It is well established that the superconductivity in the recently discovered superconducting compound MgB$_{ 2}$ resides in the quasi-two-dimensional band ($\sigma$-band) and three-dimensional band ($\pi$-band). We demonstrate that, due to such band structure, the anisotropic Ginzburg-Landau theory practically does not have region of applicability, because gradient expansion in the $c$ direction breaks down. In the case of dirty $\pi$-band we derive the simplest equations which describe properties of such superconductors near $T_{ c}$ and explore some consequences of these equations.
Categories: cond-mat.supr-con