Dissipative Soliton Excitability Induced by Spatial Inhomogeneities and Drift

Abstract

We show that excitability is generic in systems displaying dissipative solitons when spatial inhomogeneities and drift are present. Thus, dissipative solitons in systems which do not have oscillatory states, such as the prototypical Swift-Hohenberg equation, display oscillations and Type I and II excitability when adding inhomogeneities and drift to the system. This rich dynamical behavior arises from the interplay between the pinning to the inhomogeneity and the pulling of the drift. The scenario presented here provides a general theoretical understanding of oscillatory regimes of dissipative solitons reported in semiconductor microresonators. Our results open also the possibility to observe this phenomenon in a wide variety of physical systems. [6,7], or excitability [8]. DS emerge from a balance between nonlinearity and spatial coupling, and driving and dissipation. They are unique once the system parameters are fixed, and they are different from the well known conservative solitons that appear as oneparameter families.In optical cavities, DS (also known as cavity solitons) have been proposed as bits for all-optical memories [9-13], due to their spatial localization and bistable coexistence with the fundamental solution. In Ref. [8] it was reported that DS may exhibit excitable behavior. A system is said to be excitable if perturbations below a certain threshold decay exponentially, while perturbations above this threshold induce a large response before going back to the resting state. Excitability is found for parameters close to those where a limit cycle disappears [14]. Excitability mediated by DS is different from the well known dynamics of excitable media, whose behavior stems from the (local) excitability present in the system without spatial degrees of freedom. Excitability of DS is an emergent behavior, arising through the spatial interaction and not present locally. Moreover, the interaction between different excitable DS can be used to build alloptical logical gates [15].In real systems, however, typically solitons are static, so oscillatory or excitable DS are far from being generic. In this work we present a mechanism that generically induces dynamical regimes, such as oscillations and excitable behavior, in which the structure of the DS is preserved. The mechanism relies on the interplay between spatial inhomogeneities and drift, and therefore can be implemented under very general conditions. Inhomogeneities, or defects, are unavoidable in any experimental setup, and drift is also often present in many optical, fluid and chemical systems, due to misalignments of the mirrors [16,17], nonlinear crystal birefringence [18], or parameter gradients [19], in the first case, and due to the flow of a fluid in the others [20,21]. Roughly speaking, the presence of inhomogeneities and drift introduce two competing effects. On the one hand, an inhomogeneity pins a DS at a fixed position and, on the other, the drift tries to pull it out. If the drift overcomes the pinning, DS solitons are released from the in...

Dark solitons in the Lugiato-Lefever equation with normal dispersion

“…Close to BC 2 (respectively, BC 1 ) the system can exhibit behavior reminiscent of excitability [32]. Here the stable manifold of the saddle soliton S acts as a separatrix or threshold in the sense that perturbations of A t 0 across that threshold do not relax immediately to A We choose a value of ρ close to BC 2 , namely ρ = 2.7235, and modify the parameter ρ for a brief instant using a Gaussian profile of width σ and height h using the instantaneous transformation ρ → ρ+h(t) exp[−(x−L/2) 2 /σ 2 ], where ρ = 2.7235 and σ = 0.781250 with h(t) = −2.55 for 10 ≤ t ≤ 15 and h = 0 elsewhere [34]. As shown in Fig.…”

Spatial localization in heterogeneous systems

“…We expect that the results reported here will be of interest to experimentalists working with localized structures, as well as to theorists interested in identifying generic processes resulting from forced symmetry breaking. Problems of this type have numerous applications, from fluid mechanics [38] and optics [39,40], to reaction-diffusion models [41,42] and recent work on models of desertification [43,44]. In addition, localized bump forcing may serve as a model of a persistent perturbation, such as may be applied by a focused optical probe in an optics experiment, in contrast to instantaneous perturbations applied by turning the probe rapidly on and off.…”

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

“…The inhomogeneity alters the stationary solutions of the system [41]. In particular, the homogeneous solution of Eq.…”

“…However, for a more realistic description of any experimental setup, it is often necessary to take into account spatial inhomogeneities that break the translational symmetry of the system and thus change the dynamics induced by delay. Recently, the competition between a drifting localized structure and spatial inhomogeneities has been studied experimentally in [40] and theoretically in a Swift-Hohenberg model [41,42], although in the latter case the drift of the localized structure has been introduced by simply adding an advection term to the Swift-Hohenberg equation.…”