Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework

The present work studies the influence of nonlocal spatial coupling on the existence of localized structures in one-dimensional extended systems. We consider systems described by a real field with a nonlocal coupling that has a linear dependence on the field. Leveraging spatial dynamics we provide a general framework to understand the effect of the nonlocality on the shape of the fronts connecting two stable states. In particular we show that nonlocal terms can induce spatial oscillations in the front tails, allowing for the creation of localized structures, that emerge from pinning between two fronts. In parameter space the region where fronts are oscillatory is limited by three transitions: the modulational instability of the homogeneous state, the Belyakov-Devaney transition in which monotonic fronts acquire spatial oscillations with infinite wavelength, and a crossover in which monotonically decaying fronts develop spatial oscillations with a finite wavelength. We show how these transitions are organized by codimension 2 and 3 points and illustrate how by changing the parameters of the nonlocal coupling it is possible to bring the system into the region where localized structures can be formed.

Figure 1

Example of a LS in the Ginzburg-Landau equation with a Gaussian nonlocal kernel as defined in Refs. [35, 37], resulting from the pinning of two fronts with oscillatory tails. Parameters taken as $\mu =3$, $s=-6$, and $\sigma =2$.

Figure 2

Boundaries in the $(\mu ,\alpha )$ parameter space at which the spatial eigenvalues of (25) exhibit different transitions for (a) $\beta =-3$, (b) $\beta =0$, and (c) $\beta =3$. 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; $•$, DZs, $▵$, triple zeros; $\square$, QZs; and $\text{©}$, SZs).

Figure 3

Dispersion relation $\Gamma \left(k\right)$ for $\beta =-3$, $\mu =-0.2$ and (a) $\alpha =1.4$ (region 3), (b) $\alpha =1.6109$ (on the MI line), and (c) $\alpha =1.9$ (region 2).

Figure 4

Dispersion relation $\Gamma \left(k\right)$ for $\beta =-3$, $\alpha =-2$ and (a) $\mu =-0.08$ (region 7), (b) $\mu =0$ (on the Hamiltonian-pitchfork line), and (c) $\mu =0.08$ (region 1).

Figure 5

Dispersion relation $\Gamma \left(k\right)$ for $\beta =-3$, $\alpha =2$ and (a) $\mu =-0.08$ (region 2), (b) $\mu =0$ (on the Hamiltonian-pitchfork-Hopf line), and (c) $\mu =0.08$ (region 1).

Figure 6

Dispersion relation $\Gamma \left(k\right)$ for $\beta =3$, $\mu =0.3$ and (a) $\alpha =-2.0$ (region 8), (b) $\alpha =-1.7836$ (on the HH${}^{+}$ line), and (c) $\alpha =-1.6$ (region 6).

Figure 7

Dispersion relation $\Gamma \left(k\right)$ for $\beta =3$, $\mu =0.3$ and (a) $\alpha =-2.4$ (region 8), (b) $\alpha =-2.4549$ (on the HH${}^{-}$ line), and (c) $\alpha =-2.5$ (region 5).

Figure 8

Dispersion relation $\Gamma \left(k\right)$ for $\beta =3$, $\alpha =-1.5$ and (a) $\mu =-0.08$ (region 2), (b) $\mu =0$, and (c) $\mu =0.08$ (region 8).

Figure 9

Dispersion relation $\Gamma \left(k\right)$ evaluated on top of the HH manifold close to the 3DZ$\left(i\omega \right)$ point for $\beta =3$ (a) on the HH${}^{-}$ line above the 3DZ$\left(i\omega \right)$, $\alpha =-2.3$ and $\mu =0.07458$; (b) at the 3DZ$\left(i\omega \right)$, $\alpha =-9/4$ and $\mu =0$; and (c) on the MI line below the 3DZ$\left(i\omega \right)$ $\alpha =-2.2$ and $\mu =-0.07541$.

Figure 10

Dispersion relation $\Gamma \left(k\right)$ for $\beta =3$, $\alpha =-2.8$ and (a) $\mu =-0.08$ (region 4), (b) $\mu =0$ (on the Hamiltonian-pitchfork line), and (c) $\mu =0.08$ (region 5).