Derivation of the time-dependent Ginzburg-Landau equation on the line

J. Nonlin. Sci. 8 (1998) 1-15

Ian Melbourne


We give a rigorous derivation of the time-dependent one-dimensional Ginzburg-Landau equation.

As in the work of Iooss, Mielke and Demay (who derived the steady Ginzburg-Landau equation) our derivation leads to a pseudodifferential complex amplitude equation with nonlocal terms of all orders that yields the cubic order Ginzburg-Landau equation when truncated. The truncation step itself is not justified by our methods.

Furthermore, we prove that the nontruncated Ginzburg-Landau equation has a normal form SO(2) symmetry to arbitrarily high order. The normal form symmetry forces the equation to be odd with constant coefficients. This structure is broken in the tail.

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