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

Abstract

We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial gaussian pumping beam, and subjected to timedelayed feedback. The gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assumming a continous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use two approaches based on either an order parameter equation, describing the dynamics of the localized structure in the vicinity of the Hopf bifurcation or an overdamped dynamics of a particle in a potential well generated by the inhomogeneity. In the later approach, the time-delayed feedback acts as a driving force.

Bifurcation structure of cavity soliton dynamics in a vertical-cavity surface-emitting laser with a saturable absorber and time-delayed feedback

“…However, the underlying algorithms' execution times scale badly with the system dimension. In particular, calculations for a single equation in space have shown an effective limit to spatial resolution of 64 mesh points on contemporary desktop hardware [41]. For the 4d system at interest we estimate a limit of only 16 meshpoints in space which is hardly sufficient.…”

“…For a saturable absorber VCSEL a period-doubling route to temporal chaos of a single CS has been theoretically predicted for certain feedback parameters [40]. More recently, it has been shown that delayed feedback can induce pinning and depinning of cavity solitons when the resonator is illuminated by an inhomogeneous spatial gaussian pumping beam [41]. It has to be noted that time-delayed feedback in spatially extended complex systems has a broader relevance than just laser physics and nonlinear optics.…”

Stationary localized structures and the effect of the delayed feedback in the Brusselator model

“…In reaction with diffusion and with global delayed feedback, a photoemission electron microscope was used to continuously image lateral distributions of adsorbed species, and feedback is introduced by making the instantaneous dosing rate of the catalytic carbon monoxide oxidation dependent on real-time properties of the imaged concentration patterns [56]. The effect of spatial inhomogeneities for the delayed Swift-Hohenberg equation has been investigated both analytically and numerically in [57] and a transition from oscillating to depinning solutions has been characterized. In this case, the time-delayed feedback acts as a driving force [57].…”

“…The effect of spatial inhomogeneities for the delayed Swift-Hohenberg equation has been investigated both analytically and numerically in [57] and a transition from oscillating to depinning solutions has been characterized. In this case, the time-delayed feedback acts as a driving force [57]. The effect of noise responsible for the formation of dissipative structures in the delayed Swift-Hohenberg equation has been investigated recently in depth by Kuske et al [58].…”

Stabilization of localized structures by inhomogeneous injection in Kerr resonators

“…The consideration of small inhomogeneities seems inevitable, because it is difficult to prevent them in any real experimental setup. However, even small inhomogeneities can have drastic effects on the dynamical properties of a system under consideration because they break continuous symmetries of the system [27]. It is therefore necessary to include these symmetry breaking effects in a theoretical description.…”