Pulses in a complex Ginzburg–Landau system: Persistence under coupling with slow diffusion

Abstract

The Ginzburg–Landau (GL) equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolutions of small amplitude instabilities near criticality. If the instabilities are, however, driven by two coupled instability mechanisms, of which one corresponds with a neutrally stable mode, their evolution is described by a GL equation coupled to a diffusion equation. In this paper, we study the influence of an additional diffusion equation on the existence of pulse solutions in the complex GL equation. In light of recently developed insights into the effect of slow diffusion on the stability of pulses, we consider the case of slow diffusion, i.e., in which the additional diffusion equation acts on a long spatial scale. In previous work [A. Doelman, G. Hek, N. Valkhoff, Stabilization by slow diffusion in a real Ginzburg–Landau system, J. Nonlinear Sci. 14 (2004) 237–278; A. Doelman, G. Hek, N.J.M. Valkhoff, Algebraically decaying pulses in a Ginzburg–Landau system with a neutrally stable mode, Nonlinearity 20 (2007) 357–389], we restricted ourselves to a model with both real coefficients and, more importantly, a real amplitude A rather than the complex-valued A that is needed to completely describe the pattern formation near criticality. In this simpler setting, we proved that pulse solutions of the GL equation can both persist and be stabilized under coupling with a slow diffusion equation. In the current work, we no longer make these restrictions, so that the problem is higher-dimensional and intrinsically harder. By a combination of a geometrical approach and explicit perturbation analysis, we consider the persistence of the solitary pulse solution of the GL equation under coupling with the additional diffusion equation. In the two limiting situations of the nearly real GL equation and the near nonlinear Schrödinger equation, we show that the pulse solutions can indeed persist under this coupling.