Solutions of the Ginzburg-Landau equations ford-wave superconductors and a proof of their fourfold symmetry

In some high-temperature superconductors, d-wave pairing with the quadrapole symmetry dominates over the conventional spherically symmetric s-pairing. The Ginzburg-Landau theory of this superconductive state should involve both s-wave and d-wave order parameters, psis and psid. There are two critical transition temperatures for these materials Ts and Td. We study the Ginzburg-Landau equations for these d-wave superconductors. Near the d-wave vortex core, we find fourfold symmetric solutions of the equations. More specifically, as the temperature T ↑ Td in the regime Ts < T < Td, we find a locally unique solution, which is a perturbation of ((d0, A 0), 0), of the Ginzburg-Landau equations. Also, we prove that psi s = ( 1bP 2x-P2y psid) + O( 1b2 ) which is conjectured by physicists. Besides, we study the equations without magnetic field cases.