Analyticity of Ginzburg-Landau Modes

Ginzburg-Landau formalism applies for dissipative systems defined on cylindrical domains which are close to the threshold of instability and for which the unstable Fourier modes belong to non-zero wave numbers. In these situations the real part of the curve of critical eigenvalues as function drawn over the wave numbers k is positive of height (ε2) and of width (ε). Here it is shown that the set of solutions which can be described by the Ginzburg-Landau formalism is attractive. To do this we demonstrate that in Fourier space peaks appear at integer multiples of the critical wave number kc. These peaks called Ginzburg-Landau modes concentrate in time like e−k−mkc√t for 0 ≤ t ≤ (1/ε2) and m Z. The inverse Fourier transform of such a Ginzburg-Landau mode is an analytic Function in a strip of width √t. This result extends a former work of W. Eckhaus.