DERIVATION OF THE TIME-DEPENDENT GINZBURG-LANDAU EQUATION ON THE LINE

We give a rigorous derivation of the time-dependent one-dimensional Gi nzburg-landau equation. As in the work of Iooss, Mielke, and Demay [11 ] (who derived the steady Ginzburg-Landau equation), our derivation le ads to a pseudodifferential complex amplitude equation with nonlocal t erms of all orders that yields the cubic order Ginzburg-Landau equatio n when truncated. The truncation step itself is not justified by our m ethods. Furthermore, we prove that the nontruncated Ginzburg-Landau eq uation has a normal form SO(2) symmetry to arbitrarily high order. The normal form symmetry forces the equation to be odd with constant coef ficients. This structure is broken in the tail.