Coherent and Incoherent Drifting Pulse Dynamics in a Complex Ginzburg-Landau Equation

Abstract

We show that drifting pulse solutions of a 1D complex Ginzburg-Landau equation can persist for positive growth rate e in a finite system. When e is increased, two different destabilization scenarios are observed. In sufficiently large systems, fluctuations grow out to form multiple pulses. In small systems, an increase in e eventually leads to a competition between fronts and pulses that results in a sharp transition to a state where the drifting pulse leaps forward in an incoherent fashion. Similar behavior is observed in a more realistic model. PACS numbers: 47.20.Hw, 03.40.Gc, 05.40.+j, 47.54.+r A few years ago, localized or confined traveling wave states were discovered in convection experiments in binary liquids [1 -7]. These are states of which the region where the convection occurs does not fill the total experimental cell, but instead attains a well-defined width.The discovery of these states has inspired a considerable amount of theoretical work on pulse solutions of amplitude equations [8 -12]. It is by now well established that in experiments in annular geometries a localized traveling wave state drifts slowly when the inhomogeneities of the convection cell are sufficiently small [3]. This drift velocity v~i s quite different from the group velocity vg" and this can be contributed to a slow concentration field [9,10] that "traps" the pulse in its own concentration gradient. The existence of localized states and much of their behavior can be understood in terms of pulse-shaped solutions of a complex Ginzburg-Landau amplitude equation [8,11]. The fact that the pulse velocity v~d iffers so much from vg, can, however, only be obtained from a more detailed analysis of the coupling of the convection to the slow concentration field [9,10].In the experiments in an annular geometry [2,5,6], the localized traveling wave states surprisingly persist in a regime where the conducting state (A = 0) is completely unstable (e ) 0). This persistence of pulses in an unstable background is usually explained as follows. Since an annular cell is periodic and since the pulse drifts with a velocity v~d ifferent from the velocity vg, with which the fluctuations propagate, the maximum time interval during

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

“…We have done all our simulations in Section 4 and in this section with a more stable and accurate semi-implicit method [105], 24 which for the F-KPP equation amounts to the discretization For this integration scheme, it is now straightforward to find…”

Holding spatial solitons in a pumped cavity with the help of nonlinear potentials

“…Systematic simulations also demonstrate that the sweeping effect, that is, periodic suppression of locally growing perturbations by the passing soliton [27], does not stabilize the soliton either. Note that the frequency of the shuttle motion is limited by f max 2L∕C max , where C max is the largest tilt at which the moving soliton does not escape the cavity.…”

Breathing and randomly walking pulses in a semilinear Ginzburg–Landau system

“…[19]. It is not quite clear whether there is a definite border for the spontaneous generation of SPs from the chaotic background.…”

“…In section 2, we demonstrate that effectively stable pulses and sets of pulses exist in the region where the zero background is unstable. It should be mentioned that a possibility for a pulse to survive above the background-instability threshold is known in a GL model of another type [19], but in that case the stabilization is provided by the fact that the pulse is moving, due an extra symmetry-breaking term added to the GL equation, and, in a system with periodic boundary conditions, it may perform a round trip quickly enough to suppress the growing local perturbations. In our model, the situation is different: immediately above the background-instability threshold, we observe stable pulses resting on top of a background standing-wave pattern with a small amplitude.…”