PHYSICAL REVIEW E 92, 032116 (2015)

Stochastic dissipative solitons Sergio E. Mangioni* and Roberto R. Deza† Instituto de F´ısica de Mar del Plata, Universidad Nacional de Mar del Plata, and CONICET, De´an Funes 3350, B7602AYL Mar del Plata, Buenos Aires, Argentina (Received 2 November 2013; revised manuscript received 10 July 2015; published 11 September 2015) By the effect of aggregating currents, some systems display an effective diffusion coefficient that becomes negative in a range of the order parameter, giving rise to bistability among homogeneous states (HSs). By applying a proper multiplicative noise, localized (pinning) states are shown to become stable at the expense of one of the HSs. They are, however, not static, but their location fluctuates with a variance that increases with the noise intensity. The numerical results are supported by an analytical estimate in the spirit of the so-called solvability condition. DOI: 10.1103/PhysRevE.92.032116

PACS number(s): 05.40.Ca, 05.45.Yv, 05.65.+b

I. INTRODUCTION

In recent times, partly spurred by the nanotechnology and optical fiber industries, we have witnessed an upsurge of interest in localized patterns bubbles of metastable homogeneous states (HSs) [1–4]. Also known as dissipative solitons [5,6] (as long as their interaction can be disregarded), localized patterns were reported long ago [7–10]. Transcending their original context of morphogenesis [7,11–13], in the past two decades they have been observed in magnetic materials, gas discharge systems, optics, liquid crystals, and chemical and mechanical systems (see, e.g., references in [14,15]). Among the latter, parametrically driven systems [10,16–20] (and, more recently, granular media) and heterogeneous catalysis are to be highlighted. Kawasaki and Ohta [21] showed that structures composed of the kinks arising in some one-dimensional (1D) nonlinear evolution equations are unstable. A review by Coullet [15] clarified why a localized pattern (or dissipative soliton [5,6]) could not be stable in a simple 1D bistable reaction diffusion (RD) system such as ut = u(β − u2 ) + μ + uxx ,

(1)

with constant β and μ. He argued that, even in the extreme case μ = 0 (equally stable HSs), the solvability condition determining the time derivative of the soliton width could be interpreted as an attracting force between the kinks bounding the bubble, which made it shrink and disappear. Instead, chiral bubbles [15] were shown to be stable in one dimension and so were the ones arising in the parametrically driven damped nonlinear Schr¨odinger equation [5,6]. Later, however, stable localized patterns were shown to arise in a monocomponent 1D RD system (the Nagumo model of population dynamics) when a nonlocal interaction term (accounting for interaction-induced modification of the environment) is considered [22,23]. In this paper we show that in (bistable, 1D, scalar) effective RD systems obtained by considering weak enough aggregating currents [attractive lateral interactions (ALIs)], localized patterns (dissipative solitons) can be stabilized by means of a suitable multiplicative noise. The underlying

* †

[email protected] [email protected]

1539-3755/2015/92(3)/032116(10)

hypothesis builds on one of the present author’s previous work in which by pushing with a driving multiplicative noise [24] one of the HSs toward the so-called instability region [25], periodic patterns due to the aggregating effect of the ALIs, which are transient by their subdominant character, become stable [26–28]. It can be stated as follows. Let a (front-enabling) bistable RD system be supplemented with ALIs, weak enough to produce only transient periodic patterns. Then, if just one of the two HSs is pushed toward the instability region by a driving multiplicative noise, a new bistability (between a stable pattern and the remaining HS) will arise. Fronts resulting from this new bistability could in principle enable, by interacting with the pattern, a pinning mechanism giving rise to localized structures. This new route to pinning states involves thus three essential ingredients [24–28]: (a) a (front-enabling) bistable RD system, (b) an aggregating current JA (x,t), which can in certain situations (e.g., when due to a very-short-range attracting potential) be assimilated to an antidiffusive term [29], and (c) a multiplicative noise of the kind leading to entropic noise-induced phase transition [24,30,31]. Typical realistic contexts in which (a) and (b) arise together are a monolayer of adsorbates undergoing ALIs [29] and the aforementioned Nagumo model [with JA (x,t) accounting for gregarious instinct [32,33]]. The pinning mechanism we are introducing crucially relies on ingredients (b) and (c). Thus (for the sake of clarity in their exposition) we have chosen to strip down ingredient (a) to its very basics: the normal form in Eq. (1), but with |u| 0.5 in order to keep in close contact with the fully realistic model of Ref. [29] (in which the field φ = u + 0.5 is the surface covering by the adsorbate). The normal form in Eq. (1) has advantages (it is a more general front-enabling bistable RD system and its simpler form allows for an analytical estimate in the spirit of the so-called solvability condition), but as we shall see, also disadvantages (it enables kink-antikink condensation as a competing mechanism, whose effect must be disentangled). In the fully realistic model instead, no stable solitons or patterns can exist in the absence of noise. The program is thus the following. Starting from a 1D RDlike dynamical equation having Q(u) := u(β − u2 ) + μ as the reaction term, uxx as bona fide diffusion, and an ALI-driven current flow, we rewrite such a deterministic equation as a relaxation one, in terms of a functional whose explicit form we

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need not know a priori (although we do assume its existence). Subsequently, we introduce the driving multiplicative noise in such a way that it satisfies the fluctuation-dissipation theorem [34] and calculate its average effect on the reaction term. In this way, it is possible to retrieve a deterministic equation with an effective reaction term, depending on the noise intensity (this technique has already been tested in other works and contrasted with simulations of corresponding stochastic processes [25,30,31]). The crucial strategy is to tailor the noise factor so that its average effect transforms the dynamics, to allow only one of the two HSs to be destabilized by an inhomogeneous perturbation. These calculations are barely sketched, to merely explain the idea we are proposing. Then we test our hypothesis through numerical simulation of the corresponding stochastic process, showing the existence of pinning states and solitons. We also carry out an analytic estimation in the spirit of the so-called solvability condition, whereby we detect alternative ways in which solitons can be stabilized (namely, kink-antikink condensation). Regarding the article’s organization, first we introduce ingredients (a) and (b) of the model under consideration; then we add the multiplicative noise to the dynamics and describe analytically its average effect. After choosing a suitable noise factor, we find numerically the predicted pinning states and show (also numerically) that solitons are stabilized by the noise effect provided the proposed conditions are fulfilled. Subsequently, we carry out an analytical estimate to elucidate some issues arising out of the previous calculations. Since (besides validating the previous results) this estimate predicts others situations, we check the latter numerically. Finally, we draw our conclusions. II. APPLYING THE HYPOTHESIS A. Model

on which side of the ALI-modified point of the Maxwell construction the system is situated). As already argued, the ALIs promote pattern formation. If the system’s parameters are such that Deff < 0, these patterns are stable. For Deff > 0, HSs (and eventually fronts) are stable and the patterns arising because of ALIs are only transient [25,27,28]. However, if additional effects (an irreversible chemical reaction in Refs. [29,35] or a multiplicative noise in [25,27,28]) can displace one HS toward field values so that Deff < 0 (what we call the instability region), it will become unstable and the patterns will be stable at its expense. B. Adding multiplicative noise

Given any field-dependent relaxation coefficient (φ), by multiplying and dividing by −(φ), we can write Eq. (2) as ∂t φ = −(φ)

η(x,t)η(x,t) = 2σ 2 δ(x − x )δ(t − t ), √ and multiplicative factor (φ), we obtain δF[φ(x)] + (φ)η(x,t). ∂t φ = −(φ) δφ(x)

(2)

√ Constant β defines the HSs [φu (d) 0.5 ± β, with u (d) indicating high (low) coverage in the model of Ref. [29]], μ is a small parameter, and JA (x) = ε0 φ(1 − φ)∂x U [φ]

is the ALI-driven current. Here U [φ] (x) = − dx f (x − x)φ(x ), a functional of φ(x), is the average ALI effect; f (x) is the ALI potential (whose detailed form is irrelevant as long as it has midrange ra ); ε0 is a measure of the ALI strength (relative to the mean kinetic energy of the individual particles); and φ(1 − φ) limits the motion of the individual particles to free places. By defining an effective diffusion coefficient

Eq. (2) reads ∂t φ = Q(φ) + ∂x [Deff ∂x φ].

Fλ [φ(x)] = F[φ(x)] + λ

L

dx ln[(φ)], −L

which is the previous one modified by the effect of noise, with the control parameter λ proportional to the noise intensity σ 2 . (As it is well known [36], calculations in the continuum have to be done with care in the presence of multiplicative white noise. Therefore, following the standard approach in nonequilibrium critical dynamics, we regard our noise here as having a cutoff at short length scales. For instance, by regarding the interval [−L,L] as a regular lattice of spacing δx, we find λ = δx2 σ 2 .) Two features are worth noting: (i) The noise introduced in Eq. (5) fulfills automatically the fluctuation-dissipation theorem [25–28,30,31] and (ii) the average effect of this noise can be found without knowing the analytical form of F[φ(x)]. The modified average dynamics can be described by replacing Q(φ) in Eq. (3) by Qλ (φ) = Q(φ) − λ(d/dφ), namely, ∂t φ = Qλ (φ) + ∂x [Deff ∂x φ].

∂x U [φ] , ∂x φ (3)

The solutions to either Eq. (2) or (3) are stable HSs and fronts propagating toward one or the other HS (φd or φu , depending

(5)

Under these conditions, the stationary probability distribution Pst (φ) of the field is of Boltzmann type [30,31] Fλ (u) Pth (φ) ∝ exp − , σ2 in terms of an effective functional

∂t φ = Q(φ) − ∂x JA + ∂xx φ.

(4)

thus obtaining a relaxational dynamics in a free-energy functional F[φ(x)] (defined by previous operation) with a field-dependent kinetic coefficient (φ). By adding to this dynamics a Gaussian white noise with zero mean, correlation

The dynamics of the field φ(x), driven by Q(φ) := u(β − u2 ) + μ (with u = φ − 0.5) and undergoing ALI, can be described by [29,32,33]

Deff = 1 − ε0 φ(1 − φ)

δF[φ(x)] , δφ(x)

(6)

We remind the reader that the starting system is always situated outside the instability region, so in the absence of noise, the result are fronts propagating toward one or the other HS (φd or φu ) depending on which side of the ALI-modified point of the Maxwell construction the system is situated. Our previous results show that a suitable multiplicative noise can push

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the HS toward the instability region [24], resulting in the stabilization of patterns [25,27,28]. C. Seeding the solitons

Since our goal is to stabilize solitons, we introduce an initial condition favoring their formation. In particular, we propose φ0u (x) := − 12 + 0.48[tanh ρA(x − xf + δ) − tanh ρA(x − xf − δ)] in order to elicit S + solitons (or bumps) and φ0d (x) :=

1 2

− 0.48[tanh ρA(x − xf + δ)

− tanh ρA(x − xf − δ)] to elicit S − solitons (or dips). Here ρ 1, the seed’s width is 2δ, and the factor A := Ldiff /ra is introduced (whereas the equation is normalized with the diffusion length Ldiff ) because the relevant parameter is ra . The seeding position xf is taken right at the center of the domain (xf = 21 600δx, with δx = 2.5 × 10−2 ra , chosen such that δx ra L). Equation (5) is then integrated by Heun’s method. (a)

D. Choosing a suitable noise factor for solitons

The following task is to choose the simplest multiplicative noise factor 1/2 (φ) to push only one of the two HSs toward the instability region. With the choice (φ) = φ, a linear stability analysis shows that φu is effectively displaced toward values for which Deff < 0 and therefore likely to be destabilized by an inhomogeneous perturbation [25,27,28]. We solved Eq. (6) numerically for this case and got stable localized structures. However, the corresponding stochastic numerical simulations yield only regular patterns (alternating high- and low-density regions) as in [25,27,28]. We concluded that a linear dependence on φ is so mild that it affects not only the target HS but the other one too and undertook a specific study [24], whereby we found that the noise’s effectiveness to push a target HS improves as (φ) becomes steeper. Accordingly (and keeping in mind that the noise should push the system toward field values where the HS can be destabilized), we propose u := exp[b(φ − 1)] to affect φu or d := exp[−bφ] to affect φd (we use b = 10). Now, since these factors can set the target HS near its instability edge, then a too intense noise would plainly destroy it. After several trials, we found a range of σ values for which solitons are stabilized. Before exhibiting our results on solitons, it is worthwhile (b)

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FIG. 1. Evolution of a hyperbolic-tangent-like profile under multiplicative noise with d : (a) ε0 = 12 and σ = 0, (b) ε0 = 16 and σ = 0, (c) ε0 = 16 and σ = 0.2, and (d) expanded view of (c). The other parameters are ra = 1.0, μ = −0.02, β = 0.2, δx = 2.510−2 , and Deff 0 for ε0 = 16 and Deff > 0 for ε0 = 12. 032116-3

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checking our main hypothesis by putting pinning fronts into evidence.

(a)

E. Pinning states

We remind the reader that it is for the sake of clarity in the presentation that we have chosen to introduce the mechanism within the context of a normal form. However, our results are clearly applicable to all (front-enabling) bistable systems. Now, whereas in the more realistic models they can never be stabilized without an additional effect (be it an irreversible chemical reaction [35] or multiplicative noise in [25,27,28])— in a normal form, patterns can arise even in the absence of noise by choosing the parameters in such a way that at least one HS lies within the instability region (Deff < 0) [25,27,28,32]. A pushing multiplicative noise is required if the HS is located outside this region (namely, where Deff > 0) and the only excitations are fronts, propagating in one or the other direction (depending on μ). Now, even for Deff > 0 but close to the edge, transient patterns may occur. Then a multiplicative noise can push the corresponding HS toward the instability region [24] and therefore the appearance of pinning fronts is to be expected. In order to check this, we locate the system close to the edge (but outside the instability region) and turn on the multiplicative noise. This way, the pinning fronts are easily stabilized when introducing a hyperbolic-tangent-like profile as the seeding initial condition. Figure 1 illustrates these concepts√(we chose to express the noise amplitude in terms of σ := 2σ ). In Fig. 1(a) the system (without noise) is located away from the instability region and the solution is a propagating front. In Fig. 1(b) the system (without noise) is located close to the edge but outside this region: The seeding initial condition stimulates the formation of spatial oscillations, which decay, however, after a short time. Figure 1(c) shows the effect of a noise with d on the case of Fig. 1(b) and an expanded view focusing on the oscillations is shown Fig. 1(d). Our expectation is fulfilled, since we managed to stabilize a pinning front by adding noise. Looking at the parameter sets, whenever we located the system near the stability edge but outside the instability region, we could stabilize a pinning front with the proposed multiplicative noise. However, very close to the Maxwell point, pinning fronts are stabilized only if the system is seeded with a Pomeau-type profile [37], whereas only propagating fronts are obtained when seeding with a hyperbolic-tangent-like initial condition. This means that both types of solution (fronts propagating toward the less stable HSs and pinning fronts) can coexist and in fact we have observed their coexistence.

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Figure 2(a) shows the evolution of seven S − solitons (a localized φ d region surrounded by φ u ) elicited with d , with σ = 30. Only for the central one did we take φ0d (x) as the initial condition; the other six emerged spontaneously out of homogeneous initial conditions. Figure 2(b) focuses on the evolution of the first soliton at the left. Noise not only stabilizes the solitons, but also induces their (apparently stochastic) motion. For decreasing σ , the motion becomes more restricted and the number of spontaneous solitons is reduced (for σ ∼ 1

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FIG. 2. (Color online) (a) Evolution of seven solitons [the central one has been seeded with φ0d (x) and the rest arose spontaneously] and (b) expanded view, focusing on the leftmost (spontaneous) soliton. The parameters are σ = 30, ra = 0.01, ε0 = 16, μ = 0.005, and β = 0.2.

a single soliton is observed, with a motion range much less than its width). Finally, the fronts collapse when σ ∼ 0.05. On the other hand, solitons are also destabilized when σ is large enough (σ 45 for this case). Figure 3 illustrates one S + soliton (a localized φu region surrounded by φd ) elicited with u . We explored a wide range

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of parameter values and found the same results whenever our criterion was satisfied. Initially, S + (−) solitons could not be elicited with d (u) , but in a subsequent rough calculation we discovered a range of parameters for which this is possible. We also observed stable solitons not likely to be due to pinning (with the system located away from the instability region). In our opinion, they are due to the more standard kink-antikink mechanism. Hereafter we show an analytical estimate that not only explains this, but also provides a result consistent with the cases for which the pinning effect is indeed observed. III. ANALYTICAL ESTIMATE

As noted before, we find some cases resulting in stable solitons by the effect of the noise but for which no pinning fronts get stabilized in the evolution of a hyperbolic-tangentlike initial condition (this situation occurs when the system is located away from the instability region). On the other hand, near the Maxwell point and close to edge (but outside the instability region) we have found both types of front (pinning and propagating hyperbolic-tangent-like) and they can even coexist. In order to understand this result, we perform a rough calculation by appealing to the solvability condition [14,15,22,32,38]. (It will prove convenient to work here in terms of u = φ − 0.5.) For σ = 0, Eq. (6), whose stationary front solutions we call us , yields, after linearization, ψ˙ = Lψ, with ψ(x,t) a small inhomogeneous perturbation and dQ dDeff dus us . D L= + ∂ ∂ + x eff x du us du us dx In this calculation we consider a situation in which both HSs are stable, but with the system close to both the instability region and the Maxwell point (through numerical calculations we know that in this case, there are solutions resembling hyperbolic-tangent-like profiles). Then we destabilize one of the HSs with a weak noise of the above-proposed form. In this context, it is valid to apply the typical techniques of the kinkantikink case using the solubility condition. So considering ∂x U/∂x φ ≡ ∂x U/∂x u as a form factor (independent of u and x), we write ε = ε0 ∂x U/∂x u. Since L(dus /dx) = 0 and, moreover, the us have the appearance of the solutions of the original Eq. (6) with ε0 = 0 and σ = 0 (neither ALI nor noise and obviously μ = 0), we can approximate us ≈ us± = β tanh[(β/2)1/2 A(x − xf± )] and apply the solvability condition, which yields an equation for the dynamics of the soliton width := x+ − x− (always positive). We write ± u (d) , where the suffixes refer to the 1/2 multiplicative noise factors u (d) (φ = u + 0.5) and the signs ± indicate S . This equation reads ˙± u (d) 3

μ ∓ λWu (d) β

+ (β/2)1/2 exp − (2β)1/2 A± u (d)

× −15 + ε 52 − 7β + Rλ ,

±23/2

(7)

where Wu = +W and Wd = −W , with 1 exp(∓bus∓ ) b W = √ exp(−b/2) dx > 0 cosh2 x β 2 −1 and R=

b2 exp(−b/2) β

1

exp[∓(bus∓ + 2x)]

−1

cosh2 x

dx > 0.

Of course, it is the positivity of [−15 + ε( 52 − 7β) + Rλ] that guarantees solution stability. The stabilizing effect of noise (which cooperates with the ALI to stabilize the solitons) is manifest in Rλ > 0. This result is consistent with those of the numerical calculations for both cases observed (pinning states and front solutions). Moreover, the above is true regardless of the sign of μ and consistent with the fact that next to the Maxwell point (small μ), the pinning mechanism is independent of which of the HSs is more stable [37,39]. In order to analyze possible stationary solutions, we define −15 + ε(5/2 − 7β) + Rλ 1 s λ = ln , (8) (2β)1/2 A λW 1 [−15 + ε(5/2 − 7β) + Rλ]β s . (9) ln μ = (2β)1/2 A 23/2 μ Now recall that the soliton’s width ( := x+ − x− ) is positive definite. So, if the second term of Eq. (7) dominates the first one (μ small enough or λ large enough), the only possible − s s stable solutions are + u λ and d λ (other solutions cause a mathematical absurdity). This means that with u the only solutions are stable S + solitons and with d the only solutions are stable S − solitons. Henceforth we will refer to this situation as the direct case (the noise is significant only at the soliton’s heart). On the other hand, when the first term in Eq. (7) dominates, μ’s sign is indeed relevant. For this case, the normal form model would enable stable S − solitons with u and stable S + solitons with d (the noise is significant only in the soliton’s environment), solutions that are not observed in the more realistic model (henceforth we will refer to this more restrictive situation as the crossed case). Then, whenever μ > 0 the s only possible stable solutions are − u μ (crossed case) s and − d μ (direct case). On the other hand, if μ < 0 the s only possible stable solutions are + d μ (crossed case) and + s u μ (direct case). In summary, we could observe that the solitons are stabilized by a cooperation between noise and ALIs. The limit for stable solitons is determined by both effects. So, on the stability line, a decrease in ALI strength requires an increased noise intensity to sustain the soliton stability. Moreover, this estimation revealed the possible existence of solutions that were not found during the first attempts: crossed cases. We saw that there are two terms competing to dominate the possible space of solutions, one driven by the noise and the other (measured by the parameter μ) by the energy gap between the two HSs (φd and φu ). When the noise dominates, only the direct case is possible; when the energy gap dominates, both case are possible. Therefore, we can conclude that for crossed cases, the solitons will be destabilized if the noise intensity is too high. On the other hand, we note that for both cases,

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FIG. 4. (Color online) (a) Evolution of three solitons [one seeded with φ0d (x) and two spontaneous] for μ = 0.005, σ = 20, and ε0 = 13 (far from the instability region), using d . (b) Expanded view focusing on the seeded soliton. (c) Expanded view focusing on one of the spontaneous solitons. (d) Same as in (a) but for ε0 = 16 (near the instability region). The other parameters are ra = 0.01, β = 0.2, and δx = 2.510−4 .

the solitons are destabilized when the noise intensity is even higher, because the target HS ceases to exist. Recall, however, that the whole approach in this section loses validity away from the Maxwell point. In other words, crossed cases are situations in which S − solitons are elicited with u and S + solitons with d . According to our estimate (and the following numerical calculations) this is impossible for very small μ values. This implies that while in the direct case, the noise acts at the heart of the soliton; here it acts on its environment. To enable this possibility we must increase μ significantly so that the first term of Eq. (7) dominates the second one. As already pointed out, Eq. (7) requires μ > 0 for S − solitons elicited with u and μ < 0 for S + solitons elicited with d . For these tests, we use the Maxwell point (slightly affected by ALIs) as reference. In the next section we exhibit numerical results confirming all the possibilities predicted by our analytical estimate, regardless of whether the possible solutions are propagating or pinning fronts. IV. NUMERICAL TEST

In this section we submit to numerical test the different scenarios found in the analytical estimate. Since the latter is valid near the Maxwell point (slightly affected by ALIs), we always choose parameter sets satisfying this condition.

In particular for the interaction midrange, we consider two alternate values: ra = 0.01 (used thus far) and ra = 1.0 (leading to faster numerical calculations). The most noticeable difference is that for ra = 0.01 spontaneous solitons emerge, whereas for ra = 1.0 only seeded ones are observed.

A. Direct case

In this case we expect stable S − solitons with d and stable S + solitons with u . In order to ensure that the dominant term in Eq. (7) is the second one, we take μ very small (|μ| = 0.005). Figure 4(a) shows the evolution of three S − solitons [one seeded with φ0d (x) and two spontaneous] 1/2 using the noise factor d for a system located away from the instability region (μ = 0.005, σ = 20, and ε0 = 13; the remaining parameters are ra = 0.01 and β = 0.2). Figures 4(b) and 4(c) focus on the seeded and one of the spontaneous solitons, respectively. For this parameter set, any frontlike initial profile (whether hyperbolic tangent or Pomeau) will yield a propagating front (never a pinning front). For ε0 = 16 instead (the system is located close to the edge, but outside the instability region) a pinning front emerges when the initial profile is of Pomeau type (and a propagating front when it is of hyperbolic tangent type). However, both the seeded [with φ0d (x)] and spontaneous

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FIG. 5. (Color online) (a) Evolution of three solitons [one seeded with φ0u (x) and two spontaneous] for μ = 0.005, σ = 20, and ε0 = 13 (far from the instability region), using u . (b) Expanded view focusing on the seeded soliton. (c) Expanded view focusing on one of the spontaneous solitons. (d) Same as in (a) but for ε0 = 16 (near the instability region). The other parameters are ra = 0.01, β = 0.2, and δx = 2.510−4 .

S − solitons [Fig. 4(d)] are almost indistinguishable from the ones in Fig. 4(a). Figure 5(a) displays the evolution of three S + solitons [one seeded with φ0u (x) and two spontaneous] using the noise 1/2 factor u for ε0 = 13 (the regime where propagating fronts result irrespectively of whether the initial profile is hyperbolictangent-like or Pomeau type, namely, away from the instability region). Figures 5(b) and 5(c) focus on the seeded and one of the spontaneous solitons, respectively. Figure 5(d) illustrates the ε0 = 16 case (system located close to the edge but outside the instability region, where an underlying pinning effect is observed). As a closing remark on the direct case, we note that numerical short-time calculations showed that a change in the sign of μ (with the same values for the remaining parameters) is irrelevant to soliton stability. This indicates that the second term of Eq. (7) effectively dominates the first one. We also checked the direct case with ra = 1.

B. Crossed case

As expected from Eq. (7), in this case we could not elicit solitons with |μ| → 0. These could be obtained only for |μ| high enough, so that the first term of Eq. (7) dominates the second one. For practical reasons, we studied this case with ra = 1.0 (this means sacrificing spontaneous solitons and working with σ values 100 times smaller).

Figure 6 displays the evolution of an S − soliton [seeded with φ0d (x) and elicited with u ] for μ = 0.02 when σ = 0.2 [Fig. 6(a)] and σ = 0.3 [Fig. 6(b)]. As expected from Eq. (7), no S − soliton was elicited for μ < 0. Figure 7 illustrates the evolution of an S + soliton [seeded with φ0u (x) and elicited with d ] for μ = −0.02 and the same remaining parameters as in Fig. 6. As expected from Eq. (7), no S + soliton was elicited for μ > 0. The remaining parameters were in both cases ra = 1.0, ε0 = 16, and β = 0.2. Regarding the noise amplitude, only between σ = 0.15 and σ = 0.3 could we elicit solitons for this parameter set. For σ = 0.1 and σ = 0.35, even the seeded solitons turned out to be unstable. Since soliton stabilization is the result of cooperation between noise and ALIs, the stability window (the range of σ for which solitons can be stabilized) varies with ε0 (lower noise intensity is required for higher ALI strength). Figure 8 illustrates the evolution of an S + soliton [seeded with φ0u (x) and elicited with d ] for μ = −0.02, σ = 0.1, and ε0 = 17. This result shows that by increasing the ALI strength, a lower noise intensity is required to stabilize the solitons. This is consistent with what the third term in Eq. (7) predicts (namely, the larger σ and ε0 are, the more stable the solitons become). Therefore, on the border of the soliton stability region, as one decreases the other must be increased in order to maintain soliton stability. On the other hand (as already mentioned), the solitons are destabilized for higher noise intensity. There are two possible causes for this: (a) The second term in Eq. (7)

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φ0u (x)

FIG. 6. (Color online) Evolution of an S − soliton seeded with and elicited by u for (a) σ = 0.2 and (b) σ = 0.3. The parameters are μ = 0.02, ra = 1.0, ε0 = 16, and β = 0.2 (δx = 2.5 × 10−2 ). φ0d (x)

becomes dominant and (b) for very high noise, the affected HS disappears. Possibility (b) is also the reason for which, in the direct case, solitons are destabilized for high enough σ . Of course, when the first term in Eq. (7) dominates the second one, there is no difficulty in obtaining solitons in the direct case (i.e., when the noise affects only the soliton’s heart); now for μ > 0 (μ < 0), only S − (S + ) solitons are obtained. Furthermore, for all the cases considered we have verified numerically the relationship between the stability window and ALIs predicted by the third term in Eq. (7).

of the HSs will be significantly affected by the noise), pinning fronts and solitons can in fact be stabilized. Moreover, for high enough noise intensity, multiple spontaneous solitons emerge and become stable. We remark that pusher noise means noise able to push a system’s HS toward field values predetermined by its multiplicative factor. This effect was proposed in several reports [25–28,30,31], but particularly studied in [24]. In particular, options for the multiplicative √ we proposed two √ factor: d to affect φd and u to affect φu . Through the first one we obtained S − solitons (the noise affects only the

V. ANALYSIS AND CONCLUSIONS

Based on of our previous work [24–28], in which periodic patterns arising from the effect of ALIs in a (front-enabling) bistable RD system were stabilized with a multiplicative noise, we stated the hypothesis that if only one HS (out of φd and φu ) is pushed toward the instability region by a driving multiplicative noise, we obtain a HS–pattern bistability situation that could in principle enable a pinning mechanism and thus give rise to dissipative solitons, which prompted us to undertake a numerical search of dissipative solitons. We found that if (for a given noise intensity) the multiplicative noise factor depends strongly enough on the field (so as to ensure the effectiveness of the pusher noise and that only one

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soliton’s heart), while through the second one we obtained S + solitons (again, the noise affects only the soliton’s heart). In the preliminary numerical search, we did not find any trace of the crossed case (solitons with the noise affecting only their environment). Then an analytical estimate (applying the usual techniques of the kink-antikink case, namely. using the solubility condition) allowed us to find the parameter values for which the crossed case is possible. Of course, we checked this result through numerical simulation. Clearly, the aforementioned techniques were devised for front solutions of a system located near the Maxwell point, not for pinning fronts. The need to perform this estimate was prompted by the results obtained when looking for the conjectured pinning fronts. We found that the pinning mechanism can be easily enabled by the noise when the system is located close enough to the instability region. Under general conditions, it suffices to start from a hyperbolic-tangent-like profile to activate a pinning front. However, very close to the Maxwell point, pinning fronts become stable only if starting from a Pomeau-type profile. Under the same conditions as above, only propagating fronts emerge when starting from a hyperbolic-tangent-like profile. The latter means that both types of fronts are possible solutions (in fact, numerical calculations showed the coexistence of both solutions). Moreover, when the system is located away from the instability region, we found situations where the solitons become stable even though the pinning fronts cannot be activated. Briefly, we found three different situations as a possible support of the dynamics that enables soliton stabilization: (a) pinning fronts only, (b) pinning fronts coexisting with propagating ones, and (c) propagating fronts only. Cases (a) and (b) are to be expected, but not (c). This is what prompted us to perform the analytical estimate, whereby we found an equation that governs the soliton width dynamics and shows the cooperation of noise and ALIs in stabilizing the soliton. This equation also predicts different situations, whose existence we subsequently verified through numerical simulations. Concerning the dependence of the soliton behavior on the noise intensity, through numerous simulations we observed

that its effect is twofold. On the one hand, a stronger noise promotes multiple spontaneous solitons (their number increases with the noise intensity). On the other hand, it drives the solitons’ stochastic motion. On a shorter time scale, the amplitude of such motion increases monotonically with the noise intensity. On a longer time scale, it initially increases monotonically, but when the noise intensity is high enough, the interactions between neighboring solitons tends to confine them (their mobility is reduced). We also observed that if the noise intensity exceeds a certain value, the solitons are destabilized. As established earlier, in this case the HS is overly pushed by the noise until it disappears. In short, given a bistable system with front solutions and ALIs, it is possible to induce the formation of stable solitons by introducing a suitable multiplicative noise. These solitons acquire a stochastic motion that grows with the noise intensity, although it is in part limited by the interaction between solitons when they are close together. We emphasize that the solitons only exist within a (wide enough) window of noise intensity values. By increasing the noise intensity above the stability threshold, first only the seeded soliton is stabilized, but as the noise intensity is increased further, a growing number of spontaneous solitons are stabilized. Then, past the upper intensity threshold, all solitons are destabilized. Finally, we observed that the solitons are constructed by ALIs but stabilized by the cooperation of both ALIs and the driving noise. This is independent of the nature of the interacting agents, as long as their interaction is short range. Therefore, we believe our idea to be applicable to other systems with ALIs, be it within the realm of physics or not. In order to considerer agents of other nature, we only require Q(φ) to express a bistable system with front solutions. In future work, we plan to support the generality of our results by applying this approach to systems of different nature.

[1] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009). [2] Localized States in Physics: Solitons and Patterns, edited by O. Descalzi, M. Clerc, S. Residori, and G. Assanto (Springer, Berlin, 2011). [3] A. W. Liehr, Dissipative Solitons in Reaction Diffusion Systems: Mechanisms, Dynamics, Interaction (Springer, Berlin, 2013). [4] Localized Excitations in Nonlinear Complex Systems: Current State of the Art and Future Perspectives, edited by R. CarreteroGonz´alez, J. Cuevas-Maraver, D. Frantzeskakis, N. Karachalios, P. Kevrekidis, and F. Palmero-Acebedo (Springer, Berlin, 2014). [5] M. G. Clerc, S. Coulibaly, and D. Laroze, Phys. Rev. E 77, 056209 (2008). [6] M. G. Clerc, S. Coulibaly, L. Gordillo, N. Mujica, and R. Navarro, Phys. Rev. E 84, 036205 (2011). [7] A. Gierer and H. Meinhardt, Kybernetik 12, 30 (1972).

[8] M. Herschkowitz-Kaufman and G. Nicolis, J. Chem. Phys. 56, 1890 (1972). [9] Magnetic Bubbles, edited by T. H. O’Dell (Wiley, New York, 1974). [10] J. Wu, R. Keolian, and I. Rudnick, Phys. Rev. Lett. 52, 1421 (1984). [11] D. Iron, M. J. Ward, and J. Wei, Physica D 150, 25 (2001). [12] D. Iron and M. J. Ward, Math. Comput. Simul. 77, 035204 (2001). [13] A. Yochelis and A. Garfinkel, Phys. Rev. E 77, R035204 (2008). [14] P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000). [15] P. Coullet, Int. J. Bif. Chaos Appl. Sci. Eng. 12, 2445 (2002). [16] B. Denardo, B. Galvin, A. Greenfield, A. Larraza, S. Putterman, and W. Wright, Phys. Rev. Lett. 68, 1730 (1992).

ACKNOWLEDGMENTS

Financial support from CONICET and UNMdP of Argentina is acknowledged.

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PHYSICAL REVIEW E 92, 032116 (2015)

W. Zhang and J. Vi˜nals, Phys. Rev. Lett. 74, 690 (1995). S. Longhi, Phys. Rev. E 53, 5520 (1996). X. Wang and R. Wei, Phys. Rev. E 57, 2405 (1998). I. V. Barashenkov and E. V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999). K. Kawasaki and T. Ohta, Physica A 116, 19089 (1996). M. G. Clerc, D. Escaff, and V. M. Kenkre, Phys. Rev. E 82, 036210 (2010). D. Escaff, Eur. Phys. J. D 62, 33 (2011). S. E. Mangioni, Eur. Phys. J. B 88, 53 (2015). S. E. Mangioni, Physica A 389, 1799 (2010). S. E. Mangioni and H. S. Wio, Phys. Rev. E 71, 056203 (2005). S. E. Mangioni and R. R. Deza, Phys. Rev. E 82, 042101 (2010). S. E. Mangioni and R. R. Deza, Physica A 391, 4191 (2012). M. Hildebrand and A. S. Mikhailov, J. Phys. Chem. 100, 573 (1982).

[30] M. Iba˜nes, J. Garc´ıa-Ojalvo, R. Toral, and J. M. Sancho, Phys. Rev. Lett. 87, 020601 (2001). [31] O. Carrillo, M. Iba˜nes, J. Garc´ıa-Ojalvo, J. Casademunt, and J. M. Sancho, Phys. Rev. E 67, 046110 (2003). [32] S. E. Mangioni, Eur. Phys. J. B 86, 390 (2013). [33] S. E. Mangioni, Eur. Phys. J. B 87, 248 (2014). [34] K. Kitahara and M. Imada, Prog. Theor. Phys. Suppl. 64, 65 (1978). [35] M. Hildebrand, A. S. Mikhailov, and G. Ertl, Phys. Rev. E 58, 5483 (1998). [36] J. Garc´ıa-Ojalvo and J. M. Sancho, Noise in Spatially Extended System (Springer, New York, 1999). [37] Y. Pomeau, Physica D 23, 3 (1986). [38] M. G. Clerc, D. Escaff, and V. M. Kenkre, Phys. Rev. E 72, 056217 (2005). [39] N. Verschueren Van Rees, M.Sc. thesis, Universidad de Chile, 2013.

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Stochastic dissipative solitons Sergio E. Mangioni* and Roberto R. Deza† Instituto de F´ısica de Mar del Plata, Universidad Nacional de Mar del Plata, and CONICET, De´an Funes 3350, B7602AYL Mar del Plata, Buenos Aires, Argentina (Received 2 November 2013; revised manuscript received 10 July 2015; published 11 September 2015) By the effect of aggregating currents, some systems display an effective diffusion coefficient that becomes negative in a range of the order parameter, giving rise to bistability among homogeneous states (HSs). By applying a proper multiplicative noise, localized (pinning) states are shown to become stable at the expense of one of the HSs. They are, however, not static, but their location fluctuates with a variance that increases with the noise intensity. The numerical results are supported by an analytical estimate in the spirit of the so-called solvability condition. DOI: 10.1103/PhysRevE.92.032116

PACS number(s): 05.40.Ca, 05.45.Yv, 05.65.+b

I. INTRODUCTION

In recent times, partly spurred by the nanotechnology and optical fiber industries, we have witnessed an upsurge of interest in localized patterns bubbles of metastable homogeneous states (HSs) [1–4]. Also known as dissipative solitons [5,6] (as long as their interaction can be disregarded), localized patterns were reported long ago [7–10]. Transcending their original context of morphogenesis [7,11–13], in the past two decades they have been observed in magnetic materials, gas discharge systems, optics, liquid crystals, and chemical and mechanical systems (see, e.g., references in [14,15]). Among the latter, parametrically driven systems [10,16–20] (and, more recently, granular media) and heterogeneous catalysis are to be highlighted. Kawasaki and Ohta [21] showed that structures composed of the kinks arising in some one-dimensional (1D) nonlinear evolution equations are unstable. A review by Coullet [15] clarified why a localized pattern (or dissipative soliton [5,6]) could not be stable in a simple 1D bistable reaction diffusion (RD) system such as ut = u(β − u2 ) + μ + uxx ,

(1)

with constant β and μ. He argued that, even in the extreme case μ = 0 (equally stable HSs), the solvability condition determining the time derivative of the soliton width could be interpreted as an attracting force between the kinks bounding the bubble, which made it shrink and disappear. Instead, chiral bubbles [15] were shown to be stable in one dimension and so were the ones arising in the parametrically driven damped nonlinear Schr¨odinger equation [5,6]. Later, however, stable localized patterns were shown to arise in a monocomponent 1D RD system (the Nagumo model of population dynamics) when a nonlocal interaction term (accounting for interaction-induced modification of the environment) is considered [22,23]. In this paper we show that in (bistable, 1D, scalar) effective RD systems obtained by considering weak enough aggregating currents [attractive lateral interactions (ALIs)], localized patterns (dissipative solitons) can be stabilized by means of a suitable multiplicative noise. The underlying

* †

[email protected] [email protected]

1539-3755/2015/92(3)/032116(10)

hypothesis builds on one of the present author’s previous work in which by pushing with a driving multiplicative noise [24] one of the HSs toward the so-called instability region [25], periodic patterns due to the aggregating effect of the ALIs, which are transient by their subdominant character, become stable [26–28]. It can be stated as follows. Let a (front-enabling) bistable RD system be supplemented with ALIs, weak enough to produce only transient periodic patterns. Then, if just one of the two HSs is pushed toward the instability region by a driving multiplicative noise, a new bistability (between a stable pattern and the remaining HS) will arise. Fronts resulting from this new bistability could in principle enable, by interacting with the pattern, a pinning mechanism giving rise to localized structures. This new route to pinning states involves thus three essential ingredients [24–28]: (a) a (front-enabling) bistable RD system, (b) an aggregating current JA (x,t), which can in certain situations (e.g., when due to a very-short-range attracting potential) be assimilated to an antidiffusive term [29], and (c) a multiplicative noise of the kind leading to entropic noise-induced phase transition [24,30,31]. Typical realistic contexts in which (a) and (b) arise together are a monolayer of adsorbates undergoing ALIs [29] and the aforementioned Nagumo model [with JA (x,t) accounting for gregarious instinct [32,33]]. The pinning mechanism we are introducing crucially relies on ingredients (b) and (c). Thus (for the sake of clarity in their exposition) we have chosen to strip down ingredient (a) to its very basics: the normal form in Eq. (1), but with |u| 0.5 in order to keep in close contact with the fully realistic model of Ref. [29] (in which the field φ = u + 0.5 is the surface covering by the adsorbate). The normal form in Eq. (1) has advantages (it is a more general front-enabling bistable RD system and its simpler form allows for an analytical estimate in the spirit of the so-called solvability condition), but as we shall see, also disadvantages (it enables kink-antikink condensation as a competing mechanism, whose effect must be disentangled). In the fully realistic model instead, no stable solitons or patterns can exist in the absence of noise. The program is thus the following. Starting from a 1D RDlike dynamical equation having Q(u) := u(β − u2 ) + μ as the reaction term, uxx as bona fide diffusion, and an ALI-driven current flow, we rewrite such a deterministic equation as a relaxation one, in terms of a functional whose explicit form we

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need not know a priori (although we do assume its existence). Subsequently, we introduce the driving multiplicative noise in such a way that it satisfies the fluctuation-dissipation theorem [34] and calculate its average effect on the reaction term. In this way, it is possible to retrieve a deterministic equation with an effective reaction term, depending on the noise intensity (this technique has already been tested in other works and contrasted with simulations of corresponding stochastic processes [25,30,31]). The crucial strategy is to tailor the noise factor so that its average effect transforms the dynamics, to allow only one of the two HSs to be destabilized by an inhomogeneous perturbation. These calculations are barely sketched, to merely explain the idea we are proposing. Then we test our hypothesis through numerical simulation of the corresponding stochastic process, showing the existence of pinning states and solitons. We also carry out an analytic estimation in the spirit of the so-called solvability condition, whereby we detect alternative ways in which solitons can be stabilized (namely, kink-antikink condensation). Regarding the article’s organization, first we introduce ingredients (a) and (b) of the model under consideration; then we add the multiplicative noise to the dynamics and describe analytically its average effect. After choosing a suitable noise factor, we find numerically the predicted pinning states and show (also numerically) that solitons are stabilized by the noise effect provided the proposed conditions are fulfilled. Subsequently, we carry out an analytical estimate to elucidate some issues arising out of the previous calculations. Since (besides validating the previous results) this estimate predicts others situations, we check the latter numerically. Finally, we draw our conclusions. II. APPLYING THE HYPOTHESIS A. Model

on which side of the ALI-modified point of the Maxwell construction the system is situated). As already argued, the ALIs promote pattern formation. If the system’s parameters are such that Deff < 0, these patterns are stable. For Deff > 0, HSs (and eventually fronts) are stable and the patterns arising because of ALIs are only transient [25,27,28]. However, if additional effects (an irreversible chemical reaction in Refs. [29,35] or a multiplicative noise in [25,27,28]) can displace one HS toward field values so that Deff < 0 (what we call the instability region), it will become unstable and the patterns will be stable at its expense. B. Adding multiplicative noise

Given any field-dependent relaxation coefficient (φ), by multiplying and dividing by −(φ), we can write Eq. (2) as ∂t φ = −(φ)

η(x,t)η(x,t) = 2σ 2 δ(x − x )δ(t − t ), √ and multiplicative factor (φ), we obtain δF[φ(x)] + (φ)η(x,t). ∂t φ = −(φ) δφ(x)

(2)

√ Constant β defines the HSs [φu (d) 0.5 ± β, with u (d) indicating high (low) coverage in the model of Ref. [29]], μ is a small parameter, and JA (x) = ε0 φ(1 − φ)∂x U [φ]

is the ALI-driven current. Here U [φ] (x) = − dx f (x − x)φ(x ), a functional of φ(x), is the average ALI effect; f (x) is the ALI potential (whose detailed form is irrelevant as long as it has midrange ra ); ε0 is a measure of the ALI strength (relative to the mean kinetic energy of the individual particles); and φ(1 − φ) limits the motion of the individual particles to free places. By defining an effective diffusion coefficient

Eq. (2) reads ∂t φ = Q(φ) + ∂x [Deff ∂x φ].

Fλ [φ(x)] = F[φ(x)] + λ

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which is the previous one modified by the effect of noise, with the control parameter λ proportional to the noise intensity σ 2 . (As it is well known [36], calculations in the continuum have to be done with care in the presence of multiplicative white noise. Therefore, following the standard approach in nonequilibrium critical dynamics, we regard our noise here as having a cutoff at short length scales. For instance, by regarding the interval [−L,L] as a regular lattice of spacing δx, we find λ = δx2 σ 2 .) Two features are worth noting: (i) The noise introduced in Eq. (5) fulfills automatically the fluctuation-dissipation theorem [25–28,30,31] and (ii) the average effect of this noise can be found without knowing the analytical form of F[φ(x)]. The modified average dynamics can be described by replacing Q(φ) in Eq. (3) by Qλ (φ) = Q(φ) − λ(d/dφ), namely, ∂t φ = Qλ (φ) + ∂x [Deff ∂x φ].

∂x U [φ] , ∂x φ (3)

The solutions to either Eq. (2) or (3) are stable HSs and fronts propagating toward one or the other HS (φd or φu , depending

(5)

Under these conditions, the stationary probability distribution Pst (φ) of the field is of Boltzmann type [30,31] Fλ (u) Pth (φ) ∝ exp − , σ2 in terms of an effective functional

∂t φ = Q(φ) − ∂x JA + ∂xx φ.

(4)

thus obtaining a relaxational dynamics in a free-energy functional F[φ(x)] (defined by previous operation) with a field-dependent kinetic coefficient (φ). By adding to this dynamics a Gaussian white noise with zero mean, correlation

The dynamics of the field φ(x), driven by Q(φ) := u(β − u2 ) + μ (with u = φ − 0.5) and undergoing ALI, can be described by [29,32,33]

Deff = 1 − ε0 φ(1 − φ)

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(6)

We remind the reader that the starting system is always situated outside the instability region, so in the absence of noise, the result are fronts propagating toward one or the other HS (φd or φu ) depending on which side of the ALI-modified point of the Maxwell construction the system is situated. Our previous results show that a suitable multiplicative noise can push

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the HS toward the instability region [24], resulting in the stabilization of patterns [25,27,28]. C. Seeding the solitons

Since our goal is to stabilize solitons, we introduce an initial condition favoring their formation. In particular, we propose φ0u (x) := − 12 + 0.48[tanh ρA(x − xf + δ) − tanh ρA(x − xf − δ)] in order to elicit S + solitons (or bumps) and φ0d (x) :=

1 2

− 0.48[tanh ρA(x − xf + δ)

− tanh ρA(x − xf − δ)] to elicit S − solitons (or dips). Here ρ 1, the seed’s width is 2δ, and the factor A := Ldiff /ra is introduced (whereas the equation is normalized with the diffusion length Ldiff ) because the relevant parameter is ra . The seeding position xf is taken right at the center of the domain (xf = 21 600δx, with δx = 2.5 × 10−2 ra , chosen such that δx ra L). Equation (5) is then integrated by Heun’s method. (a)

D. Choosing a suitable noise factor for solitons

The following task is to choose the simplest multiplicative noise factor 1/2 (φ) to push only one of the two HSs toward the instability region. With the choice (φ) = φ, a linear stability analysis shows that φu is effectively displaced toward values for which Deff < 0 and therefore likely to be destabilized by an inhomogeneous perturbation [25,27,28]. We solved Eq. (6) numerically for this case and got stable localized structures. However, the corresponding stochastic numerical simulations yield only regular patterns (alternating high- and low-density regions) as in [25,27,28]. We concluded that a linear dependence on φ is so mild that it affects not only the target HS but the other one too and undertook a specific study [24], whereby we found that the noise’s effectiveness to push a target HS improves as (φ) becomes steeper. Accordingly (and keeping in mind that the noise should push the system toward field values where the HS can be destabilized), we propose u := exp[b(φ − 1)] to affect φu or d := exp[−bφ] to affect φd (we use b = 10). Now, since these factors can set the target HS near its instability edge, then a too intense noise would plainly destroy it. After several trials, we found a range of σ values for which solitons are stabilized. Before exhibiting our results on solitons, it is worthwhile (b)

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checking our main hypothesis by putting pinning fronts into evidence.

(a)

E. Pinning states

We remind the reader that it is for the sake of clarity in the presentation that we have chosen to introduce the mechanism within the context of a normal form. However, our results are clearly applicable to all (front-enabling) bistable systems. Now, whereas in the more realistic models they can never be stabilized without an additional effect (be it an irreversible chemical reaction [35] or multiplicative noise in [25,27,28])— in a normal form, patterns can arise even in the absence of noise by choosing the parameters in such a way that at least one HS lies within the instability region (Deff < 0) [25,27,28,32]. A pushing multiplicative noise is required if the HS is located outside this region (namely, where Deff > 0) and the only excitations are fronts, propagating in one or the other direction (depending on μ). Now, even for Deff > 0 but close to the edge, transient patterns may occur. Then a multiplicative noise can push the corresponding HS toward the instability region [24] and therefore the appearance of pinning fronts is to be expected. In order to check this, we locate the system close to the edge (but outside the instability region) and turn on the multiplicative noise. This way, the pinning fronts are easily stabilized when introducing a hyperbolic-tangent-like profile as the seeding initial condition. Figure 1 illustrates these concepts√(we chose to express the noise amplitude in terms of σ := 2σ ). In Fig. 1(a) the system (without noise) is located away from the instability region and the solution is a propagating front. In Fig. 1(b) the system (without noise) is located close to the edge but outside this region: The seeding initial condition stimulates the formation of spatial oscillations, which decay, however, after a short time. Figure 1(c) shows the effect of a noise with d on the case of Fig. 1(b) and an expanded view focusing on the oscillations is shown Fig. 1(d). Our expectation is fulfilled, since we managed to stabilize a pinning front by adding noise. Looking at the parameter sets, whenever we located the system near the stability edge but outside the instability region, we could stabilize a pinning front with the proposed multiplicative noise. However, very close to the Maxwell point, pinning fronts are stabilized only if the system is seeded with a Pomeau-type profile [37], whereas only propagating fronts are obtained when seeding with a hyperbolic-tangent-like initial condition. This means that both types of solution (fronts propagating toward the less stable HSs and pinning fronts) can coexist and in fact we have observed their coexistence.

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FIG. 2. (Color online) (a) Evolution of seven solitons [the central one has been seeded with φ0d (x) and the rest arose spontaneously] and (b) expanded view, focusing on the leftmost (spontaneous) soliton. The parameters are σ = 30, ra = 0.01, ε0 = 16, μ = 0.005, and β = 0.2.

a single soliton is observed, with a motion range much less than its width). Finally, the fronts collapse when σ ∼ 0.05. On the other hand, solitons are also destabilized when σ is large enough (σ 45 for this case). Figure 3 illustrates one S + soliton (a localized φu region surrounded by φd ) elicited with u . We explored a wide range

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of parameter values and found the same results whenever our criterion was satisfied. Initially, S + (−) solitons could not be elicited with d (u) , but in a subsequent rough calculation we discovered a range of parameters for which this is possible. We also observed stable solitons not likely to be due to pinning (with the system located away from the instability region). In our opinion, they are due to the more standard kink-antikink mechanism. Hereafter we show an analytical estimate that not only explains this, but also provides a result consistent with the cases for which the pinning effect is indeed observed. III. ANALYTICAL ESTIMATE

As noted before, we find some cases resulting in stable solitons by the effect of the noise but for which no pinning fronts get stabilized in the evolution of a hyperbolic-tangentlike initial condition (this situation occurs when the system is located away from the instability region). On the other hand, near the Maxwell point and close to edge (but outside the instability region) we have found both types of front (pinning and propagating hyperbolic-tangent-like) and they can even coexist. In order to understand this result, we perform a rough calculation by appealing to the solvability condition [14,15,22,32,38]. (It will prove convenient to work here in terms of u = φ − 0.5.) For σ = 0, Eq. (6), whose stationary front solutions we call us , yields, after linearization, ψ˙ = Lψ, with ψ(x,t) a small inhomogeneous perturbation and dQ dDeff dus us . D L= + ∂ ∂ + x eff x du us du us dx In this calculation we consider a situation in which both HSs are stable, but with the system close to both the instability region and the Maxwell point (through numerical calculations we know that in this case, there are solutions resembling hyperbolic-tangent-like profiles). Then we destabilize one of the HSs with a weak noise of the above-proposed form. In this context, it is valid to apply the typical techniques of the kinkantikink case using the solubility condition. So considering ∂x U/∂x φ ≡ ∂x U/∂x u as a form factor (independent of u and x), we write ε = ε0 ∂x U/∂x u. Since L(dus /dx) = 0 and, moreover, the us have the appearance of the solutions of the original Eq. (6) with ε0 = 0 and σ = 0 (neither ALI nor noise and obviously μ = 0), we can approximate us ≈ us± = β tanh[(β/2)1/2 A(x − xf± )] and apply the solvability condition, which yields an equation for the dynamics of the soliton width := x+ − x− (always positive). We write ± u (d) , where the suffixes refer to the 1/2 multiplicative noise factors u (d) (φ = u + 0.5) and the signs ± indicate S . This equation reads ˙± u (d) 3

μ ∓ λWu (d) β

+ (β/2)1/2 exp − (2β)1/2 A± u (d)

× −15 + ε 52 − 7β + Rλ ,

±23/2

(7)

where Wu = +W and Wd = −W , with 1 exp(∓bus∓ ) b W = √ exp(−b/2) dx > 0 cosh2 x β 2 −1 and R=

b2 exp(−b/2) β

1

exp[∓(bus∓ + 2x)]

−1

cosh2 x

dx > 0.

Of course, it is the positivity of [−15 + ε( 52 − 7β) + Rλ] that guarantees solution stability. The stabilizing effect of noise (which cooperates with the ALI to stabilize the solitons) is manifest in Rλ > 0. This result is consistent with those of the numerical calculations for both cases observed (pinning states and front solutions). Moreover, the above is true regardless of the sign of μ and consistent with the fact that next to the Maxwell point (small μ), the pinning mechanism is independent of which of the HSs is more stable [37,39]. In order to analyze possible stationary solutions, we define −15 + ε(5/2 − 7β) + Rλ 1 s λ = ln , (8) (2β)1/2 A λW 1 [−15 + ε(5/2 − 7β) + Rλ]β s . (9) ln μ = (2β)1/2 A 23/2 μ Now recall that the soliton’s width ( := x+ − x− ) is positive definite. So, if the second term of Eq. (7) dominates the first one (μ small enough or λ large enough), the only possible − s s stable solutions are + u λ and d λ (other solutions cause a mathematical absurdity). This means that with u the only solutions are stable S + solitons and with d the only solutions are stable S − solitons. Henceforth we will refer to this situation as the direct case (the noise is significant only at the soliton’s heart). On the other hand, when the first term in Eq. (7) dominates, μ’s sign is indeed relevant. For this case, the normal form model would enable stable S − solitons with u and stable S + solitons with d (the noise is significant only in the soliton’s environment), solutions that are not observed in the more realistic model (henceforth we will refer to this more restrictive situation as the crossed case). Then, whenever μ > 0 the s only possible stable solutions are − u μ (crossed case) s and − d μ (direct case). On the other hand, if μ < 0 the s only possible stable solutions are + d μ (crossed case) and + s u μ (direct case). In summary, we could observe that the solitons are stabilized by a cooperation between noise and ALIs. The limit for stable solitons is determined by both effects. So, on the stability line, a decrease in ALI strength requires an increased noise intensity to sustain the soliton stability. Moreover, this estimation revealed the possible existence of solutions that were not found during the first attempts: crossed cases. We saw that there are two terms competing to dominate the possible space of solutions, one driven by the noise and the other (measured by the parameter μ) by the energy gap between the two HSs (φd and φu ). When the noise dominates, only the direct case is possible; when the energy gap dominates, both case are possible. Therefore, we can conclude that for crossed cases, the solitons will be destabilized if the noise intensity is too high. On the other hand, we note that for both cases,

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FIG. 4. (Color online) (a) Evolution of three solitons [one seeded with φ0d (x) and two spontaneous] for μ = 0.005, σ = 20, and ε0 = 13 (far from the instability region), using d . (b) Expanded view focusing on the seeded soliton. (c) Expanded view focusing on one of the spontaneous solitons. (d) Same as in (a) but for ε0 = 16 (near the instability region). The other parameters are ra = 0.01, β = 0.2, and δx = 2.510−4 .

the solitons are destabilized when the noise intensity is even higher, because the target HS ceases to exist. Recall, however, that the whole approach in this section loses validity away from the Maxwell point. In other words, crossed cases are situations in which S − solitons are elicited with u and S + solitons with d . According to our estimate (and the following numerical calculations) this is impossible for very small μ values. This implies that while in the direct case, the noise acts at the heart of the soliton; here it acts on its environment. To enable this possibility we must increase μ significantly so that the first term of Eq. (7) dominates the second one. As already pointed out, Eq. (7) requires μ > 0 for S − solitons elicited with u and μ < 0 for S + solitons elicited with d . For these tests, we use the Maxwell point (slightly affected by ALIs) as reference. In the next section we exhibit numerical results confirming all the possibilities predicted by our analytical estimate, regardless of whether the possible solutions are propagating or pinning fronts. IV. NUMERICAL TEST

In this section we submit to numerical test the different scenarios found in the analytical estimate. Since the latter is valid near the Maxwell point (slightly affected by ALIs), we always choose parameter sets satisfying this condition.

In particular for the interaction midrange, we consider two alternate values: ra = 0.01 (used thus far) and ra = 1.0 (leading to faster numerical calculations). The most noticeable difference is that for ra = 0.01 spontaneous solitons emerge, whereas for ra = 1.0 only seeded ones are observed.

A. Direct case

In this case we expect stable S − solitons with d and stable S + solitons with u . In order to ensure that the dominant term in Eq. (7) is the second one, we take μ very small (|μ| = 0.005). Figure 4(a) shows the evolution of three S − solitons [one seeded with φ0d (x) and two spontaneous] 1/2 using the noise factor d for a system located away from the instability region (μ = 0.005, σ = 20, and ε0 = 13; the remaining parameters are ra = 0.01 and β = 0.2). Figures 4(b) and 4(c) focus on the seeded and one of the spontaneous solitons, respectively. For this parameter set, any frontlike initial profile (whether hyperbolic tangent or Pomeau) will yield a propagating front (never a pinning front). For ε0 = 16 instead (the system is located close to the edge, but outside the instability region) a pinning front emerges when the initial profile is of Pomeau type (and a propagating front when it is of hyperbolic tangent type). However, both the seeded [with φ0d (x)] and spontaneous

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FIG. 5. (Color online) (a) Evolution of three solitons [one seeded with φ0u (x) and two spontaneous] for μ = 0.005, σ = 20, and ε0 = 13 (far from the instability region), using u . (b) Expanded view focusing on the seeded soliton. (c) Expanded view focusing on one of the spontaneous solitons. (d) Same as in (a) but for ε0 = 16 (near the instability region). The other parameters are ra = 0.01, β = 0.2, and δx = 2.510−4 .

S − solitons [Fig. 4(d)] are almost indistinguishable from the ones in Fig. 4(a). Figure 5(a) displays the evolution of three S + solitons [one seeded with φ0u (x) and two spontaneous] using the noise 1/2 factor u for ε0 = 13 (the regime where propagating fronts result irrespectively of whether the initial profile is hyperbolictangent-like or Pomeau type, namely, away from the instability region). Figures 5(b) and 5(c) focus on the seeded and one of the spontaneous solitons, respectively. Figure 5(d) illustrates the ε0 = 16 case (system located close to the edge but outside the instability region, where an underlying pinning effect is observed). As a closing remark on the direct case, we note that numerical short-time calculations showed that a change in the sign of μ (with the same values for the remaining parameters) is irrelevant to soliton stability. This indicates that the second term of Eq. (7) effectively dominates the first one. We also checked the direct case with ra = 1.

B. Crossed case

As expected from Eq. (7), in this case we could not elicit solitons with |μ| → 0. These could be obtained only for |μ| high enough, so that the first term of Eq. (7) dominates the second one. For practical reasons, we studied this case with ra = 1.0 (this means sacrificing spontaneous solitons and working with σ values 100 times smaller).

Figure 6 displays the evolution of an S − soliton [seeded with φ0d (x) and elicited with u ] for μ = 0.02 when σ = 0.2 [Fig. 6(a)] and σ = 0.3 [Fig. 6(b)]. As expected from Eq. (7), no S − soliton was elicited for μ < 0. Figure 7 illustrates the evolution of an S + soliton [seeded with φ0u (x) and elicited with d ] for μ = −0.02 and the same remaining parameters as in Fig. 6. As expected from Eq. (7), no S + soliton was elicited for μ > 0. The remaining parameters were in both cases ra = 1.0, ε0 = 16, and β = 0.2. Regarding the noise amplitude, only between σ = 0.15 and σ = 0.3 could we elicit solitons for this parameter set. For σ = 0.1 and σ = 0.35, even the seeded solitons turned out to be unstable. Since soliton stabilization is the result of cooperation between noise and ALIs, the stability window (the range of σ for which solitons can be stabilized) varies with ε0 (lower noise intensity is required for higher ALI strength). Figure 8 illustrates the evolution of an S + soliton [seeded with φ0u (x) and elicited with d ] for μ = −0.02, σ = 0.1, and ε0 = 17. This result shows that by increasing the ALI strength, a lower noise intensity is required to stabilize the solitons. This is consistent with what the third term in Eq. (7) predicts (namely, the larger σ and ε0 are, the more stable the solitons become). Therefore, on the border of the soliton stability region, as one decreases the other must be increased in order to maintain soliton stability. On the other hand (as already mentioned), the solitons are destabilized for higher noise intensity. There are two possible causes for this: (a) The second term in Eq. (7)

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φ0u (x)

FIG. 6. (Color online) Evolution of an S − soliton seeded with and elicited by u for (a) σ = 0.2 and (b) σ = 0.3. The parameters are μ = 0.02, ra = 1.0, ε0 = 16, and β = 0.2 (δx = 2.5 × 10−2 ). φ0d (x)

becomes dominant and (b) for very high noise, the affected HS disappears. Possibility (b) is also the reason for which, in the direct case, solitons are destabilized for high enough σ . Of course, when the first term in Eq. (7) dominates the second one, there is no difficulty in obtaining solitons in the direct case (i.e., when the noise affects only the soliton’s heart); now for μ > 0 (μ < 0), only S − (S + ) solitons are obtained. Furthermore, for all the cases considered we have verified numerically the relationship between the stability window and ALIs predicted by the third term in Eq. (7).

of the HSs will be significantly affected by the noise), pinning fronts and solitons can in fact be stabilized. Moreover, for high enough noise intensity, multiple spontaneous solitons emerge and become stable. We remark that pusher noise means noise able to push a system’s HS toward field values predetermined by its multiplicative factor. This effect was proposed in several reports [25–28,30,31], but particularly studied in [24]. In particular, options for the multiplicative √ we proposed two √ factor: d to affect φd and u to affect φu . Through the first one we obtained S − solitons (the noise affects only the

V. ANALYSIS AND CONCLUSIONS

Based on of our previous work [24–28], in which periodic patterns arising from the effect of ALIs in a (front-enabling) bistable RD system were stabilized with a multiplicative noise, we stated the hypothesis that if only one HS (out of φd and φu ) is pushed toward the instability region by a driving multiplicative noise, we obtain a HS–pattern bistability situation that could in principle enable a pinning mechanism and thus give rise to dissipative solitons, which prompted us to undertake a numerical search of dissipative solitons. We found that if (for a given noise intensity) the multiplicative noise factor depends strongly enough on the field (so as to ensure the effectiveness of the pusher noise and that only one

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soliton’s heart), while through the second one we obtained S + solitons (again, the noise affects only the soliton’s heart). In the preliminary numerical search, we did not find any trace of the crossed case (solitons with the noise affecting only their environment). Then an analytical estimate (applying the usual techniques of the kink-antikink case, namely. using the solubility condition) allowed us to find the parameter values for which the crossed case is possible. Of course, we checked this result through numerical simulation. Clearly, the aforementioned techniques were devised for front solutions of a system located near the Maxwell point, not for pinning fronts. The need to perform this estimate was prompted by the results obtained when looking for the conjectured pinning fronts. We found that the pinning mechanism can be easily enabled by the noise when the system is located close enough to the instability region. Under general conditions, it suffices to start from a hyperbolic-tangent-like profile to activate a pinning front. However, very close to the Maxwell point, pinning fronts become stable only if starting from a Pomeau-type profile. Under the same conditions as above, only propagating fronts emerge when starting from a hyperbolic-tangent-like profile. The latter means that both types of fronts are possible solutions (in fact, numerical calculations showed the coexistence of both solutions). Moreover, when the system is located away from the instability region, we found situations where the solitons become stable even though the pinning fronts cannot be activated. Briefly, we found three different situations as a possible support of the dynamics that enables soliton stabilization: (a) pinning fronts only, (b) pinning fronts coexisting with propagating ones, and (c) propagating fronts only. Cases (a) and (b) are to be expected, but not (c). This is what prompted us to perform the analytical estimate, whereby we found an equation that governs the soliton width dynamics and shows the cooperation of noise and ALIs in stabilizing the soliton. This equation also predicts different situations, whose existence we subsequently verified through numerical simulations. Concerning the dependence of the soliton behavior on the noise intensity, through numerous simulations we observed

that its effect is twofold. On the one hand, a stronger noise promotes multiple spontaneous solitons (their number increases with the noise intensity). On the other hand, it drives the solitons’ stochastic motion. On a shorter time scale, the amplitude of such motion increases monotonically with the noise intensity. On a longer time scale, it initially increases monotonically, but when the noise intensity is high enough, the interactions between neighboring solitons tends to confine them (their mobility is reduced). We also observed that if the noise intensity exceeds a certain value, the solitons are destabilized. As established earlier, in this case the HS is overly pushed by the noise until it disappears. In short, given a bistable system with front solutions and ALIs, it is possible to induce the formation of stable solitons by introducing a suitable multiplicative noise. These solitons acquire a stochastic motion that grows with the noise intensity, although it is in part limited by the interaction between solitons when they are close together. We emphasize that the solitons only exist within a (wide enough) window of noise intensity values. By increasing the noise intensity above the stability threshold, first only the seeded soliton is stabilized, but as the noise intensity is increased further, a growing number of spontaneous solitons are stabilized. Then, past the upper intensity threshold, all solitons are destabilized. Finally, we observed that the solitons are constructed by ALIs but stabilized by the cooperation of both ALIs and the driving noise. This is independent of the nature of the interacting agents, as long as their interaction is short range. Therefore, we believe our idea to be applicable to other systems with ALIs, be it within the realm of physics or not. In order to considerer agents of other nature, we only require Q(φ) to express a bistable system with front solutions. In future work, we plan to support the generality of our results by applying this approach to systems of different nature.

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ACKNOWLEDGMENTS

Financial support from CONICET and UNMdP of Argentina is acknowledged.

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