Stable solutions and their spatial structure of the Ginzburg-Landau equation

Recent papers [7], [8] and [10] studies the Ginzburg-Landau equation A<& + A(l — l^l)^ = 0,$ == u\ +iu^ in a bounded domain fl C IR with the homogeneous Neumann boundary condition. Those works revealed the instability of non-constant solutions in any convex domain and the existance of stable non-constant solutions in topologically non-trivial domains. This report surveys these studies together with introduction of a new result.