Exploring Vortex Dynamics in the Presence of Dissipation: Analytical and Numerical Results

In this paper, we systematically examine the stability and dynamics of vortices under the effect of a phenomenological dissipation used as a simplified model for the inclusion of the effect of finite temperatures in atomic Bose-Einstein condensates. An advantage of this simplified model is that it enables an analytical prediction that can be compared directly (and favorably) to numerical results. We then extend considerations to a case of considerable recent experimental interest, namely that of a vortex dipole and observe good agreement between theory and numerical computations in both the stability properties (eigenvalues of the vortex dipole stationary states) and the dynamical evolution of such configurations.Comment: 12 pages, 5 figures, accepted by PR

Recovering a Potential from Cauchy Data via Complex Geometrical Optics Solutions

This paper is devoted to the problem of recovering a potential q in a domain in ℝd for d≥3 from the Dirichlet to Neumann map. This problem is related to the inverse Calder\'on conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of W2,d/2 conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvester and Uhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of the second result, in the work of Ferreira et al. [10], is based on the construction of CGO solutions via Carleman estimates. The main goal of the paper is to understand whether or not an approach which is based on the construction of CGO solutions in the spirit of Sylvester and Uhlmann and involves only standard Sobolev spaces can be used to obtain these results. In fact, we are able to obtain a new proof of uniqueness for the Calder\'on problem for 1) a slightly different class as the one in [11], and for 2) the class of W2,d/2 conductivities. The proof of statement 1) is based on a new estimate for CGO solutions and some averaging estimates in the same spirit as in [11]. The proof of statement 2) is on the one hand based on a generalized Sobolev inequality due to Kenig et al. [14] and on another hand, only involves standard estimates for CGO solutions [22]. We are also able to prove the uniqueness of a potential for 3) the class of Ws,3/s (⫌W2,3/2) conductivities with 3/2<s<2 in three dimensions. As far as we know, statement 3) is new

Analysis of Nematic Liquid Crystals with Disclination Lines

We investigate the structure of nematic liquid crystal thin films described by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the elasticity constant goes to zero a limiting uniaxial texture forms with disclination lines corresponding to a finite number of defects, all of degree 1/2 or all of degree -1/2. We also state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs model without magnetic field that follows from a similar proof.Comment: 40 pages, 1 figur

Ginzburg-Landau vortex dynamics with pinning and strong applied currents

We study a mixed heat and Schr\"odinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or \ep \to 0, when the intensity of the electric current and applied magnetic field on the boundary scale like \lep. We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg-Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg-Landau vortices.Comment: 48 pages; v2: minor errors and typos correcte

Juxtaposing a cultural reading of landscape with institutional boundaries: the case of the Masebe Nature Reserve, South Africa

The article explores theoretically the juxtaposition of local stories about landscape with institutional arrangements and exclusionary practices around a conservation area in South Africa. The Masebe Nature Reserve is used as a case study. The article argues that the institutional arrangements in which the nature reserve is currently positioned are too static, and consequently exclusionary, in their demarcation of boundaries. This stifles local communities’ sense of belonging to these landscapes. Hence, they strongly resent and feel alienated by the nature reserve. Their opposition and alienation often manifests in poaching. The empirical material is based on how local people living adjacent to the Masebe Nature Reserve have historically named and interpreted the area’s impressive sandstone mountains, in the process creating a sense of belonging. Juxtaposing this mostly tranquil cultural reading of the landscape to the institutional practices of boundary demarcation gives the analysis an immediate critical edge regarding issues of social justic

Dynamics for Ginzburg-Landau vortices under a mixed flow

Abstract. We consider a complex Ginzburg-Landau equation that contains a Schrödinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time. 1. introduction We study the asymptotic behavior of the complex Ginzburg-Landau equation (αε + i)∂tuε = ∆uε + 1 ε2 uε(1 − |uε | 2 (1.1) uε(·, 0) = u 0 (1.2) ε(·) as ε → 0, where uε(t) : Ω → C, Ω ⊂ R 2 a bounded and simply connected C 1 domain, with either Dirichlet boundary conditions (1.3) uε = g on ∂Ω, where g ∈ C 1 (∂Ω, S 1), or Neumann boundary conditions