Delay-Induced Depinning of Localized Structures in a spatially inhomogeneous Swift-Hohenberg Model

TL;DR

This paper investigates how inhomogeneous spatial pumping and time-delayed feedback influence the behavior of localized structures in a Swift-Hohenberg model, revealing mechanisms for pinning, oscillations, and depinning in optical pattern formation.

Contribution

It introduces a novel analysis of delay-induced depinning of localized structures under inhomogeneous pumping, combining stability analysis, numerical continuation, and analytical approaches.

Findings

Localized structures can be pinned by inhomogeneity, preventing delay-induced drift.

Identification of instability regimes including spiral formation and oscillations.

Transition from oscillating to depinning states analyzed via numerical continuation.

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 time-delayed 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.