Global modes for the complex Ginzburg-Landau equation

Linear global modes, which are time-harmonic solutions with vanishing boundary conditions, are analysed in the context of the complex Ginzburg-Landau equation with slowly varying coefficients in doubly infinite domains. The most unstable modes are shown to be characterized by the geometry of their Stokes line network: they are found to generically correspond to a configuration with two turning points issued from opposite sides of the real axis which are either merged or connected by a common Stokes line. A region of local absolute instability is also demonstrated to be a necessary condition for the existence of unstable global modes.