Traveling waves in the complex Ginzburg-Landau equation

Abstract

Summary. In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speed v and (temporal) frequency ~o; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real GinzburgLandau equation. We explicitly determine a region in the (wave speed v, frequency ~)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e, reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to oo. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.

Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms

“…Observe that there is no peak at K = 0 due to the subtraction '−C(A)'. Since T 0 becomes unbounded as k ↑ 1 4 a 2 (see [4]), we see that the n = ±1 peaks approach the K = O(δ) region if k approaches 1 4 a 2 . In other words, the periodic orbits satisfy the extra condition (1.7) as long as…”

“…Below we will show that χ is a monotonic function of k, so we can conclude that 0 < χ < a (since χ(0) = 0 and lim k↑ 1 4 a 2 χ(k) = a). Although this result is a special case of a more general result proved in [4], we will sketch the derivation of the monotonicity result: χ(k) is an important quantity which will also appear in subsequent sections. Note that ∂χ ∂k = ∂ ∂k…”

“…First we search for spatially periodic solutions. For both systems, the analysis is based on the fact that the (stationary) equation for A is integrable when B is fixed at a constant value (see [4] for references to the stationary problem of the (uncoupled) 'real' Ginzburg-Landau equation). Thus, the stationary problem associated with (1.5) is a (singularly) perturbed integrable system; periodic orbits in the fast field can be found by constructing a Poincaré map.…”

“…The first case leads to the stationary uncoupled Ginzburg-Landau equation for A (which is integrable; see below and [4]). The second case leads to the following equation for A:…”

“…Therefore, due to the phase shift invariance in (1.6), one can say that we restrict ourselves to studying real solutions of this system. We refer to [4] for a detailed discussion of the relation between = 0 and = 0 in the single real Ginzburg-Landau case.…”