  The Ginzburg Landau theory for d_{x^2-y^2}-wave superconductors is
constructed, by starting from the Gor'kov equation with including correction
terms up to the next order of ln(T_c/T). Some of the non-local correction terms
are found to break the cylindrical symmetry and lead to the fourfold symmetric
core structure, reflecting the internal degree of freedom in the pair
potential. Using this extended Ginzburg Landau theory, we investigate the
fourfold symmetric structure of the pair potential, current and magnetic field
around an isolated single vortex, and clarify concretely how the vortex core
structure deviates from the cylindrical symmetry in the d_{x^2-y^2}-wave
superconductors.
