2D Complex Ginzburg-Landau Equation Research Articles

Frequency modes of unstable spiral waves in two-dimensional Rosenzweig–MacArthur ecological networks

The existence of two frequency regimes in a two-dimensional (2D) Rosenzweig–MacArthur ecological network is debated. The semi-discrete approximation differentiates the two regimes, each described by a 2D complex Ginzburg–Landau equation. Using the standard theory of the linear stability analysis, a generalized expression for the modulational instability growth rate is derived for each frequency mode. The parametric study of the growth rate of modulational instability reveals its sensitivity to the changes in the recruitment rate of the resources. Moreover, direct numerical simulations are carried out to confirm our analytical results. Over the prolonged evolution of the perturbed plane wave solution, the high-frequency mode entertains spiral wave patterns. In contrast, the appearance of target waves manifests the low-frequency regime. In that context, we further explore the impact of the recruitment rate of resources and give the qualitative meaning of the obtained dynamical behaviors and their ecological implications. This work may additionally provide more insight into the mechanism leading to spiral and target waves in environmental systems.

Application of bifurcation method and rational sine-Gordon expansion method for solving 2D complex Ginzburg–Landau equation

This paper implements bifurcation method and the rational sine-Gordon expansion method to investigate the dynamical behavior of traveling wave solutions of a 2D complex Ginzburg–Landau equation. By varying the parameters, we obtained traveling wave solutions including the periodic wave solutions, solitary wave solution, kink and anti-kink wave solution and in addition by using the rational sine-Gordon expansion method, we determined bright and dark soliton which have a great contribution in the long distance telecommunication system.

Pattern formation by kicked solitons in the two-dimensional Ginzburg-Landau medium with a transverse grating

We consider the kick- (tilt-) induced mobility of two-dimensional (2D) fundamental dissipative solitons in models of bulk lasing media based on the 2D complex Ginzburg-Landau equation including a spatially periodic potential (transverse grating). The depinning threshold, which depends on the orientation of the kick, is identified by means of systematic simulations and estimated by means of an analytical approximation. Various pattern-formation scenarios are found above the threshold. Most typically, the soliton, hopping between potential cells, leaves arrayed patterns of different sizes in its wake. In the single-pass-amplifier setup, this effect may be used as a mechanism for the selective pattern formation controlled by the tilt of the input beam. Freely moving solitons feature two distinct values of the established velocity. Elastic and inelastic collisions between free solitons and pinned arrayed patterns are studied too.

Size transition of spiral waves using the pulse array method

The domain size of spiral waves is an important issue in studies of two-dimensional (2D) spatiotemporal patterns. In this work, we use the 2D complex Ginzburg–Landau equation (CGLE) as our model and find that an initially big spiral can successfully transfer to several small spirals by applying a pulse array method. The impacts of several important factors, such as array density, controlling intensity and pulsing time, are investigated. This control approach may be useful for the control of 2D spatiotemporal patterns and has potential applications in the control of some realistic systems, such as meteorological and cardiac systems.

The structure of spiral-domain patterns and shocks in the 2D complex Ginzburg-Landau equation

Cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field. In the case of the complex Ginzburg-Landau equation, disordered cells of effectively frozen spirals appear, separated by thin walls (shocks), on a scale much larger than the basic wavelength of the spirals. We show that these structures can be understood in very simple terms. In particular, we show that the walls are, to a good approximation, segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex. The fact that the phase is only defined up to an integer multiple of 2π introduces a quantization condition on the sizes of the smallest spiral domains. The transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations. The structure depends on the angle the wall makes with the local phase contours, and the behavior can be either monotonic or oscillatory, depending on the parameters.

Synchronized disorder in a 2D complex Ginzburg-Landau equation

The phenomenon of spiral pair synchronization by oscillating strips (Nozaki-Bekki solution) in a 2D CGLE is investigated analytically. The equations describing the interaction of the strips with one another and with spirals are derived. Analysis of the equations shows that under certain conditions the strips lead to frequency and phase locking of the spirals. In this case the spiral pair (dipole) is aligned parallel to the strips, with the position along the strips being arbitrary. Thus, the interaction with strips may transform the spatio-temporal chaos of spirals to the regime of periodically oscillating spatial disorder. The dynamics of circular strips is investigated and their lifetime is estimated. The behavior of the spirals bounded by circular stript is analysed.

The screening of a spiral field in a 2D complex Ginzburg-Landau equation

The stability of one-dimensional dark strips, i.e. Nozaki-Bekki type solutions, relative to small and finite perturbations in a 2D complex Ginzburg-Landau equation is investigated analytically and numerically. It is shown that in a linear approximation in the region -∞< x<∞, 0⩽ y⩽ L, Nozaki-Bekki strip (NBS) solutions are stable in a definite range of parameters. The regime of coexistence of NBS and chaotically walking spirals — bounded defect mediated turbulence — is found. In this regime the time averaged defect density in a rectangle bounded by parallel NBS solutions depends only on the value of supercriticality. Such a turbulent regime transforms, in a certain region of parameters, to a regime of spatio-temporal synchronization when the spirals are arranged periodically. The phenomenon of spiral field screening by a NBS field is shown. This phenomenon is responsible for spatial localization of turbulent regimes.