the first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. this method can be applied to non integrable equations as well as to integrable ones. in this paper, the first integral method is used to construct exact solutions of the 2d ginzburg-landau equation.

We describe a Sobolev gradient method for finding minima of the Ginzburg–Landau functional for superconductivity. This method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the Ginzburg–Landau equations.

We study the asymptotic limit of solutions of the Ginzburg-Landau equations in two dimensions with or without magnetic field. We first study the Ginzburg-Landau system with magnetic field describing a superconductor in an applied magnetic field, in the “London limit” of a Ginzburg-Landau parameter κ tending to ∞. We examine the asymptotic behavior of the “vorticity measures” associated to the v...

Suppose that ω ⊂ Ω ⊂ R. In the annular domain A = Ω \ ω̄ we consider the class J of complex valued maps having degree 1 on ∂Ω and ∂ω. It was conjectured in [5] that the existence of minimizers of the Ginzburg-Landau energy Eκ in J is completely determined by the value of the H-capacity cap(A) of the domain and the value of the Ginzburg-Landau parameter κ. The existence of minimizers of Eκ for al...

In a simply connected two dimensional domain Ω, we consider Ginzburg-Landau minimizers u with zero degree Dirichlet boundary condition g ∈ H1/2(∂Ω;S1). We prove uniqueness of u whenever either the energy or the Ginzburg-Landau parameter are small. This generalizes a result of Ye and Zhou requiring smoothness of g. We also obtain uniqueness when Ω is multiply connected and the degrees of the vor...

The so-called Ginzburg-Landau formalism applies for parabolic systems which are defined on cylindrical domains, which are close to the threshold of instability, and for which the unstable Fourier modes belong to non-zero wave numbers. This formalism allows to describe an attracting set of solutions by a modulation equation, here the Ginzburg-Landau equation. If the coefficient in front of the c...

We study a mixed heat and Schrödinger 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...