Exact Global Solutions Research Articles

A remark on the existence of a class of stretched [formula omitted] Magnetohydrodynamics flow

We consider the existence of a class of stretched solutions of 212D Magnetohydrodynamics equations in R3, which are sometimes called the columnar or two and half dimensional flows. The third components of the state variables have the form u3=x3γ1(x1,x2)+φ1(x1,x2) and B3=x3γ2(x1,x2)+φ2(x1,x2) and the first two components of the state variables depend on x1 and x2 only. We prove the local existence of such a flow in Sobolev spaces and give the regularity criteria in terms of 2D vorticity ω or 2D magnetic density j (but not both). Next, we identify the global existence of a class of axisymmetric flow without swirl for both velocity and magnetic field as a corollary of the main theorem. Finally, we present some exact global solutions as well as singular solutions with a special structure for viscous case and some exact global solutions for full inviscid case.

Moment-SOS Approach to Interval Power Flow

Intermittent renewable sources and market-driven operation have brought many uncertainties into modern power systems. Power flow analysis tools are expected to be able to incorporate uncertainties into the solution process. Interval power flow (IPF) analysis which aims at obtaining the upper and lower bounds of power flow solutions under interval uncertainties, thereby emerges as a promising framework to meet such expectation. This paper describes a novel optimization-based method to obtain high-accuracy or even exact global solutions to IPF problems. At first, the IPF problems are formulated as polynomial optimization problems probably with rational objective functions. Then Lasserre’s hierarchy, or moment-SOS approach, is introduced to relax the non-convex problems to convex semidefinite programming (SDP) problems. Correlative sparsity in the polynomial optimization problems is exploited to improve numerical tractability and efficiency. Finally, case studies on IEEE 6-bus, 9-bus and 14-bus systems demonstrate the second-order moment relaxation is capable of obtaining exact global interval solutions on small-scale systems, and numerical results on IEEE 57-bus, 118-bus and 300-bus systems show the proposed method can significantly improve the interval solutions compared with recent Linear Programming (LP) relaxation method on larger systems.

Outage Constrained Robust Secure Transmission for MISO Wiretap Channels

In this paper, we consider the robust secure beamformer design for multiple-input-single-output wiretap channels. Assuming that the eavesdroppers' channels are only partially available at the transmitter, we seek to maximize the secrecy rate under the transmit power and the secrecy rate outage probability constraint. The outage probability constraint requires that the secrecy rate exceed certain thresholds with high probability. Therefore, including such constraint in the design naturally ensures the desired robustness. Unfortunately, the presence of the probabilistic constraints makes the problem nonconvex and, hence, difficult to solve. In this paper, we investigate the outage probability constrained secrecy rate maximization problem using a novel two-step approach. Under a wide range of uncertainty models, our developed algorithms can obtain high-quality solutions, sometimes even exact global solutions, for the robust secure beamformer design problem. Simulation results are presented to verify the effectiveness and robustness of the proposed algorithms.

SOME EXACT BLOWUP SOLUTIONS TO MULTIDIMENSIONAL SCHRÖDINGER MAP EQUATION ON HYPERBOLIC SPACE AND CONE

Exact solutions for the multidimensional Schrödinger map equation (SM for short) on hyperbolic 2-space [Formula: see text] cone are obtained. Consequently, we show the non-traveling wave solution on [Formula: see text] is a finite energy solution on the finite spacial domain. The question of whether a solution of SM can develop a finite time singularity on [Formula: see text] with smooth initial data is not clear. Our result show that blowup can really happen on this initial data. In addition, some exact global smooth solutions are constructed.

A convex relaxation method for computing exact global solutions for multiplicative noise removal

We propose a convex relaxation technique for computing global solutions for the nonconvex multiplicative noise model. The method is based on functional lifting by introducing an additional dimension. We employ a primal–dual-based gradient-type algorithm in numerical implementations to overcome the nondifferentiability of the total variation term. Numerical results show that our algorithm is highly efficient. Furthermore, global solutions of the original model can be obtained with no dependence on the initial guess.

Near optimal routing and capacity management for PWCE-based survivable WDM networks

Protected Working Capacity Envelope (PWCE) has been proposed to simplify resource management and traffic control for survivable WDM networks. In a PWCE-based network, part of the link capacity is reserved for accommodating working routes, and the remaining capacity is reserved for backup routes. The shortest path routing is applied in PWCE-based networks. An arrival call is accepted only when each link along the shortest path has a free working channel. Such a working path routing scheme greatly simplifies the call admission control process for dynamic traffic, and it is especially suitable for implementation in a distributed manner among network nodes. In this article, we investigate two protection strategies: Bundle Protection (BP) and Individual Protection (IDP). In BP, only one backup path can be used to protect a failure component, whereas multiple backup paths can be used in IDP. We formulate four mixed integer non-linear programming (MINLP) problems using BP and IDP strategies for single link and single node failure protection. Each model is designed to determine link metrics for shortest working path routing, working and backup channel assignments, and backup path planning. Our objective is to minimize call-blocking probability on the bottleneck link. Since these models are highly non-linear and non-convex, it is difficult to obtain exact global optimal solutions. We propose a Simulated Annealing-based Heuristic (SAH) algorithm to obtain near optimal solutions. This SAH adopts the concepts of simulated annealing as well as the bi-section technique to minimize call-blocking probabilities. To evaluate the performance, we made simulation comparisons between SAH and the unity link weight assignment scheme. The results indicate that SAH can greatly reduce call-blocking probabilities on benchmark and the randomly generated networks.

Numerical Examination of the Stability of an Exact Two‐dimensional Solution for Flux Pile‐up Magnetic Reconnection

The Kelvin--Helmholtz (KH) and tearing instabilities are likely to be important for the process of fast magnetic reconnection that is believed to explain the observed explosive energy release in solar flares. Theoretical studies of the instabilities, however, typically invoke simplified initial magnetic and velocity fields that are not solutions of the governing magnetohydrodynamic (MHD) equations. In the present study, the stability of a reconnecting current sheet is examined using a class of exact global MHD solutions for steady state incompressible magnetic reconnection, discovered by Craig & Henton. Numerical simulation indicates that the outflow solutions where the current sheet is formed by strong shearing flows are subject to the KH instability. The inflow solutions where the current sheet is formed by a fast and weakly sheared inflow are shown to be tearing unstable. Although the observed instability of the solutions can be interpreted qualitatively by applying standard linear results for the KH and tearing instabilities, the magnetic field and plasma flow, specified by the Craig--Henton solution, lead to the stabilization of the current sheet in some cases. The sensitivity of the instability growth rate to the global geometry of magnetic reconnection may help in solving the trigger problem in solar flare research.

Stationary perturbation configurations in a composite system of stellar and coplanarly magnetized gaseous singular isothermal discs

We construct aligned and unaligned stationary perturbation configurations in a composite system of stellar and coplanarly magnetized gaseous singular isothermal discs (SIDs) coupled by gravity. This study extends recent analyses on (magnetized) SIDs by Shu et al., Lou and Lou & Shen. By this model, we intend to provide a conceptual framework to gain insights for multiwavelength large-scale structural observations of disc galaxies. Both SIDs are approximated to be razor thin and are in a self-consistent axisymmetric background equilibrium with power-law surface mass densities and flat rotation curves. The gaseous SID is embedded with a coplanar azimuthal magnetic field B θ (r) of a radial scaling r -1/2 that is not force-free. In comparison with the SID problems studied earlier, there are three possible classes of stationary solutions allowed by more dynamic freedoms. To identify physical solutions, we explore parameter space involving three dimensionless parameters: ratio λ of Alfven speed to sound speed in the magnetized gaseous SID; ratio β of the square of the stellar velocity dispersion to the gas sound speed; and ratio δ of the surface mass densities of the two SIDs. For both aligned and unaligned spiral cases with azimuthal periodicities |m| ≥ 2, one of the three solution branches is always physical, while the other two branches might become invalid when β exceeds certain critical values. For the onset criteria from an axisymmetric equilibrium to aligned secular bar-like instabilities, the corresponding T/|W - M| ratio, which varies with λ, β and δ, may be considerably lower than the oft-quoted value of T/|W| ∼ 0.14, where T is the total kinetic energy, W is the total gravitational potential energy and M is the total magnetic energy. For unaligned spiral cases, we examine marginal instabilities for axisymmetric (|m| = 0) and non-axisymmetric (|m| > 0) disturbances. The resulting marginal stability curves differ from the previous ones. The case of a composite partial magnetized SID system is also investigated to include the gravitational effect of an axisymmetric dark matter halo on the SID equilibrium. We further examine the phase relationship among the mass densities of the two SIDs and azimuthal magnetic field perturbation. Our exact global perturbation solutions and critical points are valuable for testing numerical magnetohydrodynamic codes. For galactic applications, our model analysis contains more realistic elements and offers useful insights into the structures and dynamics of disc galaxies consisting of stars and magnetized gas.

Exact global solutions of brane universes and big bounce

Exact global solutions of five-dimensional cosmological models compactified on a S1/Z2 orbifold with two 3-branes are presented, and evolution of a simple model is studied. It is found that on all 4D spacetime hypersurfaces, except a singular one, the expanding universe was started not from a big bang but from a big bounce, and before the bounce the universe was in a deflationary contracting phase. It is also found that whether the energy density on the second brane is positive, zero, or negative depends on the size of the fifth dimension.

Some exact nontrivial global solutions with values in unit sphere for two-dimensional Landau–Lifshitz equations

We present some exact global solutions with values in unit sphere for two-dimensional Landau–Lifshitz equations with initial-boundary conditions, and obtained a continuum which can be made from those solutions of a tuft of Landau–Lifshitz equations.

Imperfection-Sensitivity of Classical Bifurcation Models Loaded by Configuration-Dependent Devices1*

An overall global imperfection-sensitivity analysis of classical bifurcation models is presented. A modification of the classical imperfection-sensitivity functions of the stable-symmetric, unstable-symmetric, and asymmetric bifurcation problems under the effect of configuration-dependent conservative loading device is analyzed. The deformation-sensitive characteristics of loading are considered an imperfection of the dead loading device. By considering the interaction of several types of imperfection, imperfection-sensitivity surfaces are obtained. Geometric, material, and loading imperfections are considered here and applied to classical bifurcation models. To obtain the imperfection-sensitivity surfaces of each model, exact global graphical solutions and approximate local analytical solutions are presented.

Counterexamples to Parker’s theorem

Two families of exact global solutions to the equations of plasma equilibrium are derived. These solutions have no singularities, are localized, and are quasiperiodic in variable z; each family depends on an arbitrary number of free parameters. The solutions model a wide variety of magnetic field phenomena that might occur in laboratory plasmas, in astrophysical jets, and in solar corona. The same solutions describe global equilibria of an ideal incompessible fluid. In addition, these families provide counterexamples to a well-known theorem of Parker.

Astrophysical Jets as Exact Plasma Equilibria

Two families of exact global solutions to the equations of plasma equilibrium are derived. The solutions model astrophysical jets and solar prominences and provide counterexamples to Parker's hypothesis.

Global Structure and Dynamics of Advection‐dominated Accretion Flows around Black Holes

We present global solutions that describe advection-dominated accretion flows around black holes. The solutions are obtained by solving numerically a set of coupled ordinary differential equations corresponding to a steady axisymmetric height-integrated flow. The solutions satisfy consistent boundary conditions at both ends. On the inside, the flow passes through a sonic point and falls supersonically into the black hole with a zero-torque condition at the horizon. On the outside, the flow attaches to a normal thin accretion disk. We obtain consistent transonic solutions for a wide range of values of the viscosity parameter ?, from 0.001 to 0.3. We do not find any need for shocks in our solutions, and we disagree with previous claims that viscous accretion flows with low values of ? must have shocks. We compare the exact global solutions of this paper with a local self-similar solution that has been studied in the past. Although the self-similar solution makes significant errors close to the boundaries, we find that it provides nevertheless a reasonable description of the overall properties of the flow. We compare also two different forms of viscosity: one is based on a diffusion prescription, while the other takes the shear stress to be simply proportional to the pressure. The results with the two prescriptions are similar. We see a qualitative difference between solutions with low values of the visocity parameter, ? 0.01, and those with large values, ? 0.01. The solutions with low ? have their sonic transitions occurring close to the radius of the marginally bound orbit. These flows are characterized by regions of super-Keplerian rotation and have pressure maxima outside the sonic point. The solutions are similar in many respects to the hydrostatic thick tori developed previously as models of active galactic nuclei. In contrast, the solutions with large ? have sonic transitions farther out, close to, or beyond the marginally stable orbit and have no super-Keplerian rotation or pressure maxima. We believe these flows will be nearly quasi-spherical down to the sonic radius and will not have empty funnels along the rotation axis. The large-? solutions are more likely to be representative of real systems, since most observations of astrophysical systems are best fit, within the context of advection-dominated theories, with values of ? 0.1.

Steady State Solutions of the Fokker-Planck Equations

A new way of constructing the steady state solutions of the Fokker-Planck Equations(FPE's) is described in which the concept of a vortex field plays an important role. The steady state solutions can be classified according to the type of . In particular, one easily obtains almost all the exact global solutions known so far for the stationary FPE's, as well as the exact local solutions near the fixed points of the corresponding deterministic equations. In this way some new properties of FPE are explored.

Evanescent waves and complex rays

Evanescent wave theory (EWT) is an extension of the geometrical theory of diffraction (GTD) to the tracking of high-frequency plane wave fields with complex phase. The tracking takes place along phase paths perpendicular to the local phase fronts. The determination of the generally complicated configuration of phase paths constitutes a difficulty in the implementation of EWT. The viability of an approximate scheme is examined whereby the phase paths emanating from the initial surface, on which the field is prescribed, are taken locally as hyperbolas; these are known to represent exact global solutions of a particular canonical evanescent wave problem. Numerical comparisons to be presented elsewhere reveal, however, that the local hyperbolic phase path matching may generally be inadequate even for weakly evanescent fields, although this procedure is effective for certain types of initial conditions. To overcome this difficulty, the solution is obtained from an exact formulation involving field integration over the intital surface and subsequent asymptotic evaluation by the saddle point method. Since the saddle points are complex, the latter procedure is equivalent to an analysis in terms of complex rays, from which one may derive the phase paths of EWT. Therefore complex rays should be used as an algorithm for solving the phase path equations. The rigorous analysis also shows that complex ray theory, or EWT, cannot account for strongly evanescent fields, and that there may exist relevant ray contributions from deep within the complex space that cannot be found by the real-space tracking of EWT.