Complex Ginzburg - Landau Equations in HighDimensions and Codimension two
نویسنده
چکیده
Complex Ginzburg-Landau Equations were originated in the theory of Super-conductivity, 13]. When Ginzburg-Landau parameter chosen to be a special constant, the equations are called self-dual vortex equations that were carefully studied by Jaae and Taubes 16]. For the vortex equation on a Riemannian surface , one considers an open, smooth domain with, possibly empty, smooth boundary @. Let L be a complex line bundle over equipped with a Hermetian metric < :; : >. For a section u of L we write ju(x)j 2 =< u(x); u(x) >. Then the Ginzburg-Landau functionals are deened for a section u of L and a unitary connection A on L. The self-dual case of this functional is given by E(u; A;) = Z jdAj 2 + jr A uj 2 + 1 4 (1 ? juj 2) 2 dx (I.1) where dx is the volume form of some xed KK ahler metric on. As usual we adopt the following notations : r A u = (d ? iA)u ;
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