Traveling waves and defects in the complex Swift-Hohenberg equation

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves investigated numerically. A number of distinct dynamical regimes is identified and analyzed using appropriate phase equations describing the evolution of long-wavelength instabilities of both the homogeneous oscillating state and constant amplitude traveling waves.

Figure 1

The growth rates ${\sigma }_{\pm }$ of infinitesimal perturbations of the nontrivial homogeneous solution (2, 3) as functions of the wave number $k$. Only ${\sigma }_{+}$ leads to an instability. Parameters: $r=1$, $b=-0.5$, $k_0=0.2$, $\beta =1$, and $\zeta =0,-0.1,0.1$ in (a)–(c), respectively. The long-wavelength instability is suppressed for $b\zeta >0$ (b) or enhanced for $b\zeta <0$ (c).

Figure 2

Stability analysis of the TW solution (8, 9). Parameters: $r=1$, $b=-1$, $K=k_0$ $=$ $0.2$, $\zeta =0$, and (a) $\beta =0.9$ and (b) $\beta =1.1$. When $K=k_0$ and $\zeta =0$ there exists a long-wavelength instability for $1+b\beta <0$.

Figure 3

Temporal simulation of the CSHE with complex coefficients starting from a TW with wave number $K_0=0.063$ and superposed small amplitude noise. (a)–(d) Space-time plots of the time evolution of $\mathrm{Re}u$, $\mathrm{Im}u$, the local wave number $K\equiv {\phi }_x$, and the amplitude $R$, respectively. Panels (a) and (b) show the last 25 time steps of the much longer simulation shown in panels (c) and (d) that show the details of the slow coarsening process ultimately resulting in a single source and a single sink. White/black regions correspond to high/low values of $K$ or $R$. (e)–(h) The profiles at the last time step of (a)–(d). Parameters: $r=1$, $b=-0.5$ and $k_0$ $=$ $0.2$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 4

Temporal simulation of the CSHE with complex coefficients. (a)–(c) Space-time plots of the time evolution of the local wave number $K$. (d)–(f) The profiles $K\left(x\right)$ at the last time step of (a)–(c). White/black regions correspond to high/low values of $K$. Parameters: $b=-0.5$, $\beta =-1$, $\zeta =-1$, $L=300$, $N=1024$, $r_a=0.125$. (a) $k_0=0.2$, $r=0.1<r_a$. (b) $k_0=0.2$, $r=0.3>r_a$. (c) $k_0=1$, $r=0.3>r_a$.

Figure 5

The velocity of a front in the derivative of the phase (a shocklike structure in the local wave number $K\equiv {\phi }_x$) as a function of the parameter $b$ obtained by numerical continuation of the solution in Figs. 3g and 3h for a fixed spatial average of $K$ ($\overline{K}=0.063$). Other parameters are as in Fig. 3. The right panels show the resulting $K$ profiles for three different values of $b$.

Figure 6

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of the local wave number $K\equiv {\phi }_x$ [(a), (c), (e)] and the amplitude $R$ [(b), (d), (f)] for increasing values of $\left|b\right|$. White/black regions correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-1$ [(a), (b)],$b=-1.5$ [(c), (d)], $b=-2$ [(e), (f)] and $k_0$ $=$ $0.2$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 7

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a) the local wave number $K\equiv {\phi }_x$ and (b) the amplitude $R$. White/black correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-1$, $\beta =-2$, and $k_0$ $=$ $0.2$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 8

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a) $\mathrm{Re}u$, (b) $\mathrm{Im}u$, (c) the local wave number $K\equiv {\phi }_x$, and (d) the amplitude $R\equiv \left|u\right|$. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-1$, $\beta =1$, and $k_0$ $=$ $0.2$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 9

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a), (c) the local wave number $K\equiv {\phi }_x$ and (b), (d) the amplitude $R$. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-1$, $\beta =2$(a), (b), (e), (f), $\beta =3$ (c), (d), (g), (h), and $k_0$ $=$ $0.2$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 10

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of (a), (c) the local wave number $K\equiv {\phi }_x$ and (b), (d) the amplitude $R$. (e)–(h) The profiles at the last time step of (a)–(d). White/black regions correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-0.5$ (a), (b), (e), (f), $b=-1$ (c), (d), (g), (h), and $k_0=1$, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 11

Temporal simulation of the CSHE with complex coefficients. Space-time plots of the time evolution of the local wave number $K\equiv {\phi }_x$ [(a), (c), (e)] and the amplitude $R$ [(b), (d), (f)], for decreasing values of $b$. Panels (c) and (d) represent a zoom of the black box in (a) and (b). White/black regions correspond to high/low values of $K$ or $R$. Sinks (Si) are stationary and stable for long time intervals, while sources (So) oscillate wildly. Nucleation points creating or destroying a source/sink pair are denoted by N${}_1$, N${}_2$. Parameters: $r=1$, $b=-3$ (a)–(d) and $b=-5$ (e) and (f), and $k_0$ $=$ 1, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 12

Temporal simulation of the CSHE with complex coefficients in terms of space-time plots showing the time evolution of the local wave number $K\equiv {\phi }_x$ and the amplitude $R$. White/black regions correspond to high/low values of $K$ or $R$. Parameters: $r=1$, $b=-1$, $\beta =-5$, (a) $-3$, (b) $-2.5$, (c) 1, (d) 3, (e) $4.1$, (f) $4.5$, (g) and 5, (h) $k_0$ $=$ 1, $L=300$, $N=1024$ (parameters not mentioned are zero).

Figure 13

(a) A soliton solution in the perturbed Korteweg-de Vries equation (60) for $b=-1$, $\beta =0.9$, $\zeta =-0.1$, $K_0=0.505$, $k_0=0.1$, and $r=1$, corresponding to the coefficient values $S=-0.7062$, $U=1.199$, $\widehat{Q}=-K_0=-0.505$, $V=-0.6916$, $Z=-0.5397$, $\epsilon =0.098$. (b) The solution (a) interpreted within the CSHE as a perturbation of the wave number $K$ (here a localized wavelength dilation).

Figure 14

The quantities $Q$, $\rho$, and $N_0$ characterizing soliton solutions of the perturbed Korteweg-de Vries equation (60) as functions of $\zeta$ and $K_0$, with $b=-1$, $\beta =0.9$, $k_0=0.1$, and $r=1$ kept fixed.

Figure 15

(a) A cnoidal wave solution consisting of seven wavelengths in the perturbed Korteweg-de Vries equation (60) for $b=-1$, $\beta =0.9$, $\zeta =-0.1$, $K_0=0.505$, $k_0=0.1$, $r=1$, and $L=80$, corresponding to the coefficient values $S=-0.7062$, $U=1.199$, $\widehat{Q}=-K_0=-0.505$, $V=-0.6916$, $Z=-0.5397$, $\epsilon =0.098$. (b) The solution (a) interpreted within the CSHE as a perturbation of the wave number $K_0$.

Figure 16

Temporal simulation of the nonlinear phase equation (70) in terms of space-time plots of the time evolution of the local wave number $v$. Red (light gray)/blue (dark gray) correspond to high/low values of $v$. Parameters: $L=60$, $N=128$, $\beta =0$, $b=-0.15$, (a) $-0.3$, (b) $-0.6$, (c) $-2$ (d) and (e), and $-3$ (f).

Figure 17

(a) The coefficients $\gamma \equiv -2k_0^2(1+b\beta )$ and $D/2\equiv 2k_0^2(\beta -b)$ in Eq. (69) as functions of the parameter $\beta$ for $k_0=0.2$ and $b=-1$. (b) Transition from coarsening behavior to spatiotemporal intermittency in the ($b,\beta$) plane (gray dots), together with the locus of the Benjamin-Feir (BF) instability Eq. (7) and the locus of the points where $\gamma$ and $D/2$ are equal in magnitude. The transition from coarsening behavior to spatiotemporal intermittency has been determined numerically with an accuracy $\Delta \beta =0.1$.

Figure 18

(a) Temporal simulation of the nonlinear phase equation (70). The top panel shows a space-time plot of the time evolution of the local wave number $v$. Red (light gray)/blue (dark gray) correspond to high/low values of $v$. The bottom panel shows the profile of $v$ at the final time step $T=500$. Parameters: $L=60$, $N=128$, $b=-1$, $\beta =0.8$. Panels (b) and (c) show, for the same parameter values, the space-time evolution of the wave number $K$ and the amplitude $R$ for the CSHE (1). The bottom panels show the profiles at the final time $t=3\times {10}^5$. Parameters: $L=300$, $N=1024$, $b=-1$, $\beta =0.8$, $k_0=0.2$.

Figure 19

Temporal simulation of the perturbed Korteweg-de Vries equation (60) in terms of space-time plots showing the time evolution of the local wave number $w$. Red (light gray)/yellow-blue (dark gray) correspond to high/low values of $w$. The middle and bottom panels show the intermediate solution at $T=300$ and the final solution at $T=2000$, respectively. Parameters: $S=-0.7062$, $U=1.199$, $\widehat{Q}=-K_0=-0.505$, $V=-0.6916$, $Z=-0.5397$, $\epsilon =0.098$, and $N=256$. (a) $L=30$, (b) $L=50$, (c) $L=80$. The final state in case (c) resembles closely the analytical prediction depicted in Fig. 15a.

Figure 20

Temporal simulation of the CSHE for different values of the domain length $L$ showing the profile of the local wave number $K$ after $t=3\times {10}^5$ (top panels) and $t=2\times {10}^6$ (bottom panels). Parameters: $b=-1$, $\beta =0.9$, $\zeta =-0.1$, $K_0=0.505$, $k_0=0.1$, $r=1$, and $N=1024$. $L\approx 300$, (a) 500, (b) and 800 (c).