Spatial localization in heterogeneous systems

Hsien-Ching Kao, Cédric Beaume, and Edgar Knobloch

Phys. Rev. E 89, 012903 – Published 6 January 2014

Abstract

We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.

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Received 18 October 2013

DOI:https://doi.org/10.1103/PhysRevE.89.012903

Authors & Affiliations

Hsien-Ching Kao^*

Wolfram Research Inc., Champaign, Illinois 61820, USA

Cédric Beaume^† and Edgar Knobloch^‡

Department of Physics, University of California, Berkeley, California 94720, USA

^*hkao@wolfram.com
^†ced.beaume@gmail.com
^‡knobloch@berkeley.edu

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Vol. 89, Iss. 1 — January 2014

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Images

Figure 1
(a) Bifurcation diagram showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for homogeneous forcing ( $α = 0$ ) in SH23. The black [red (or gray)] snaking branch corresponds to solutions with $ϕ = 0$ ( $ϕ = π$ ). (b) Solution profiles at the bottom five saddle nodes along each snaking branch. The asymmetric rung states [22] are omitted.
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Figure 2
(a) Bifurcation diagram showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for homogeneous forcing ( $α = 0$ ) in SH35. The black [red (or gray)] snaking branch corresponds to even (odd) solutions. (b) Solution profiles at the bottom five saddle nodes along the left end of the snaking branches. The asymmetric rung states [18] are omitted.
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Figure 3
Bifurcation diagrams for SH23 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for (a) $Δ ϕ = 0$ steady states and (b) $Δ ϕ = π$ steady states. The forcing amplitude $α = 0.1$ (solid lines) and $α = - 0.1$ (dashed lines).
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Figure 4
Solution profiles at successive saddle nodes along the $Δ ϕ = π$ branches from Fig. 3 for (a) $α = 0.1$ and (b) $α = - 0.1$ , proceeding upward along the branch (profiles below the horizontal separation line) and then following it further as it starts to snake back down (profiles above the horizontal separation line). The bottom panel shows the forcing function $f_{p}$ (thin solid line).
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Figure 5
Location of the saddle nodes of the periodic branches (solid lines) and the associated Maxwell points (dashed lines) in the $(r, α)$ plane for SH23. The thick (thin) lines represent $Δ ϕ = 0$ ( $Δ ϕ = π$ ) solutions.
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Figure 6
(a) Bifurcation diagrams for SH23 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for $Δ ϕ = 0$ steady states. The forcing amplitude $α = 0.1$ (solid lines) and $α = - 0.1$ (dashed lines). Dash-dotted lines: rung states. (b) Sample solution profiles along the lower rung branch in (a) from left (bottom) to right (top).
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Figure 7
Bifurcation diagrams and solution profiles for SH35 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for even parity branches when (a), (c) $α = \pm 0.1$ , (b), (d) $α = \pm 0.5$ . (a), (b) Branches of localized states for positive (negative) $α$ are shown in solid (dashed) lines. (c), (d) Solution profiles at the first five saddle nodes along the snaking branches. Solutions with positive (negative) $α$ are shown in the upper (lower) subpanels. The thin solid lines indicate the spatial profile $f_{p}$ of the heterogeneity. Because of the symmetry of SH35 with respect to $u \to - u$ profiles obtained by reflection in the horizontal axis are also solutions of SH35.
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Figure 8
(a) Bifurcation diagram for SH35 with the forcing $f_{p}$ and $δ = 1$ , $α = \pm 0.5$ showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for the odd parity branch. Localized branches of positive (negative) $α$ are shown in solid (dashed) lines. (b) Solution profiles along the odd parity branch with $α > 0$ proceeding upward along the branch (profiles below the thick horizontal separation line) and then following it further as it starts to snake back down (profiles above the thick horizontal separation line). The thin solid line in the lowest panel indicates the spatial profile $f_{p}$ of the heterogeneity. Because of the symmetry of SH35 with respect to $u \to - u$ profiles obtained by reflection in the horizontal axis are also solutions.
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Figure 9
Bifurcation diagram for SH35 with the forcing $f_{p}$ , $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for (a) even parity solutions and (b) odd parity solutions. Thin line: $α = 0$ . Thick solid (dashed) line: $α = 0.03$ $(- 0.03)$ .
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Figure 10
Bifurcation diagrams for SH35 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for even parity steady states [cf. Fig. 9] and the corresponding rung states (dashed-dotted lines). The forcing amplitude (a) $α = \pm 0.1$ and (b) $α = \pm 0.5$ . The rung states are the result of destabilization of the phase mode.
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Figure 11
Bifurcation diagram for SH35 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for even parity steady states [cf. Fig. 9] and the corresponding rung states (dashed-dotted lines). The forcing amplitude $α = 0.1$ . The S-shaped branch of rung states is the result of destabilization of the translation mode.
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Figure 12
Bifurcation diagram for SH35 with the forcing $f_{p}$ and $δ = 1$ , showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for odd parity steady states [cf. Fig. 9] and the corresponding rung states (dashed-dotted lines). The forcing amplitude $α = \pm 0.1$ .
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Figure 13
Location of the saddle node of the periodic branches (solid lines) and the Maxwell points (dashed lines) in the $(r, α)$ plane for SH35. The black [red (or gray)] lines correspond to even (odd) parity states.
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Figure 14
Eigenfunctions at (a) the left end and (b) the right end of the upper Z-shaped rung branch shown in Fig. 10. Both eigenfunctions are localized at the fronts at either end of the localized structure indicating that the Z-shaped branch is created as a result of the destabilization of the $α = 0$ phase mode.
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Figure 15
Eigenfunctions at (a) the lower end and (b) the upper end of the S-shaped rung branch shown in Fig. 11. Both eigenfunctions are spatially extended indicating that the S-shaped branch is created as a result of the destabilization of the $α = 0$ translation mode.
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Figure 16
(a), (b) Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L^{2}}$ as a function of $r$ for SH23 with the forcing $f_{b}$ and $α = \pm 0.5$ . Solution branches with $Δ ϕ = 0$ (resp. $π$ ) are shown in (a) [resp. (b)]. Localized branches for positive (negative) $α$ are shown in solid (dashed) lines. (c), (d) Solution profiles at the first five saddle nodes along the snaking branches. The profiles for positive (negative) $α$ are shown in the upper (resp. lower) subpanels. The thin solid line at the bottom of each subpanel indicates the spatial profile of the heterogeneity.
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Figure 17
(a), (b) Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L^{2}}$ as a function of $r$ for SH35 with the forcing $f_{b}$ and $α = \pm 0.5$ . Solution branches with even (resp. odd) parity are shown in (a) [resp. (b)]. Localized branches for positive (negative) $α$ are shown in solid (dashed) lines. (c), (d) Solution profiles at the first five saddle nodes along the snaking branches. The profiles for positive (negative) $α$ are shown in the upper (resp. lower) subpanels. The thin solid line at the bottom of each subpanel indicates the spatial profile of the heterogeneity.
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Figure 18
(a) Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L^{2}}$ as a function of $r$ for SH23 with the forcing $f_{b}$ and $α = - 0.5$ . Dashed line: even parity $Δ ϕ = 0$ localized states. Dashed-dotted line: asymmetric rung states. (b) A stable asymmetric state with $r = - 0.307$ and $N = 0.248$ on the middle part of the rung [reached in Fig. 26]. The bump profile is shown in the lowest panel.
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Figure 19
(a) Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L^{2}}$ as a function of $r$ for SH35 with the forcing $f_{b}$ and $α = - 0.5$ . Dashed line: odd parity localized states. Dashed-dotted line: asymmetric rung states. (b) A stable asymmetric state with $r = - 0.686$ and $N = 0.305$ on the middle part of the rung [reached in Fig. 26]. The bump profile is shown in the lowest panel.
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Figure 20
The translation eigenvalue of (a) $Δ ϕ = 0$ and (b) $Δ ϕ = π$ localized states in SH23 with the forcing $f_{p}$ and $δ = 1$ as a function of $N \equiv {∥ u ∥}_{L_{2}}$ . Solid (dashed) line: $α = 10^{- 2}$ $(α = - 10^{- 2})$ . $+$ : Prediction from Eq. (14).
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Figure 21
The translation eigenvalue of (a) even and (b) odd localized states in SH35 with the forcing $f_{p}$ and $δ = 1$ as a function of $N \equiv {∥ u ∥}_{L_{2}}$ . Solid (dashed) line: $α = 10^{- 4}$ $(α = - 10^{- 4})$ . $+$ : Prediction from Eq. (14).
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Figure 22
The linear stability of even localized states in SH35 with the forcing $f_{p}$ and $δ = 1$ . (a) The first five leading eigenvalues of the even parity stationary solutions in Fig. 9 as functions of $N \equiv {∥ u ∥}_{L_{2}}$ , and (b) enlargement of the translation eigenvalue. The thin lines correspond to $α = 0$ , while the thick solid (dashed) lines correspond to $α = 0.03$ $(α = - 0.03)$ .
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Figure 23
The linear stability of odd localized states in SH35 with the forcing $f_{p}$ and $δ = 1$ . (a) The first five leading eigenvalues of the odd parity stationary solutions in Fig. 9 as functions of $N \equiv {∥ u ∥}_{L_{2}}$ , and (b) enlargement of the translation (lower curve) and phase (upper curve) eigenvalues when $α = 0.03$ . In (a) the thin lines correspond to $α = 0$ , while the thick solid (dashed) lines correspond to $α = 0.03$ $(α = - 0.03)$ .
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Figure 24
Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L_{2}}$ as a function of $r$ for (a), (b) SH23 and (c), (d) SH35 with the forcing $f_{p}$ and $δ = 0.05$ . Panels (a) and (c) correspond to $α = \pm 0.1$ , while (b) and (d) are for $α = \pm 0.5$ . In (a) and (b) the black [red (or gray)] snaking branch corresponds to solutions with $ϕ = 0$ ( $ϕ = π$ ); in (c) and (d) the black [red (or gray)] lines correspond to even (odd) parity states.
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Figure 25
Bifurcation diagrams showing $N \equiv {∥ u ∥}_{L_{2}}$ from Fig. 24 as a function of $a_{c} \equiv r [1 + α cos (x_{c})$ ], for (a) SH23 and (b) SH35 with the heterogeneity $f_{p}$ and $δ = 0.05$ . Left panels correspond to $α = \pm 0.1$ , while the right panels correspond to $α = \pm 0.5$ .
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Figure 26
Time evolution of (a) an unstable $Δ ϕ = π$ localized solution in SH23 with forcing $f_{b}$ and $α = - 0.5$ , (b) an unstable even localized solution in SH35 with forcing $f_{p}$ with $α = 0.1, δ = 1$ , and (c) an unstable odd localized solution in SH35 with forcing $f_{b}$ with $α = - 0.5$ .
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Figure 27
The quantity $- F_{α}$ ( $= \frac{1}{2} \int_{- Γ / 2}^{Γ / 2} r f_{b} (x + Δ x) u_{0} {(x)}^{2}$ ) in SH23 as a function of the displacement $Δ x$ of the forcing $f_{b}$ relative to the symmetry point of an $α = 0$ , $Δ ϕ = π$ localized state with $r = - 0.306$ and $N = 0.217$ , showing the presence of nearby stable asymmetric states corresponding to local minima with $Δ x \neq 0$ .
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Figure 28
(a) Time evolution of an unstable $ϕ = 0$ localized solution in SH23 with $f_{p}$ forcing and $α = - 0.01$ , $δ = 0.05$ . (b) Location of the maximum of the solution as a function of $t$ . The solid line represents the results of direct numerical simulation while the $+$ signs are computed from the asymptotic result in Eq. (12).
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