surface waves are also obtained for forcing frequencies where no stable stationary solutions exist. Non-stationary solutions are either unidirectionally or bidirectionally rotating waves. For , chaotic waves and bidirectionally rotating waves are observed more frequently for smaller . For , for sufficiently small fluid depth, regular non-rotating waves are expected to occur for any forcing frequency. Moreover, stable stationary and non-stationary solutions obtained for are found to agree fairly well with the experimental results in a preceding study.

Quasi-periodic non-stationary solutions of 3D Euler equations for incompressible flow

A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: one consisting of the spatial component and the other being related to the time dependent part.Spatial part of a solution could be determined if we substitute such a solution to the equations of motion (equation of momentum) with the requirement of scale-similarity in regard to the proper component of spatial velocity. So, the time-dependent part of equations of momentum should depend on the time-parameter only.The main result, which should be outlined, is that the governing (time-dependent) ODE-system consists of 2 Riccati-type equations in regard to each other, which has no solution in general case. But we obtain conditions when each component of time-dependent part is proved to be determined by the proper elliptical integral in regard to the time-parameter t, which is a generalization of the class of inverse periodic functions.

Fortran and C programs for the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

Many of the static and dynamic properties of an atomic Bose–Einstein condensate (BEC) are usually studied by solving the mean-field Gross–Pitaevskii (GP) equation, which is a nonlinear partial differential equation for short-range atomic interaction. More recently, BEC of atoms with long-range dipolar atomic interaction are used in theoretical and experimental studies. For dipolar atomic interaction, the GP equation is a partial integro-differential equation, requiring complex algorithm for its numerical solution. Here we present numerical algorithms for both stationary and non-stationary solutions of the full three-dimensional (3D) GP equation for a dipolar BEC, including the contact interaction. We also consider the simplified one- (1D) and two-dimensional (2D) GP equations satisfied by cigar- and disk-shaped dipolar BECs. We employ the split-step Crank–Nicolson method with real- and imaginary-time propagations, respectively, for the numerical solution of the GP equation for dynamic and static properties of a dipolar BEC. The atoms are considered to be polarized along the z axis and we consider ten different cases, e.g., stationary and non-stationary solutions of the GP equation for a dipolar BEC in 1D (along x and z axes), 2D (in x–y and x–z planes), and 3D, and we provide working codes in Fortran 90/95 and C for these ten cases (twenty programs in all). We present numerical results for energy, chemical potential, root-mean-square sizes and density of the dipolar BECs and, where available, compare them with results of other authors and of variational and Thomas–Fermi approximations. Program summaryProgram title: (i) imag1dZ, (ii) imag1dX, (iii) imag2dXY, (iv) imag2dXZ, (v) imag3d, (vi) real1dZ, (vii) real1dX, (viii) real2dXY, (ix) real2dXZ, (x) real3dCatalogue identifier: AEWL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWL_v1_0.htmlProgram obtainable from: CPC Program Library, Queens University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 111384No. of bytes in distributed program, including test data, etc.: 604013Distribution format: tar.gz

Stability of nonstationary solutions of the generalized KdV-Burgers equation

The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

Asymptotic model-based estimation of a wake oscillator for a tethered sphere in uniform flow

We derive and investigate the nonlinear dynamics of a tethered sphere in uniform flow. A Lagrangian based model augmented by wake oscillators for the streamwise vertical and transverse directions enables model-based estimation of both structural and aeroelastic parameters via asymptotic analysis of internal resonance conditions between the transverse wake frequency and its structural counterpart. Validation of the proposed methodology is demonstrated by comparison of results with those of a benchmark experiment for a light sphere in water conducted by Govardhan and Williamson (1997) and by analysis of a spherical aerostat experiment conducted by Coulomb-Pontbriand and Nahon (2009). A numerical bifurcation analysis of the validated system reveals existence of possible quasiperiodic and non-stationary solutions that are consistent with documented instabilities of aerostats in severe environmental conditions and shed light on control mechanisms required for suppression of finite amplitude limit-cycle oscillations.

Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid

We have determined and analyzed exact stationary and nonstationary solutions to the laminar Marangoni convection problem, which is an overdetermined boundary value problem. The numerical solution of this problem belongs to the class of Birikh solutions. The overdetermination of the resolving system of equations results from the zeroness of the velocity parallel to the applicate axis. Cases of thermal and solutal convection of a viscous incompressible fluid are considered. To make the boundary value problem solvable, the class of exact solutions is proposed for use, where velocities are one-dimensional in coordinates, the pressure and temperature fields are three-dimensional. Identical equality to zero of the convective derivative in the impulse conservation equation is typical of the class presented. The convective derivative remains in the caloric equation. The discussed boundary value problem is shown to be irreducible to the one-dimensional problem when the temperature gradient is specified on both boundaries of the fluid layer, as distinct from the classical Birikh solution and its numerous generalizations. The obtained stationary and nonstationary solutions have a stagnation point for velocities, thus suggesting the presence of counter flows in the moving fluid. It is demonstrated by localization of the polynomial roots of the stationary solutions that there exists such a value of layer thickness that the tangential stress can become zero on the lower boundary of the fluid layer only under thermal Marangoni convection. The nonstationary solutions obtained by the boundary element method, which can be treated as an exact method, tend to become stationary. The application of the boundary element method extends the class of exact nonstationary solutions considerably, since this method enables one to study not only invariant exact solutions.

Remarks on the Internal Exponential Stabilization to a Nonstationary Solution for 1D Burgers Equations

The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the control, a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examples are presented illustrating the stabilizing effect of the feedback control and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible.

Response to “Comment on ‘General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation’” [Phys. Fluids 26, 119101 (2014)

In R. A. Van Gorder, “General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation,” Phys. Fluids 26, 065105 (2014) I discussed properties of generalized vortex filaments exhibiting purely rotational motion under the low-temperature Svistunov model of the local induction approximation. Such solutions are stationary in terms of translational motion. In the Comment [N. Hietala, “Comment on ‘General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation’ [Phys. Fluids 26, 065105 (2014)],” Phys. Fluids 26, 119101 (2014)], the author criticizes my paper for not including translational motion (although it was clearly stated that the filament motion was assumed rotational). As it turns out, if one is interested in studying the geometric structure of solutions (which was the point of my paper), one obtains the needed qualitative results on the structure of such solutions by studying the purely rotational case. Nevertheless, in this Response I shall discuss the vortex filaments that have both rotational and translational motions. I then briefly discuss why one might want to study such generalized rotating filament solutions, in contrast to simple the standard helical or planar examples (which are really special cases). I also discuss how one can study the time evolution of filaments which exhibit more complicated dynamics than pure translation and rotation. Doing this, one can study non-stationary solutions which initially appear purely rotational and gradually display other dynamics as the filaments evolve.

Exact convoluted solutions in higher-dimensional Einstein-Maxwell theory

We construct two classes of exact solutions to six and higher dimensional Einstein-Maxwell theory in which the metric functions can be written as convolution-like integrals of two special functions. The solutions are regular everywhere and show a bolt structure on a single point in any dimensionality. Moreover, we find the exact nonstationary solutions to the Einstein-Maxwell theory with positive cosmological constant. We show that the cosmological solutions are expanding patches in asymptotically de Sitter spacetime.

Heat Supply Optimization

Due to the energetic evaluation of buildings, the solutions for optimal heat supply are a problem of current interest. The topicality of this problem is underlined by the European Union requirements for reducing the energetic demands of buildings by the year 2020.From the operational point of view, most of the heating systems seem to be optimal; however, with the use of the results of this detailed analysis it is possible to make even further improvement. The objective is not to supply more heat energy than necessary in order to achieve energy saving. This reduction of heat supply can be achieved by the use of the heat accumulation potential of the building by means of appropriately chosen heating mode. By operation modes are meant a mode with constant heating and a mode with interrupted heating.In this sample case of non-stationary simulation during the interrupted heating regime it is possible to demonstrate the potential saving achieved by the chosen interrupted heating mode. The case study is conducted on a secondary school building. In this project, supported by the specific investigation project on the Faculty of Civil Engineering, I carried out an experiment of an interrupted mode, which was also supported by a theoretical part, in which a non-stationary calculation was used.This contribution suggests the possibility to save energy by optimization of the heat supply. It also presents the difference between a stationary and non-stationary solution to this problem.

Decay Results of Higher-Order Norms for the Navier–Stokes Flows in 3D Exterior Domains

Using the nonstationary Stokes solution formula and the regularity theory of the steady Stokes equations, we establish the decay results (including the weighted cases) of higher-order spatial derivatives for the Navier–Stokes flows in 3D exterior domains, which is a long standing problem. The higher-order norm (weighted) decay results in this article are inspired by the work in Bae and Jin (Proc R Soc Edinburgh Sect A 135:461–477, 2005), Schonbek and Wiegner (Proc R Soc Edinburgh Sect A 126:677–685, 1996), respectively, which are considered in the whole space.

Causal structures in Gauss-Bonnet gravity

We analyze causal structures in Gauss-Bonnet gravity. It is known that Gauss-Bonnet gravity potentially has superluminal propagation of gravitons due to its noncanonical kinetic terms. In a theory with superluminal modes, an analysis of causality based on null curves makes no sense, and thus, we need to analyze them in a different way. In this paper, using the method of the characteristics, we analyze the causal structure in Gauss-Bonnet gravity. We have the result that, on a Killing horizon, gravitons can propagate in the null direction tangent to the Killing horizon. Therefore, a Killing horizon can be a causal edge as in the case of general relativity; i.e. a Killing horizon is the ``event horizon'' in the sense of causality. We also analyze causal structures on nonstationary solutions with $(D\ensuremath{-}2)$-dimensional maximal symmetry, including spherically symmetric and flat spaces. If the geometrical null energy condition, ${R}_{AB}{N}^{A}{N}^{B}\ensuremath{\ge}0$ for any null vector ${N}^{A}$, is satisfied, the radial velocity of gravitons must be less than or equal to that of light. However, if the geometrical null energy condition is violated, gravitons can propagate faster than light. Hence, on an evaporating black hole where the geometrical null energy condition is expected not to hold, classical gravitons can escape from the ``black hole'' defined with null curves. That is, the causal structures become nontrivial. It may be one of the possible solutions for the information loss paradox of evaporating black holes.

Modulational instability and associated rogue structures of slow magnetosonic wave in Hall magnetohydrodynamic plasmas

The modulational instability and associated rogue structures of a slow magnetosonic wave are investigated for a Hall magnetohydrodynamic plasma. Nonlinear Schrodinger equation is obtained by using the multiple scale method, which shows a modulationally unstable slow magnetosonic mode evolving into bright wavepackets. The dispersive effects induced by the Hall electron current increase with the increase in plasma β and become weaker as the angle of propagation increases. The growth rate of the modulational instability also increases with the increase in plasma β. The growth rate is greatest for the parallel propagation and drops to zero for perpendicular propagation. The envelope wavepacket of a slow magnetosonic is widened with less oscillations as plasma β increases. But the wavepacket becomes slightly narrower and more oscillatory as the angle of propagation increases. Further a non-stationary envelope solution of the Peregrine soliton is analyzed for rogue waves. The Peregrine soliton contracts temporally and expands spatially with increase in plasma β. However, the width of a slow magnetosonic Peregrine soliton decreases both temporally and spatially with increase of the propagation angle.

Linear Eсkman friction in the mechanism of the cyclone-anticyclone vortex asymmetry and in a new theory of rotating superfluid

The observed experimental and natural phenomenon of cyclone-anticyclone vortex asymmetry implies that a relatively more stable and showing a longer life, as well as a relatively more intense mode of rotation with an anticyclonic circulation direction (opposite to the direction of rotation of the medium as a whole) is realized as compared with an oppositely directed rotation of the cyclonic vortex mode. Until now, however, it was not a success to identify a universal triggering mechanism responsible for the formation of the corresponding breaking of chiral vortex symmetry. In this paper we reveal the said linear universal instability mechanism of breaking of chiral symmetry in the sign of vortex circulation in the rotating medium in the presence of linear Eckman friction. Results Obtained is a condition for the linear dissipative - centrifugal instability (DCI), which leads (only when considering the external linear Eckman friction for an abovethreshold value of rotation frequency of the underlying boundary surface of fluid) to the breaking of chiral symmetry in the Lagrangian fluid particle dynamics and the corresponding realization of the cyclone-anticyclone vortex asymmetry. Conclusion A new non-stationary solution to the problem for the disc which carries out weak axial-torsional oscillations in fluid with the frequency which are superimposed on its rotation with the previously considered frequency 0 ω in connection with the experimental data on the rotating superfluid helium II (see [ 16, 17 ]) has been found. It gives the possibility to conclude that the effects of external, linear on velocity, friction forces must be important to include into consideration for the solve of any fundamental problems of hydrodynamics in bounded systems (as for the blood dynamics in cardiovascular system, for example).

Nonlinear Schrödinger equation containing the time-derivative of the probability density: A numerical study

A nonlinear Schrödinger equation that contains the time-derivative of the probability density is investigated, which is motivated by the attempt to include the recoil effect of radiation. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigen-states of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the normalization of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential (1D ISWP) and for several time-dependent potentials that tend to the former as time increases. Results show that for various initial states, the wave function always evolves into some eigen-state of the corresponding linear Hamiltonian of the 1D ISWP. For a small time-dependent perturbation potential, solutions present the process similar to the spontaneous transition between stationary states. For a periodical potential with an appropriate frequency, solutions present the process similar to the stimulated transition. This nonlinear Schrödinger equation thus presents the state evolution similar to the wave-function reduction.

New class of N-dimensional braneworlds

The new class of the non-stationary solutions to the system of N-dimensional equations for coupled gravitational and massless scalar field is found. The model represents a single (N-1)-brane in a space-time with one large (infinite) and (N-5) small (compact) space-like extra dimensions. In some particular cases the model corresponds to the gravitational and scalar field standing waves bounded by the brane. These braneworlds can be relevant in string and other higher dimensional models.

Space charge emission in cylindrical diode

In this paper, a mathematical model to describe cylindrical electron current emissions through a physics approximation method is presented. The proposed mathematical approximation consists of analyzing and solving the nonlinear Poisson's equation, with some determined mathematical restrictions. Our findings tackle the problem when charge-space creates potential barrier that disable the steady-state of the beam propagation. In this problem, the potential barrier effects of electron's speed with zero velocity emitted through the virtual cathode happens. The interaction between particles and the virtual cathode have been to find the inter-atomic potentials as boundary conditions from a quantum mechanics perspective. Furthermore, a non-stationary spatial solution of the electrical potential between anode and cathode is presented. The proposed solution is a 2D differential equation that was linearized from the generalized Poisson equation. A single condition was used solely, throughout the radial boundary conditions of the current density formation.

The evolution of the thermocapillary motion of three fluids in a plane layer

The unidirectional motion of three immiscible incompressible viscous heat-conducting liquids in a plane layer is considered. It is assumed that the motion occurs only under the action of thermocapillary forces from a state of rest. The analysis of the motion is reduced to solving linear conjugate initial boundary value problems for a system of parabolic equations. A non-stationary solution is sought by the Laplace transformation method and is obtained in the form of finite analytical expressions in transforms. It is proved that, as the time increases, the solution always reaches the steady state obtained earlier and an exponential estimate of the rate of convergence is given with an indicator which depends on the physical properties of the media and the layer thicknesses. The evolution of the velocity and temperature perturbation fields to a steady state for specific liquid media is obtained by numerical inversion of the Laplace transformation.

Vaidya black hole in non-stationary de Sitter space: Hawking’s temperature

In this paper we present a class of non-stationary solutions of Einstein’s field equations describing embedded Vaidya-de Sitter black holes with a cosmological variable function Λ(u). The Vaidya-de Sitter black hole is interpreted as the radiating Vaidya black hole is embedded into the non-stationary de Sitter space with variable Λ(u). The energy-momentum tensor of the Vaidya-de Sitter black hole is expressed as the sum of the energy-momentum tensors of the Vaidya null fluid and that of the non-stationary de Sitter field, and satisfies the energy conservation law. We study the energy conditions (like weak, strong and dominant conditions) for the energy-momentum tensor. We find the violation of the strong energy condition due to the negative pressure and leading to a repulsive gravitational force of the matter field associated with Λ(u) in the space-time. We also find that the time-like vector field for an observer in the Vaidya-de Sitter space is expanding, accelerating, shearing and non-rotating. It is also found that the space-time geometry of non-stationary Vaidya-de Sitter solution with variable Λ(u) is Petrov type D in the classification of space-times. We also find the Vaidya-de Sitter black hole radiating with a thermal temperature proportional to the surface gravity and entropy also proportional to the area of the cosmological black hole horizon.