Daniel Spirn

SCALING LIMITS OF THE CHERN–SIMONS–HIGGS ENERGY

We continue our study in [16] of the Gamma limit of the Abelian Chern–Simons–Higgs energy on a bounded, simply connected, two-dimensional domain where e → 0 and μe → μ ∈ [0, +∞]. Under the critical scaling, Gcsh ≈ |log e2, we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H1 = H1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μμ → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(ue) = det(∇ ue) in terms of the Chern–Simons–Higgs energy Ecsh.

Electrically and magnetically charged vortices in the Chern-Simons-Higgs theory

In this paper, we prove the existence of finite-energy electrically and magnetically charged vortex solutions in the full Chern–Simons–Higgs theory, for which both the Maxwell term and the Chern–Simons term are present in the Lagrangian density. We consider both Abelian and non-Abelian cases. The solutions are smooth and satisfy natural boundary conditions. Existence is established via a constrained minimization procedure applied on indefinite action functionals. This work settles a long-standing open problem concerning the existence of dually charged vortices in the classical gauge field Higgs model minimally extended to contain a Chern–Simons term.

Vortex Motion Law for the Schrödinger--Ginzburg--Landau Equations

In the Ginzburg--Landau model for superconductivity a large Ginzburg--Landau parameter

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

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A two-dimensional model for slow convection at infinite Marangoni number

corresponds to the formation of tight, stable vortices. These vortices are located where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large-

Hydrodynamic Limit of the Gross-Pitaevskii Equation

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Γ-Stability and Vortex Motion in Type II Superconductors

solutions blows up near each vortex, which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of a renormalized energy. We consider the motion of such vortices in a dynamic model for superconductivity that couples a U(1) gauge-invariant Schrodinger-type Ginzburg--Landau equation to a Maxwell-type equation under the limit of large Ginzburg--Landau parameter

Gross--Pitaevskii Vortex Motion with Critically Scaled Inhomogeneities

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Petviashvilli's Method for the Dirichlet Problem

. It is shown that under an almost-energy-minimizing condition each vortex moves in the direction of the net supercurrent located at the vortex posit...

Linear dispersive decay estimates for the 3+1 dimensional water wave equation with surface tension

In the present work, we consider the problem of a system of few vortices N ≤ 5 as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a two-pronged approach in order to reveal the configuration space of the systems preferred dynamical states. We use a Monte Carlo method parametrizing the vortex particles by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for N=2,...,5 and different vortex particle angular momenta. We then complement this picture with a dynamical system analysis of the possible rigidly rotating states. The latter reveals a supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise, exposing the full wealth of the problem even for such low-dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte Carlo approach selects for different values of the angular momentum that is used as a bifurcation parameter.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.