Singular Line Research Articles

Events in fields of optical vortices: rings and reconnection

It is known (Berry and Dennis 2007 J. Phys. A: Math. Theor. 40 65–74; Berry and Dennis 2012 Eur. J. Phys. 33 723–731) that only one kind of reaction between wave vortices can occur generically in a monochromatic optical field. It appears either in elliptic form as the birth and death of vortex rings or in hyperbolic form as reconnection between separate vortex lines. To make it occur the field must be changed, and, since the codimension is one, it suffices to adjust a single external parameter. The paper analyses a model in which the initial field is produced by superposing n plane waves of the same frequency but different random amplitudes, directions and phases. This is perturbed by an additional plane wave of variable amplitude. The field necessarily obeys the Helmholtz equation and, in spite of the randomness, there is systematic behaviour for n = 3 and 4, which leads to some understanding of the more complicated results for higher values of n. Three plane waves of equal amplitude, perturbed by a fourth, provide a surprising special case, and the remarkable succession of events discovered by (O’Holleran et al 2006a J. Eur. Opt. Soc. Rapid Publ. 1 06008; O’Holleran et al 2006b Opt. Express 14 3039–3044) is fully explained. This is a central point of the paper. Looking at the singularity itself, and initially following Berry and Dennis, the simplest model that satisfies the Helmholtz equation is presented and also the most general local model that uses ‘polynomial waves’. We also consider waves that are described simply by a polynomial without any exponential factor. The inclusion of time in the polynomial allows explicitly for quasi-monochromatic waves in which the events occur spontaneously, rather than by adjusting an external control. The circulating phase structure around a simple wave vortex is its most distinctive feature. But in reconnection two such singular vortex lines cross one another and the phase pattern around them must reflect this higher singularity. How it does so is illustrated in the paper.

Analysis of singular stresses at a vertex and along a singular line in three-dimensional bonded joints using a conservative integral

Abstract In the present study, singular stress fields in three-dimensional dissimilar material joints are investigated at a vertex and along a free edge (a singular line) of an interface. A conservative integral based on Betti's reciprocal principle is used to calculate the intensities of the singularities in three-dimensional dissimilar material joints. Two geometries in which the angle between the side surface of the vertex equals 60° and 90° and a range of material combinations are used to investigate the characteristics of the stress singularity. The relationships between the angular functions of the stresses at the vertex and along the singular line are studied. The intensities of the singularities at several positions along the singular line of the interface are determined. Then, the relationship between the intensity of the singularity at the vertex and that at a point located on the singular line is investigated. Finally, the intensities of the singularities along the singular line can be described as a function of the distance from the vertex and the singular stress at the vertex.

Ground-state phase diagram of the repulsive fermionict−t′Hubbard model on the square lattice from weak coupling

We obtain a complete and exact in the weak-coupling limit ($U \rightarrow 0$) ground state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors $0 < n < 2$ and next-nearest-neighbour hopping amplitudes $0 \le t^{\prime} \le 0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high --- higher than that of the fundamental mode of the corresponding irreducible representation --- number of nodes. The effective coupling strength in the Cooper channel $\lambda$, which determines the critical temperature $T_c$ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\lambda$ are identified. It is shown that besides the expected increase of $\lambda$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\lambda$ can be found at quarter filling and $t^{\prime}=0.5$ due to the presence of a line of nesting at $t^{\prime} \ge 0.5$. The results can serve as benchmarks for controlled non-perturbative methods and guide the ongoing search for high-$T_c$ superconductivity in the Hubbard model.

Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

Edge pinning and transformation of defect lines induced by faceted colloidal rings in nematic liquid crystals.

Nematic colloids exhibit a large diversity of topological defects and structures induced by colloidal particles in the orientationally ordered liquid crystal host fluids. These defects and field configurations define elastic interactions and medium-mediated self-assembly, as well as serve as model systems in exploiting the richness of interactions between topologies and geometries of colloidal surfaces, nematic fields, and topological singularities induced by particles in the nematic bulk and at nematic-colloidal interfaces. Here we demonstrate formation of quarter-strength surface-pinned disclinations, as well as a large variety of director field configurations with splitting and reconnections of singular defect lines, prompted by colloidal particles with sharp edges and size large enough to define strong boundary conditions. Using examples of faceted ring-shaped particles of genus g=1, we explore transformation of defect lines as they migrate between locations in the bulk of the nematic host to edge-pinned locations at the surfaces of particles and vice versa, showing that this behavior is compliant with topological constraints defined by mathematical theorems. We discuss how transformation of bulk and surface defect lines induced by faceted colloids can enrich the diversity of elasticity-mediated colloidal interactions and how these findings may impinge on prospects of their controlled reconfigurable self-assembly in nematic hosts.

Anisotropic density fluctuations, plasmons, and Friedel oscillations in nodal line semimetal

Motivated by recent experimental efforts on three-dimensional semimetals, we investigate the static and dynamic density response of the nodal line semimetal by computing the polarizability for both undoped and doped cases. The nodal line semimetal in the absence of doping is characterized by a ring-shape zero energy contour in momentum space, which may be considered as a collection of Dirac points. In the doped case, the Fermi surface has a torus shape and two independent processes of the momentum transfer contribute to the singular features of the polarizability even though we only have a single Fermi surface. In the static limit, there exist two independent singularities in the second derivative of the static polarizability. This results in the highly anisotropic Friedel oscillations which show the angle-dependent algebraic power law and the beat phenomena in the oscillatory electron density near a charged impurity. Furthermore, the dynamical polarizability has two singular lines along and , where η is the angle between the external momentum and the plane where the nodal ring lies. From the dynamical polarizability, we obtain the plasmon modes in the doped case, which show anisotropic dispersions and angle-dependent plasma frequencies. Qualitative differences between the low and high doping regimes are discussed in light of future experiments.

Two-layer spin coating flow of Newtonian liquids: A computational study

Spin coating is the most commonly used method in industries to form coating films of desired thickness and functionality. In case of two-layer spin coating process, stratified layers of two immiscible liquids are deposited onto the substrate simultaneously, which spread and thin to form two-layer coating film of finite thickness. Questions concerning the effect of surface/interfacial tension on uniformity of these films and contact line evolution are relevant and need thorough investigation. Therefore, an axisymmetric model governing the flow of two-layer spin coating process is developed here. Liquids used for the study are assumed to be Newtonian and fully wetting. The contact line singularity is resolved using a precursor layer model and the governing equations are simplified using lubrication approximation. A Galerkin finite-element method (G/FEM) based scheme is developed to solve the resulting fourth order non-linear PDEs. Simulation results reveal that the fluid properties like ratio of the viscosity of upper layer fluid to lower layer and ratio of the upper gas-liquid surface tension to inner liquid-liquid interfacial tension have profound impact on the time evolution of the film profile, contact radius and shape of the capillary ridges. It is observed that a uniform two-layer film surrounded by thin single layer film is formed when the viscosity ratio is small. On the contrary, when viscosity ratio is large, a thin two-layer film surrounded by bulky capillary ridges is formed. Similarly, the results also show that sharpness of capillary ridge increases with decrease in the surface tension ratio. Further, it is found that increase in the precursor layer thickness increases the spreading rate, thereby making the film more uniform. Finally, the uniformity of the final two-layer film does not get affected by the initial volume of fluid present in the upper layer.

Thermal singularity and contact line motion in pool boiling: Effects of substrate wettability.

The dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] is employed to model the growth of a single vapor bubble in a superheated liquid on a flat homogeneous substrate. The bubble spreading dynamics in the pool boiling regime has been numerically investigated for one-component van der Waals fluids close to the critical point, with a focus on the effect of the substrate wettability on bubble growth and contact line motion. The substrate wettability is found to control the apparent contact angle and the rate of bubble growth (the rate of total evaporation), through which the contact line speed is determined. An approximate expression is derived for the contact line speed, showing good agreement with the simulation results. This demonstrates that the contact line speed is primarily governed by (1) the circular shape of interface (for slow bubble growth), (2) the constant apparent contact angle, and (3) the constant bubble growth rate. It follows that the contact line speed has a sensitive dependence on the substrate wettability via the apparent contact angle which also determines the bubble growth rate. Compared to hydrophilic surfaces, hydrophobic surfaces give rise to a thinner shape of bubble and a higher rate of total evaporation, which combine to result in a much faster contact line speed. This can be linked to the earlier formation of a vapor film and hence the onset of boiling crisis.

Nature of the many-body excitations in a quantum wire: Theory and experiment

The natural excitations of an interacting one-dimensional system at low energy are hydrodynamic modes of Luttinger liquid, protected by the Lorentz invariance of the linear dispersion. We show that beyond low energies, where quadratic dispersion reduces the symmetry to Galilean, the main character of the many-body excitations changes into a hierarchy: calculations of dynamic correlation functions for fermions (without spin) show that the spectral weights of the excitations are proportional to powers of $\mathcal{R}^{2}/L^{2}$, where $\mathcal{R}$ is a length-scale related to interactions and $L$ is the system length. Thus only small numbers of excitations carry the principal spectral power in representative regions on the energy-momentum planes. We have analysed the spectral function in detail and have shown that the first-level (strongest) excitations form a mode with parabolic dispersion, like that of a renormalised single particle. The second-level excitations produce a singular power-law line shape to the first-level mode and multiple power-laws at the spectral edge. We have illustrated crossover to Luttinger liquid at low energy by calculating the local density of state through all energy scales: from linear to non-linear, and to above the chemical potential energies. In order to test this model, we have carried out experiments to measure momentum-resolved tunnelling of electrons (fermions with spin) from/to a wire formed within a GaAs heterostructure. We observe well-resolved spin-charge separation at low energy with appreciable interaction strength and only a parabolic dispersion of the first-level mode at higher energies. We find structure resembling the second-level excitations, which dies away rapidly at high momentum in line with the theoretical predictions here.

Bifurcations of Traveling Wave Solutions of a Nonlinear Wave Model Created by Diffraction in Periodic Media

In this paper, we consider a model created by diffraction in periodic media. The study of the traveling wave solutions for this model derives a planar dynamical system with a singular straight line. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, periodic peakon solutions, compactons, etc.) under different parameter conditions.

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.

保存積分を用いた3次元ピエゾ接合体の界面角部近傍の応力特異線上の特異応力解析

In the present study, the conservative integral formulation is developed for analyzing the intensity of singularity at a point located on the singular line (along a free edge) of three-dimensional piezoelectric bonded joint. The intensities of singularities at several positions along the singular line of the interface are examined. The relationship between the stress singularity at the vertex and that at a point located on the singular line is discussed.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (III)

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions.

Topological defects in cholesteric liquid crystal shells.

We investigate experimentally and numerically the defect configurations emerging when a cholesteric liquid crystal is confined to a spherical shell. We uncover a rich scenario of defect configurations, some of them non-existent in nematic shells, where new types of defects are stabilized by the helical ordering of the liquid crystal. In contrast to nematic shells, here defects are not simple singular points or lines, but have a large structured core. Specifically, we observe five different types of cholesteric shells. We study the statistical distribution of the different types of shells as a function of the two relevant geometrical dimensionless parameters of the system. By playing with these parameters, we are able to induce transitions between different types of shells. These transitions involve interesting topological transformations in which the defects recombine to form new structures. Surprisingly, the defects do not approach each other by taking the shorter distance route (geodesic), but by following intricate paths.

Influence of the initial parameters of the magnetic field and plasma on the spatial structure of the electric current and electron density in current sheets formed in helium

The influence of the initial parameters of the magnetic field and plasma on the spatial structure of the electric current and electron density in current sheets formed in helium plasma in 2D and 3D magnetic configurations with X-type singular lines is studied by the methods of holographic interferometry and magnetic measurements. Significant differences in the structures of plasma and current sheets formed at close parameters of the initial plasma and similar configurations of the initial magnetic fields are revealed.

Application of the Methods of the Theory Similarity and Dimensional Analysis for Research the Local Stress-strain State in the Neighborhood of an Irregular Point of the Boundary

In this paper, the methods of similarity theory and dimensional analysis analyzes the features of solving the problem of elasticity, caused by a form of boundary or “geometric factor” and the finite break of specified internally strains emerging in an irregular point of the boundary. Given the similarity criteria for the self-similar solution of the elasticity problem in a neighborhood of irregular points on a singular line of the elastic body of the border.Due to self-solve the elastic problem, stress, strain, displacement in a neighborhood of an irregular point of the boundary admit of the group similarity and functions possess the property of homogeneity, characterized by the fact that these functions can be represented in the form of power complexes. The properties of similarity and homogeneity must have an experimental solution, resulting in the model as the fringe pattern by photoelasticity. Therefore, sequence stripes in some neighborhood of irregular point of the boundary should have the property of similarity, homogeneity as well as stress and be represented in the form of power complexes, m ∼ Cλr, which is confirmed by research of experimental data.

Cuspidal quintics and surfaces with , and 5-torsion

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

Singular dynamics of a q-difference Painlevé equation in its initial-value space

We construct the initial-value space of a q-discrete first Painlevé equation explicitly and describe the behaviours of its solutions w(n) in this space as with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.

Solutions of some Monge–Ampère equations with isolated and line singularities

In this paper, we study existence, regularity, classification, and asymptotic behaviors of solutions of some Monge–Ampère equations with isolated and line singularities. We classify all solutions of det⁡∇2u=1 in Rn with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having k non-removable singular points for k>1, modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi–Yau metrics.

Hölder Singular Metrics on Big Line Bundles and Equidistribution

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact K\"ahler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are H\"older with singularities we also estimate the speed of convergence.