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By Haim Brezis & Tatsien Li

The Ginzburg-Landau equation as a mathematical version of superconductors has turn into an exceptionally useful gizmo in lots of components of physics the place vortices sporting a topological cost seem. The extraordinary development within the mathematical realizing of this equation includes a mixed use of mathematical instruments from many branches of arithmetic. The Ginzburg-Landau version has been an grand resource of latest difficulties and new rules in research, geometry and topology. This assortment will meet the pressing wishes of the experts, students and graduate scholars operating during this zone or comparable parts.

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Ginzburg-Landau Vortices

The Ginzburg-Landau equation as a mathematical version of superconductors has turn into an exceptionally great tool in lots of components of physics the place vortices wearing a topological cost look. The amazing development within the mathematical figuring out of this equation includes a mixed use of mathematical instruments from many branches of arithmetic.

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For fixed a, w is of order 1, hence is of lower order than the previous 34 Amandine Aftalion terms. 16) The location of vortices is determined by the minimum of w or equivalent ly by the minimum of under the constraint YLi&i + vf = constant. For a = 1, this has been studied by [GS]. When a ^ 1, it is an open question. When n = 2 or 3 (and a not close to 1), the vortices are on the longest axis of the ellipse. 13). We will do it in the case us has a single vortex tube tending to the curve 7. t. 17) and assume that Ae is small, A being our matching parameter to be fixed later on.

Nucleation of superconductivity in decreasing fields II. European J. Appl. Math. 5 (1994), 469-494. Gunzburger. A Ginzburg-Landau type model of superconducting / normal junctions including Josephson junctions. European J. Appl. Math. 6 (1995), 97-114. Ockendon. Normal superconducting transitions in Landau-Ginzburg theory. Proc. Royal Soc. Edinburgh, 119 (1991), 117-124. Ockendon. Macroscopic models of superconductivity. SIAM Review, 34 4 (1992), 529-560. Richardson. Vortex pinning by inhomogeneities in type II superconductors.

In this scaling the length unit is the penetration depth. The object of our study is therefore J{u, A) = \f \cur\A - He|2 + \ f - iA) u\\22 + - L (i _ ?? 4) where d£u — Dimension reduction A natural special case is that where the domain is an infinite cylinder in R3 and He is parallel to the axis (think of an infinitely long insulated wire). Assuming translational invariance of (u,A) and invariance with respect to reflections across a plane perpendicular to the axis we have, taking the third coordinate axis as the cylinder's axis He = /iext(O,0,1), u(x,y,z) = u(x,y), A{x,y,z) = (A1(x,y),A2{x,y),0).

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