Optical Computation of Divergence Operation for Vector Fields

We obtain a smooth and analytic local classification of pairs of foliations of theplane assuming that one of the foliations is defined by a nonsingular vector field and theother foliation is defined by a singular vector field.

A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.

We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .

A specific singularity of a vector field on is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Šil'nikov bifurcations, which have codimension 1. Special attention is paid to the C∞-flatness of this subordinate phenomenon.

In this paper an expository account on singularities of reversible vector fields on manifolds and boundary singularities is presented. Also we present the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane {(x, y)∈R2:x>-0} whose topological unfolding is given by the quadratic three parameter family\(y\frac{\partial }{{\partial x}} + (x^2 + ax + c + \alpha x(x + b))\frac{\partial }{{\partial y}}\),\(\alpha = \pm 1\). This study can be applied to the analysis of the behavior of singularity of the germ of vector fieldX0(x, y)=(y, 2x(x4+x2y)) in the class of reversible vector fields.

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are exhaustively described. As examples, properties of singular reduction operators of (1 + 1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.

We study germs of singular holomorphic vector fields at the origin of ℂ n of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey order of the latter is linked to the diophantine type of the small divisors.

Polarization singularities of vectorial electromagnetic fields locate at the positions where properties of polarization ellipses are not defined. First observed for conical diffraction in 1830s, polarization singularities have been studied systematically with the underlying concepts being reshaped and deepened by many pioneers of wave optics. Here we review the recent results on the generation and observation ofpolarization singularities in metaphotonics. We start with the discussion of polarization singularities in the Mie theory, where both electric and magnetic multipoles are explored from perspectives of local and global polarization properties. We then proceed with the discussion of various photonic-crystal structures, for which both near- and far-field patterns manifest diverse polarization singularities characterized by the integer Poincaré or more general half-integer Hopf indices (topological charges). Next, we review the most recent studies of conversions from polarization to phase singularities in scalar wave optics, demonstrating how bound states in the continuum can be exploited to generate directly optical vortices of various charges. Throughout our paper, we discuss and highlight several fundamental concepts and demonstrate their close connections and special links to metaphotonics. We believe polarization singularities can provide novel perspectives for light-matter manipulation for both fundamental studies and their practical applications.

Singularities of Holomorphic Vector Fields in Dimensions $$\geq 3$$: Results and Problems

On singularities of discontinuous vector fields

Singularities of reversible vector fields

We present an approach for monitoring the positions of vector field singularities and related structural changes in time-dependent datasets. The concept of singularity index is discussed and extended from the well-understood planar case to the more intricate three-dimensional setting. Assuming a tetrahedral grid with linear interpolation in space and time, vector field singularities obey rules imposed by fundamental invariants (Poincare index), which we use as a basis for an efficient tracking algorithm. We apply the presented algorithm to CFD datasets to illustrate its purpose. We examine structures that exhibit topological variations with time and describe some of the insight gained with our method. Examples are given that show a correlation in the evolution of physical quantities that play a role in vortex breakdown.

The accelerated expansion of the universe is interpreted as an effect of a defect in space-time treated as a four-dimensional continuum endowed with physical properties. The analogy is with texture defects in material continua, like dislocations and disclinations, described in terms of a singular displacement vector field. A Lagrangian for empty space-time is proposed exploiting one further analogy between the phase space of a Robertson-Walker universe and the phase space of a point particle moving across an homogeneous isotropic medium. The model, named Cosmic Defect theory, produces, as a byproduct, also inflation near the initial singularity. The theory has been applied to fit the luminosity data of 192 type Ia supernovae. The results are satisfying and comparable with the ones obtained by means of the Λ Cold Dark Matter standard model.

We introduce a new notion of Reeb invariant structure Jacobi operator and two kinds of singular normal vector field N for a real hypersurface M in the complex quadric Qm, m ? 3. If the unit normal N is U-isotropic, we give a classification of Hopf real hypersurfaces with Reeb invariant structure Jacobi operator in the complex quadric Qm, for m ? 3.