arXiv subjects

Amandine Aftalion

Publications and source records attributed to Amandine Aftalion.

41 records · Page 3Linked to original sources

arXiv2001

Bose-Einstein condensate: critical velocities and energy diagrams in the Thomas-Fermi regime

For a Bose-Einstein condensate placed in a rotating trap and confined in the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas Fermi regime. We investigate an asymptotic development of the energy, the critical velocities of nucleation of vortices with respect to a small parameter $\ep$ and the location of vortices. The limit $\ep$ going to zero corresponds to the Thomas Fermi regime. The non-dimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-kappa high-field limit and our estimates rely on techniques developed for this latter problem. We also take the advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.

cond-mat.stat-mech↗

arXiv2001

The bifurcation diagrams for the Ginzburg-Landau system for superconductivity

In this paper, we provide the different types of bifurcation diagrams for a superconducting cylinder placed in a magnetic field along the direction of the axis of the cylinder. The computation is based on the numerical solutions of the Ginzburg-Landau model by the finite element method. The response of the material depends on the values of the exterior field, the Ginzburg-Landau parameter and the size of the domain. The solution branches in the different regions of the bifurcation diagrams are analyzed and open mathematical problems are mentioned.

cond-mat.supr-con↗

arXiv2000

On the symmetry and uniqueness of solutions of the Ginzburg-Landau equations for small domains

In this paper, we study the Ginzburg-Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero, that is no vortex. More precisely, we show that the order parameter $Ψ$ is almost constant. Additionnally, we obtain that if the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincaré inequality.

cond-mat.supr-con↗

arXiv2000

Pinning phenomena in the Ginzburg-Landau Model of Superconductivity

We study the Ginzburg-Landau energy of superconductors with a term $a_\ep$ modelling the pinning of vortices by impurities in the limit of a large Ginzburg-Landau parameter $κ=1/\ep$. The function $a_\ep$ is oscillating between 1/2 and 1 with a scale which may tend to 0 as $κ$ tends to infinity. Our aim is to understand that in the large $κ$ limit, stable configurations should correspond to vortices pinned at the minimum of $a_\ep$ and to derive the limiting homogenized free-boundary problem which arises for the magnetic field in replacement of the London equation. The method and techniques that we use are inspired from those of Sandier-Serfaty (in which the case $a_\ep \equiv 1$ was treated) and based on energy estimates, convergence of measures and construction of approximate solutions. Because of the term $a_\ep(x)$ in the equations, we also need homogenization theory to describe the fact that the impurities, hence the vortices, form a homogenized medium in the material.

cond-mat.supr-con↗

arXiv1999

On the solutions of the one dimensional Ginzburg-Landau equations for superconductivity

This paper gives a complete description of the solutions of the one dimensional Ginzburg-Landau equations which model superconductivity phenomena in infinite slabs. We investigate this problem over the entire range of physically important parameters: $a$ the size of the slab, $κ$ the Ginzburg-Landau parameter, and $h_0$, the exterior magnetic field. We do extensive numerical computations using the software AUTO, and determine the number, symmetry and stability of solutions for all values of the parameters. In particular, our experiments reveal the existence of two key-points in parameter space which play a central role in the formation of the complicated patterns by means of bifurcation phenomena. Our global description also allows us to separate the various physically important regimes, to classify previous results in each regime according to the values of the parameters and to derive new open problems. In addition, our investigation provides new insight into the problem of differentiating between the types of superconductors in terms of the parameters.

cond-mat.supr-con↗