Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg-Landau equation

We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical 1-dimensional real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.

Figure 1

Representations of the three nonlocal interaction kernels used in this work ($\sigma =1)$: (a) Gaussian, (b) mod-exponential (Laplacian), (c) Mexican hat ($b=1/2$). The right panels (d)–(f) show the Fourier transforms of the kernels in panels (a)–(c), respectively.

Figure 2

Boundaries in the $(s,{\mu }^{\prime })$ plane at which the leading spatial eigenvalues of the GLE with a Gaussian kernel exhibit different transitions for $\sigma =2$. Sketches indicate the location of the leading eigenvalues in the [$\mathrm{Re}\left(\lambda \right),\mathrm{Im}\left(\lambda \right)$] plane ($\times$, signal simple eigenvalues; $•$, double eigenvalues; and $\square$, quadruple eigenvalues). Dotted lines labeled as MIm4 and BDm4 show the MI and BD transitions given by the fourth moment expansion (18).

Figure 3

Boundaries in the $(\sigma ,s)$ plane at which the leading spatial eigenvalues of the GLE with a Gaussian kernel exhibit different transitions. Panel (a) shows the QZ manifold at ${\mu }^{\prime }=0$. Panels (b) and (c) show the BD and MI manifolds at ${\mu }^{\prime }=-1$ and ${\mu }^{\prime }=-6$, respectively. The short-dashed curves show the BD and MI transitions as predicted by the fourth moment expansion [Eq. (21)]. Long-dashed horizontal lines show the asymptotic values for $\sigma \to \infty$. Sketches represent the location of the leading eigenvalues.

Figure 4

Location of the first spatial eigenvalues for the GLE with a Gaussian nonlocal kernel in the complex $\lambda$ plane (shown as black dots) for ${\mu }^{\prime }=-6$. For comparison, the two gray dots show the location of the eigenvalues for the local GLE ($s=0$). The hyperbola, given by Eq. (36), in grayscale represents an approximation for the location of the spatial eigenvalues. For panels (a)–(c) on the top row, $s=1$, for (d)–(f) on the middle row $s=-1$ and for (g)–(i) on the bottom row $s=-7.5$. For panels (a), (d), and (g) on the left column $\sigma =0.3$; for (b), (e), and (h) on the middle column $\sigma =2$; and for (c), (f), and (i) on the right column $\sigma =3$. For all values of $\sigma$ the number of spatial eigenvalues is infinite: The plot just presents the region around the origin in the complex plane.

Figure 5

The solid line shows front profile for the GLE with a Gaussian nonlocal kernel for ${\mu }^{\prime }=-6$, $s=-1$, and (a) $\sigma =0.5$ and (b) $\sigma =1$. The red dashed lines show the approximations given by Eqs. (32) and (33) (see text).

Figure 6

Bifurcation diagram of the HSS of the nonlocal GLE with a Gaussian kernel and for $\sigma =2$: (a) $s=0$ (local case); (b) $s=-1$. Stable solutions are shown by a solid line, while the unstable ones are shown by a dashed line.

Figure 7

Location of the MI in parameter space for the GLE with Gaussian kernel as a function of the nonlocal strength $s$ for different values of the interaction range: The solid, long-dashed, and short-dashed lines correspond, respectively, to $\sigma =2,3,5$, while the dotted line represents the $\sigma \to \infty$ limit.

Figure 8

Boundaries in the $(\sigma ,s)$ plane separating the regions of monotonic and oscillatory tails in the GLE with a mod-exponential kernel for ${\mu }^{\prime }=-6$ obtained from (47). The horizontal dashed line at $s=-6$ shows the asymptotic limit of MI and BD lines for $\sigma \to \infty$.

Figure 9

Boundaries in the $({\mu }^{\prime },s)$ parameter space at which the leading spatial eigenvalues of the GLE with a Mexican-hat nonlocal kernel exhibit different transitions for $\sigma =1$ and (a) $b=0.1$, (b) $b=b_{\mathrm{SZ}}=0.2$, (c) $b=0.3$, and (d) $b=0.5$. Sketches indicate the position of the zeros of ${\Gamma }_s\left(\lambda \right)$ in the [$\mathrm{Re}\left(\lambda \right),\mathrm{Im}\left(\lambda \right)$] plane ($\times$, signal simple zeros; $•$, double zeros; $▵$, triple zeros; $\square$, QZs; and $\text{©}$, sextuple zeros).

Figure 10

Boundaries in the $(\sigma ,s)$ plane separating the regions of monotonic and oscillatory tails when using a Mexican-hat shaped kernel in the nonlocal GLE. ${\mu }^{\prime }=-6$.

Figure 11

Location of the first spatial eigenvalues for the GLE with a Mexican-hat nonlocal kernel in the complex $\lambda$ plane (shown as black dots) for $b=1/2$ and ${\mu }^{\prime }=-6$. The top row corresponds to $s=30$ with (a) $\sigma =0.1$, (b) $\sigma =0.5$, and (c) $\sigma =2$. The middle row corresponds to $s=-1$ with (d) $\sigma =1$, (e) $\sigma =2$, and (f) $\sigma =3$. The bottom row corresponds to $s=-8$ with (g) $\sigma =1$, (h) $\sigma =2$, and (i) $\sigma =3$. For comparison the two gray dots show the location of the eigenvalues for the local GLE ($s=0$). In all the cases the number of spatial eigenvalues is infinite: The plot just presents the region around the origin in the complex plane.

Figure 12

Velocity at which two fronts connecting the two equivalent HSSs approach each other for the GLE equation with a Mexican-hat nonlocal kernel with $b=1/2$, ${\mu }^{\prime }=6$, $s=-1$ and different values for the interaction range $\sigma$ within parameter region 4.

Figure 13

Spatial front profile for the GLE with a Mexican-hat nonlocal kernel for $b=1/2$, ${\mu }^{\prime }=-6$, $s=-1$ and (a) $\sigma =1$ and (b) $\sigma =3$. The red dashed lines show the approximations given by Eqs. (32) and (67) (see text).

Figure 14

A plot of the real branches of Lambert's $W$ function, two-valued in the range $[-1/e,0]$. The dashed line corresponds to $W_0\left(x\right)$ and the solid line to $W_{-1}\left(x\right)$.