Related papers: Automatic exploration techniques for the numerical…
This paper considers the extreme type-II Ginzburg-Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, it…
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…
We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a…
This paper considers the extreme type-II Ginzburg--Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned…
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…
A numerical approach to Ginzburg-Landau (GL) theory is demonstrated and we review its applications to several examples of current interest in the research on superconductivity. This analysis also shows the applicability of the…
We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We…
We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are…
Recently, a novel bifurcation technique known as the deflated continuation method (DCM) was applied to the single-component nonlinear Schr\"odinger (NLS) equation with a parabolic trap in two spatial dimensions. The bifurcation analysis…
A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method…
We study the Ginzburg-Landau energy of a superconductor with a variable magnetic field and a pinning term in a bounded smooth two dimensional domain $\Omega$. Supposing that the Ginzburg-Landau parameter and the intensity of the magnetic…
In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and…
pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on…
We describe how some differential geometric bifurcation problems can be treated with the MATLAB continuation and bifurcation toolbox pde2path. The basic setup consists in solving the PDEs for the normal displacement of an immersed surface…
We derive bifurcation test equations for A-series singularities of nonlinear functionals and, based on these equations, we propose a numerical method for detecting high codimensional bifurcations in parameter-dependent PDEs such as…
We use the $\mathbb T^2$-equivariant degree to establish the existence of unbounded branches of rotating spiral wave solutions with any number of arms for the complex Ginzburg Landau equation GLe on the planar unit disc, leveraging the…
Some new global results are given about solutions to the boundary value problem for the Euler-Lagrange equations for the Ginzburg-Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range…
We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand,…
We study two superconducting systems using the Landau-Ginzburg equations. The first is a superconducting half-space with an applied magnetic field parallel to the surface. We calculate the maximum applied field that still supports…
We examine the behavior of a one-dimensional superconducting wire exposed to an applied electric current. We use the time-dependent Ginzburg-Landau model to describe the system and retain temperature and applied current as parameters.…