Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems

We investigate the behavior of traveling waves in a defect turbulence regime with the periodic boundary conditions by using the lowest-order complex Ginzburg-Landau equation (CGLE), and we show the effect of the nonlinear gradient terms in the system. It is found that the nonlinear gradient terms which appear at the same order as the quintic term can change the behavior of the wave patterns. The presence of the nonlinear gradient terms can cause major changes in the behavior of the solution. They can be considered like the stabilizing terms. The system which was initially unstable or chaotic can become stable by including the nonlinear gradient terms.

Figure 1

Defect turbulence regime observed for $\chi =0.6$, $c_1=2.5$, $c_3=0.5$, $c_5=1.1$, and $L=150$, with $q_1=q_2=q_3=0.0+0.0i$.

Figure 2

Variation of $(k,\lambda )$ for $\chi =0.6$, $c_1=2.5$, $c_3=0.5$, $c_5=1.1$, and $L=150$, with (a) $q_1=q_2=q_3=0.0+0.0i$, (b) $q_1=0.1+0.1i$ and $q_2=q_3=0.4+0.4i$, and (c) $q_1=0.5+0.5i$ and $q_2=q_3=1.0+1.0i$.

Figure 3

Localized pattern energy $Q$ in defect turbulence regime with $q_1=q_2=q_3=0.0+0.0i$.

Figure 4

(a) Running wave regime for $\chi =0.6q_1=0.5+0.9i$, $q_2=0.6+0.6i$, and $q_3=0.9+0.9i$. (b) Phase portrait of running waves.

Figure 5

Oscillating regime in system for $\chi =0.6$, $q_1=1.5+0.5i$, $q_2=0.6+0.6i$, and $q_3=1.5+0.9i$. (a) Space-time plot of $\left|A\right|$ propagating in one dimension. (b) Phase portrait of wave patterns.

Figure 6

(a) Laminar regime and (b) pattern energy $Q$, as a function of time for $q_1=1.5+0.5i$ and $q_2=q_3=1.5+1.5i$.

Figure 7

(a) Bifurcation diagram of $Q_{\max }$ and (b) largest Lyapunov exponent ${\lambda }_{\max }$ as a function of ${\chi }_r$, without nonlinear gradient terms.

Figure 8

(a) Bifurcation diagram of $Q_{\max }$ and (b) largest Lyapunov exponent ${\lambda }_{\max }$ as a function of ${\chi }_r$, with $q_1=0.1+0.1i$, $q_2=0.2+0.2i$, and $q_3=0.3+0.3i$.

Figure 9

(a) Bifurcation diagram of $Q_{\max }$ and (b) largest Lyapunov exponent ${\lambda }_{\max }$ as a function of ${\chi }_r$, with $q_1=0.8+0.8i$, $q_2=0.5+0.5i$, and $q_3=0.7+0.7i$.

Figure 10

Phase portrait of Fig. 7 showing (a) the chaotic case and (b) the stationary case.