arXiv research

The article extends vortex filament theory to Hermitian reductive Lie algebras.

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such a…

The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such …

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a…

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

New flow generates surfaces with constant curvature.

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold of constant curvature gives rise under Hasimoto's transformation to the non-linear Schrodinger equation. We also give a natural interpretatio…

The Skew Mean Curvature Flow(SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.

Study on elastic curves with variable stiffness, derived from bending energy.

The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior p…

We define a class of geometric flows on a complete Kähler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrödinger equations etc. Furthermore, we consider the existence for these flows from $S^1$ into a complet…

The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests the appearance of singularity in nearb…

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation. In this paper, we present its discrete analogue, namely, a model of defor…

In this mostly pedagogical tutorial article a brief introduction to modern geometrical treatment of fluid dynamics and electrodynamics is provided. The main technical tool is standard theory of differential forms. In fluid dynamics, the approach is based on general theory of integral invariants (due to Poincare and Car…

We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equa…

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^…

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for …

The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, wa…

In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used …

This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely when there is no filamentation of the knot, as described by Hrencecin and Kauffm…

In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. W…

Generalizes Hasimoto transformation to arbitrary flows on space curves.

3D filament plots visualize curves in datasets, avoiding visual clutter.

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …

Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.

If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlin…

Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infin…

We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambjørn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2…

The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.

Study vortex loops as coadjoint orbits of diffeomorphisms.

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

New minimal surfaces found from vortex crystals.

The paper studies point vortex dynamics on specific Kähler twistor spaces.

Solved Demailly systems for Vortex bundles on manifolds.

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex …

Alternative construction of quasi-Fuchsian flows using vortex equations.

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solut…

Theory of point vortices extended to closed surfaces.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Topologically protected vortex knots and links are proposed and proven.

Tensor measures chirality for curves, even those with rough edges.

Inhomogeneous plasmas filaments instabilities are investigated by using the techniques of classical differential geometry of curves where Frenet torsion and curvature describe completely the motion of curves. In our case the Frenet frame changes in time and also depends upon the other coordinates taking into account th…

Researchers create topologically protected knots in a realizable system.

This paper uses Ghrist barcodes to track persistent shapes in video frames.