Showing papers on "Singularity published in 1994"

Ginzburg-Landau Vortices

Abstract: The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research. The text is concerned with the study in two dimensions of stationary solutions uE of a complex valued Ginzburg-Landau equation involving a small parameter E. Such problems are related to questions occuring in physics, such as phase transistion phenomena in superconductors and superfluids. The parameter E has a dimension of a length, which is usually small. Thus, it should be of interest to study the asymptotics as E tends to zero. One of the main results asserts that the limit u* of minimizers uE exists. Moreover, u* is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree - or winding number - of the boundary condition. Each singularity has degree one - or, as physicists would say, vortices are quantized. The singularities have infinite energy, but after removing the core energy we are led to a concept of finite renormalized energy. The location of the singularities is completely determined by minimizing the renormalized energy among all possible configurations of defects. The limit u* can also be viewed as a geometrical object. It is a minimizing harmonic map into S1 with prescribed boundary condition g. Topological obstructions imply that every map u into S1 with u=g on the boundary must have infinite energy. Even though u* has infinite energy one can think of u* as having "less" infinite energy than any other map u with u=g on the boundary. The material presented in this book covers mostly recent and original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations and complex functions. It is designed for researchers and graduate students alike and can be used as a one-semester text.

The crack tip region in hydraulic fracturing

Abstract: We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

Wave-field phase singularities: The sign principle

Abstract: Phase singularities (topological charges, dislocations, defects, vortices, etc.), which may be either positive or negative in sign, are found in many different types of wave fields. We show that on every zero crossing of the real or imaginary part of the wave field, adjacent singularities must be of opposite sign. We also show that this ``sign principle,'' which is unaffected by boundaries, leads to the surprising result that for a given set of zero crossings, fixing the sign of any given singularity automatically fixes the signs of all other singularities in the wave field. We show further how the sign of the first singularity created during the evolution of a wave field determines the sign of all subsequent singularities and that this first singularity places additional constraints on the future development of the wave function. We show also that the sign principle constrains how contours of equal phase may thread through the wave field from one singularity to another. We illustrate these various principles using a computer simulation that generates a random Gaussian wave field.

Inelastic collapse in two dimensions

Abstract: Molecular dynamic simulations show that a two-dimensional gas of inelastic disks, started from random initial conditions, has a finite time singularity in which a group of particles spontaneously forms a straight line. The inelastic disks then collide infinitely often in a finite time along their joint line of centers. The upshot of this process is a multiparticle collision that occurs through the accumulation of an infinite sequence of binary encounters.

Singularities and similarities in interface flows

Abstract: The onset of singularities in systems of nonlinear partial differential equations is an important issue in fields ranging from general relativity [27], to thermodynamic phase transitions [10], to fluid dynamics [13] The development of a mathematical singularity, when some quantity associated with the PDE “blows up,” reflects the creation of a new structure in the physical system which in turn forces the mathematical formulation to change Whether or not such singularities are possible for a given system can be a difficult question A famous problem from the theory of homogeneous incompressible fluids is the question of finite time singularity development in the three-dimensional Navier-Stokes equation: It is unknown if an initially smooth solution can develop a finite time singularity in which the vorticity becomes unbounded [23] To date, no rigorous proof or counterexample exists; neither numerical nor physical experiments have produced definitive answers [22, 25] When a particular system allows finite time singularities, many related questions become relevant For example, do all singularities have universal characteristics, or are there many possible behaviors? Which quantities are unbounded at the singular time?

The Analysis of Space-Time Singularities

Abstract: The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

Singularity analysis and localized coherent structures in (2+1)‐dimensional generalized Korteweg–de Vries equations

Abstract: In this article, a singularity structure analysis of a (2+1)‐dimensional generalized Korteweg–de Vries equation studied originally by Boiti et al., admitting a weak Lax pair, is carried out and it is proven that the system satisfies the Painleve property. Its bilinear form is constructed in a natural way from the P analysis and then it is used to generate ‘‘multidromion’’ solutions (exponentially decaying solutions in all directions). The same analysis can be extended to construct the multidromion solutions of the generalized Nizhnik–Novikov–Veselov (NNV) equation from which the NNV equation follows as a special case.

Fermi-edge singularity in resonant tunneling.

Abstract: We have observed a Fermi-edge singularity in the tunneling current between a two-dimensional electron gas (2DEG) and a zero-dimensional localized state. A sharp peak in the tunnel current is observed when the energy of the localized state matches the Fermi energy of the 2DEG. The peak grows and becomes sharper as the temperature is decreased to our lowest temperature of 70 mK. We attribute the singularity to the Coulomb interaction between the tunneling electron on the localized site and the Fermi sea.

Nonlinear evolution of the stress-driven morphological instability in a two-dimensional semi-infinite solid

Abstract: We consider the nonlinear evolution of the stress-driven morphological instability on the surface of a two dimensional semi-infinite solid. We track the branch of steady state solutions numerically and find that the solution branch terminates. The surface shape at the point of termination has a cusp singularity that points into the solid. We also consider the time-dependent evolution of the instability from small disturbances as a function of disturbance wavenumber. We find that the formation of singular grooves in the solid is a general feature of the nonlinear evolution of the instability. Various physical factors which may affect formation of the singularity are discussed.

Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics

Abstract: This paper presents a new generalized approach to the singularity analysis of a general mechanism (arbitrary kinematic chain), considered as a non-redundant input-output device with equal number of inputs and outputs. The instantaneous kinematics of a mechanism is described by means of a velocity equation, explicitly including not only the input and output velocities but also the passive-joint velocities. A precise definition of singularity of a general mechanism is provided. On the basis of the six types of singular configurations introduced in the paper singularity classifications are proposed. >

Scalar Green's functions in an Euclidean space with a conical-type line singularity

Abstract: In an Euclidean space with a conical-type line singularity, we determine the Green's function for a charged massive scalar field interacting with a magnetic flux running through the line singularity. We give an integral expression of the Green's function and a local form in the neighbourhood of the point source, where it is the sum of the usual Green's function in Euclidean space and a regular term. As an application, we derive the vacuum energy-momentum tensor in the massless case for an arbitrary magnetic flux.

Differential analysis of bifurcations and isolated singularities for robots and mechanisms

Abstract: This article develops a general technique for differential analysis that can be applied to singularities of three related problems: path tracking for nonredundant robots, self-motion analysis for robots with one degree of redundancy, and displacement analysis of single-loop mechanisms. For each of these problems, the locus of displacement solutions generally forms a set of one-dimensional manifolds in the space of variable parameters. However, if singularities occur, the manifolds may degenerate into isolated points, or into curves that include bifurcations at the singular points. Higher-order equations, derived from Taylor series expansion of the matrix equation of closure, are solved to identify singularity type and, in the case of bifurcations, to determine the number of intersecting branches as well as a Taylor series expansion of each branch about the point of bifurcation. To avoid unbounded mathematics, branch expansions are derived in terms of an introduced curve parameter. The results are useful for identifying singularity type, for numerical curve tracking with continuation past bifurcations on any chosen branch, and for determining exact rate relations for each branch at a bifurcation. The noniterative solution procedure involves configuration-dependent systems of equations that are evaluated by recursive algorithm, then solved using singular value decomposition, polynomial equation solution, and linear system solution. Examples show applications to RCRCR mechanisms and the Puma manipulator. >

Boundary Integral and Singularity Methods for Linearized Viscous Flow (C. Pozrikidis)

Abstract: (4) Mathematical problems arising in differential modelsfor viscoelastic fluids, by C. Guillop6 and J.-C. Saut (29 pages). The characteristic property of viscoelastic fluids is that the stress depends not only on the current velocity field, but on the history of the motion. One way to model such a history dependence is by a system of ordinary differential equations along particle trajectories which relate stress and velocity gradient. Such models are referred to as \"differential\" as opposed to \"integral\" models that express the stress in terms of one or several integrals along particle trajectories. The paper reviews the current state of the mathematical theory of the equations resuiting from differential models. The topics discussed include loss of evolution and change of type, existence of steady flows, inflow boundary conditions, existence for initial value problems, and flow stability. (5) Weak solutionsfor thermoconvectiveflows ofBoussinesq-Stefan type, by J.-E Rodrigues (24 pages). This paper considers a solid-liquid interface problem with a phase transition occurring at the interface. The equations are the heat equation in each phase and the Navier-Stokes equation in the fluid phase. A generalized Newtonian constitutive law with temperature dependent viscosity is allowed for the fluid. The existence of steadystate solutions is established under appropriate assumptions. (6) Boundaryand initial-boundary valueproblemsfor the Navier-Stokes equations in domains with noncompact boundaries, by V. A. Solonnikov (46 pages). In comparison to flows in interior or exterior domains, there are two new issues when the boundary extends to infinity. First, in addition to the usual initial and boundary conditions there needs to be some prescription of fluxes or pressure drops when the flow domain has several \"exits to infinity.\" Secondly, the solutions of interest often have infinite energy integrals. The paper discusses a recently developed technique of integral estimates to deal with this problem. These estimates are called \"Saint Venant’s type\" because the method was first used in the study of Saint Venant’s principle in elasticity. (7) Stability problems in electrohydrodynamics, ferrohydrodynamics and thermoelectric magnetohydrohynamics, by B. Straughan (30 pages). This paper studies the stability of a number of problems that involve the coupling of fluid mechanics with electrodynamic and thermal effects. Both linear and nonlinear stability are considered. The main technique for the nonlinear case is the energy method. (8) Mathematical results for compressible flows, by A. Valli (37 pages). Results on the existence of smooth solutions for compressible fluid mechanics are reviewed. Both inviscid and Newtonian fluids are considered. The paper discusses results on initial value problems as well as steady and time periodic flows. The book contains the following contributed papers: O. Ban, Une solution numrique des quations de Navier-Stokes stationnaires; L. Consiglieri, Stationary solutions for a Binghamflow with nonlocalfriction; S. Fabre, E’tude de la stabilit du couplage des quations d’Euler et Maxwell ?t tree dimension d’espace; E James, Diphasic equilibrium and chemical engineering; M. R. Levitin, Vibrations ofa viscous compressible fluid in bounded and unbounded domains; Yu.I. Skrynnikov, Shock wave in resonant despersion media; A. Skvortsov, Solitary vorticesna new exact solution ofhydrodynamic equations. The book will be of interest to researchers in partial differential equations who have an interest in fluid mechanics. It differs in a pleasant way from the flood of proceedings volumes that look like they are written with a xerox machine and a stapler. The reader will find wellwritten expository articles on various areas of current research. They will be of value to anyone who would like to work in one of these areas or who would simply like to find out about the current state of the art.

The Fermi edge singularity and boundary condition changing operators

Abstract: The boundary conformal field theory approach to quantum impurity problems is used to study the Fermi edge singularity, occurring in the X-ray adsorption probability. The deep-hole creation operator, in the effective low-energy theory, changes the boundary condition on the conduction electrons. By a conformal mapping, the dimension of such an operator is related to the ground-state energy for a finite system with different boundary conditions at the two ends. The Fermi edge singularity is solved, using this method, for the Luttinger liquid including backscattering and for the multi-channel Kondo problem.

Conical singularities and torsion

Abstract: Various metrics with different kinds of conical singularities, including one which does not seem to have appeared in the literature previously, are obtained by identification of flat space. A consideration of holonomy, both linear and affine in the terminology of Kobayashi and Nomizu, suggests an intepretation of the metrics in terms of torsion.

Static solutions of the spherically symmetric Vlasov-Einstein system

Abstract: We consider the Vlasov—Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular centre and have isotropic or anisotropic pressure, and solutions which have a Schwarzschild-singularity at the centre.

A density function and the structure of singularities of the mean curvature flow

Abstract: We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ℝ n+1, primarily in the boundaryless setting. We concentrate on the so-called “Type I” case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of “slow-forming singularities” of the flow. The main new ingredient introduced, to address these issues, is a certain “density function”, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.

Exact solutions and singularities in string theory.

Abstract: We construct two new classes of exact solutions to string theory which are not of the standard plane wave of gauged WZW type. Many of these solutions have curvature singularities. The first class includes the fundamental string solution, for which the string coupling vanishes near the singularity. This suggests that the singularity may not be removed by quantum corrections. The second class consists of hybrids of plane wave and gauged WZW solutions. We discuss a four-dimensional example in detail.

Singularity analysis of mechanisms and robots via a motion-space model of the instantaneous kinematics

Abstract: This paper investigates the kinematic singularities of a general mechanism (arbitrary kinematic chain), considered as a non-redundant input-output device with equal number of inputs and outputs. The instantaneous kinematics of a mechanism is described by the orientation of a linear subspace, the motion space, inside the velocity space of all potential instantaneous motions. The definition of singularity for a general mechanism is provided. On the basis of the six types of singular configurations introduced in the paper a comprehensive singularity classification is developed. >

Periodic orbit theory of diffraction.

Abstract: An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creeping waves. A new family of resonances in the two-disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used.

On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions

Abstract: The dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n‐) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrodinger equation. A Painleve singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n‐) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the form f(r)=e1r−2(n−1)+e2r−(n−2). This is confirmed by obtaining the associated Lax pair, Backlund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one‐dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher‐dimensional inhomogeneous spherically symmetric spin models can be formally mapped.

Proposal for a Simple Model of Dynamical SUSY Breaking

Abstract: We discuss supersymmetric $SU(2)$ gauge theory with a single matter field in the $I=3/2$ representation. This theory has a moduli space of exactly degenerate vacua. Classically it is the complex plane with an orbifold singularity at the origin. There seem to be two possible candidates for the quantum theory at the origin. In both the global chiral symmetry is unbroken. The first is interacting quarks and gluons at a non-trivial infrared fixed point -- a non-Abelian Coulomb phase. The second, which we consider more likely, is a confining phase where the singularity is simply smoothed out. If this second, more likely, possibility is realized, supersymmetry will dynamically break when a tree level superpotential is added. This would be the simplest known gauge theory which dynamically breaks supersymmetry.

Mesh-dependence and stress singularity in finite element analysis of creep crack growth by continuum damage mechanics

Abstract: A series of numerical analyses are first performed for a plate specimen of 316 stainless steel with a central crack to show the characteristics of the mesh dependence. It is shown that the effects of mesh size appear mainly in the initial stage of crack growth. Then, strain localization is discussed in relation to the mesh dependence phenomena. Finally, stress singularity due to boundary constraints such as observed at crack tip is shown to be a controlling factor in the present mesh dependence problem. Two simple approximate relations are proposed to describe the mesh dependent behavior for crack initiation time and crack growth rates, respectively

Coupled higher-order nonlinear Schrödinger equations in nonlinear optics: Painlevé analysis and integrability.

Abstract: A set of coupled higher-order nonlinear Schr\"odinger equations which can be derived from the electromagnetic pulse propagations in coupled optical waveguides and in a weakly relativistic plasma with nonlinear coupling of two polarized transverse waves is proposed. Using the Painlev\'e singularity structure analysis, we show that it admits the Painlev\'e property and hence we expect that it will exhibit soliton-type lossless propagations.