One-dimensional Symmetry Research Articles

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg–de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved H s norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.

The thermal analysis of fibers in the twenty first century: From textile, industrial and composite to nano, bio and multi-functional

The one-dimensional symmetry and low linear density of fibers make them a special case for process–structure–property performance definition, simplified by the symmetry and complicated by meta-stable morphology and unwieldy sample geometry. The thermal analysis of fibers always reflects the total process history of the fiber, from synthesis through storage environment prior to sampling. Thermal analysis provides a convenient platform for quantifying fiber performance (or fabric performance) in terms of the fiber phase stability (DSC, DMA, TSC), dimensional stability (TMA), chemical stability (TGA, DSC) and mechanical property stability (DMA) as monitored under both prior history and end-use relevant conditions. Of special interest is the application of these techniques to the characterization of functional fibers, and fibers designed for in vivo use such that the resulting data is reflective of desired performance targets.

A Mean Field Equation on a Torus: One-Dimensional Symmetry of Solutions

ABSTRACT We study the equation for u ∈ E, where E = {u ∈ H 1(Ωϵ): u is doubly periodic, ∈ t Ωϵ u = 0} and Ωϵ is a rectangle of ℝ2 with side lengths 1/ϵ and 1, 0 < ϵ ≤ 1. We establish that every solution depends only on the x-variable when λ ≤ λ*(ϵ), where λ*(ϵ) is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters ϵ and λ in which every solution is identically zero. This range is optimal for ϵ ≤ 1/2.

A SIMPLE HAMILTONIAN MODEL OF RUNAWAY PARTICLE WITH SINGULAR INTERACTION

We consider a Hamiltonian system given by a charged particle under the action of a constant electric field and interacting with a medium, which is described as a Vlasov fluid. We assume that the action of the charged particle on the fluid is negligible and that the latter has one-dimensional symmetry. We prove that if the singularity of the particle/medium interaction is integrable and the electric field intensity is large enough, then the particle escapes to infinity with a quasi-uniformly accelerated motion. A key tool in the proof is a new estimate on the growth in time of the fluid particle velocity for one-dimensional Vlasov fluids with bounded interactions.

Atomic structure of Bi-dimer row selectively adsorbed on Si(5512)-2×1 surface

In order to understand the atomic structure of nanostructures self-assembled on the template with one-dimensional symmetry, Bi/Si(5 5 12) system has been chosen and Bi-adsorption steps have been studied by STM. With Bi adsorption, the clean Si(5 5 12) is transformed to (3 3 7) terraces with disordered boundary due to mismatched periodicities between (3 3 7) and (5 5 12), and Bi-dimer rows are formed inside the (3 3 7) unit as follows: Initially, when Bi atoms are deposited at the adsorption temperature of about 450 °C, they selectively replace Si-dimers and Si-adatoms and form adsorbed Bi-dimers and Bi-adatoms, respectively. If additional Bi is supplied, the Bi-dimers adsorb on the Bi-dimers and Bi-adatoms in the first layer. These adsorbed dimers in the second layer are facing each other to form a Bi-dimer pair with relatively stable p3 bonding. Finally, a single Bi-dimer adsorbs above the Bi-dimer pair in the second layer, at which point the Bi layer thickness saturates. It has been concluded that the Bi-dimer pair with stable p3 bonding is the composing element in the second layer and such site-selective adsorption is possible due to the substrate-strain relaxation through inserting Bi-buffer layer limited to specific sites of the substrate.

Spatial Bifurcations in the Generic N−NDR Electrochemical Oscillator with Negative Global Coupling: Theory and Surface Plasmon Experiments

Pattern formation during the oscillatory reduction of IO4- at a rectangular gold film electrode was investigated by means of surface plasmon microscopy. Several one-dimensional spatial symmetry breakings were observed when a negative global coupling (NGC) was imposed by partially compensating the electrolyte resistance, using a negative impedance device. Respective theoretical investigations of the prototype electrochemical oscillator with an N-shaped current−potential characteristic (N−NDR), corresponding to the IO4- reduction, are presented. A complete picture of the spatial bifurcations reproducing the symmetry breakings at a fixed NGC strength is given. The agreement between experiment and model is further emphasized by numerical simulations that revealed a state with double broken symmetry, which we termed asymmetric standing wave. Furthermore, the theoretical studies allow for identification of patterns that arise, owing to a complex interplay of the nonlinear kinetics and unavoidable nonuniformitie...

Similarity solutions for magnetic bubble expansion

A model for an expanding magnetic bubble or plasmoid is introduced, corresponding to a large aspect ratio torus, having one-dimensional (cylindrical) symmetry but with three dimensional expansion, with the length of the cylinder expanding in time in the same manner as the radius. This model has a general class of similarity equations in ideal magnetohydrodynamics (MHD) for spherical expansion. There are two parameters c, d characterizing the similarity solutions, depending on boundary conditions and conservation relations. These solutions exhibit either tangential discontinuities or shocks at the boundary, depending on the values of the constants c and d. Some of the solutions have magnetic fluxes within the bubble increasing with time, but with smaller or zero magnetic fields outside the bubble, requiring a shock and a dynamo in the shock region. The results of simulations of one class of solutions with a Lagrangian MHD code show good agreement. Some of the properties of fully toroidal solutions of the similarity equations are derived. This model has applications to a magnetic bubble from an accretion disk around an active galactic nucleus (AGN), appropriate to the phase in which the bubble has expanded to a size much greater than the disk field length scales but much smaller than any exterior scales. At this stage the magnetic reconnection and flux conversion stage associated with setting up the expanding bubble is completed. The model may also apply to a plasmoid formed in the solar corona.

A negative answer to a one-dimensional symmetry problem in the Heisenberg group

De Giorgi conjectured that bounded monotone solutions to the Ginzburg-Landau equation are constant along hyperplanes, in this paper we prove that in the Heisenberg group this conjecture doesn't hold true with respect to the center direction.

Non-spherical evolution of the line-driven wind instability

In this paper, we study the structure and stability of line-driven winds using numerical hydrodynamic simulations. We calculate the radiation force from an explicit non-local solution of the radiation transfer equation, rather than a Sobolev approximation, without restricting the flow to one-dimensional symmetry. We find that the solutions that result have complex and highly variable structures, including dense condensations, which we compare with observed variable absorption features in the spectra of early-type stars.

Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions

(1.2) div Φσ(u,Du) = Φξ(u,Du). Using compactness methods based on the translation invariance of the equation (1.2), and a priori estimates in C1 norm, we prove various properties of bounded entire solutions of (1.2), such as a sharp inequality for the gradient, energy monotonicity and optimal growth, Liouville type results, and one-dimensional symmetry. An important role in this program is played by the function

A simple model for interconnect design of planar solid oxide fuel cells

Two geometries of solid oxide fuel cells (SOFCs) are currently under development, tubular and planar. Both types of cells are configured in a stack using an interconnect, which electrically connects the anode of one cell to the cathode of the adjacent cell, while also physically isolating fuel from oxidant. Proper design of the interconnect in conjunction with single-cells is critical to minimizing the overall stack resistance. This work quantitatively examines the dependence of total SOFC stack resistance as a function of interconnect contact spacing, interconnect contact area, cathode thickness, electrolyte thickness, anode thickness, and transport properties associated with each region and at interfaces (charge transfer resistance). Both one-dimensional (channels) and two-dimensional (dimples) symmetries of interconnect geometry are analyzed for planar cells. Analytical expressions are derived for the area-specific resistance (ASR) of a repeat unit consisting of a cell and interconnect, for both geometries, as a function of cell parameters, interconnect contact area and interconnect contact spacing. It is found that the one-dimensional interconnect symmetry leads to lower values of stack-repeat unit area-specific resistance (ASR) than the two-dimensional symmetry. Thus, based on the analysis presented here the one-dimensional interconnect geometry is preferred over the two-dimensional one.

One-dimensional broken translation symmetry and pseudo-Goldstone excitation

In a class of potentials U(x) = A(Acosh 2x-cosh x-2A)(1 + Acosh x)-2 the ground level is E = 0, and the corresponding wavefunction is known exactly. This excitation is a manifestation of broken translation symmetry in one-dimensional, nonlinear models with a symmetric double-Morse potential. Broken translation symmetry in a one-dimensional model with an asymmetric double-Morse potential results in an exotic class of double-well potentials with exactly determined first excited level. Such a property, exact determination of ground or first excited level in some classes of potentials, is common for one-dimensional models with, respectively, corresponding symmetric or asymmetric double-well potentials.

One-dimensional symmetry of bounded entire solutions of some elliptic equations

This paper is about one-dimensional symmetry properties for some entire and bounded solutions of ∆u + f(u) = 0 in IR. We consider solutions u such that −1 < u < 1 and u(x1, · · · , xn) → ±1 as xn → ±∞, uniformly with respect to x1, · · · , xn−1. Under some conditions on f , we prove that the solutions only depend on the variable xn. We also discuss more general elliptic operators. The qualitative properties then strongly depend on the coefficients of the operator. These results extend to higher dimensions and to more general operators a result of Ghoussoub and Gui [21] proved for n ≤ 3. AMS Classification : 35B05, 35B40, 35B50, 35J60.

On universality in metallic nanocohesion

A semiclassical trace formula for the mesoscopic oscillations of the cohesive force in a metallic nanowire of constant cross section is derived. It is shown that the force oscillations are universal, in the sense that their r.m.s. amplitude is independent of the area of the cross section, only for cross sections with a continuous one-dimensional symmetry. For a wire of rectangular cross section, the r.m.s. amplitude of the force oscillations is shown to be proportional to the aspect ratio of the cross section.

A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry

This work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions. Cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations. Written in Lagrangian form, these equations are discretized utilizing compatible, control volume differencing with a staggered-grid placement of the spatial variables. The concept of “compatibility” means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form. Multiple pressures are utilized in each zone; they produce forces that resist spurious vorticity generation. This difficulty can severely limit the utility of the Lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions. An edge-centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks. The particular difficulty of exactly preserving one-dimensional spherical symmetry in three-dimensional geometry is solved. This problem has both practical and pedagogical significance. The algorithm is suitable for both structured and unstructured grids. Limitations that symmetry preservation imposes on the latter type of grids are delineated.

On the thermodynamical interpretation of perfect fluid solutions of the Einstein equations with no symmetry

The Gibbs–Duhem equation dU+pdV=TdS imposes restrictions on the perfect fluid solutions of Einstein equations that have a one-dimensional symmetry group or no symmetry at all. In this paper, we investigate the restrictions imposed on the Stephani Universe and on the two classes of models found by Szafron. Upon the Stephani Universe and the β≠0 class of Szafron symmetries are forced. We find the most general subcases of the β=0 model of Szafron that are consistent with the Gibbs–Duhem equation and have no symmetry.

(Weakly) three-dimensional caseology

The singular eigenfunction technique of Case for solving one-dimensional planar symmetry linear transport problems is extended to a restricted class of three-dimensional problems. This class involves planar geometry, but with forcing terms (either boundary conditions or internal sources) which are weakly dependent upon the transverse spatial variables. Our analysis involves a singular perturbation about the classic planar analysis, and leads to the usual Case discrete and continuum modes, but modulated by weakly dependent three-dimensional spatial functions. These functions satisfy parabolic differential equations, with a different diffusion coefficient for each mode. Representative one-speed time-independent transport problems are solved in terms of these generalized Case eigenfunctions. Our treatment is very heuristic, but may provide an impetus for more rigorous analysis.

Band-structure engineering in strained semiconductor lasers

The influence of both compressive and tensile strain on semiconductor lasers and optical amplifiers is reevaluated in the light of recent experimental and theoretical work. Strain reduces the three-dimensional symmetry of the lattice and helps match the wave functions of the holes to the one-dimensional symmetry of the laser beam. It can also decrease the density of states at the valence band maximum and so reduce the carrier density required to reach threshold. These two effects appear to adequately explain the TE and TM gain in compressive and tensile structures, including polarization-independent amplifiers, the behavior of visible lasers and the improved frequency characteristics of InGaAs/GaAs lasers. In 1.5 /spl mu/m InGaAsP/InP lasers phonon-assisted Auger recombination appears to remain the dominant current path and can explain why the temperature sensitivity parameter to remains >

A Modern Approach to Quantum Mechanics

Preface Stern-Gerlach Experiments Rotation of Basis States and Matrix Mechanics Angular Momentum Time Evolution A System of Two Spin-1/2 Particles Wave Mechanics in One Dimension The One-Dimensional Harmonic Oscillator Path Integrals Translational and Rotational Symmetry in the Two-Body Problem Bound States of Central Potentials Time-Independent Perturbations Identical Particles Scattering Photons and Atoms Appendices Index

Symmetries of variable coefficient Korteweg–de Vries equations

The Lie point symmetries of the equation ut+f(x, t)uux+g(x,t)uxxx=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two-dimensional and one-dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.