Breathing dissipative solitons in three-component reaction-diffusion system

Origin of jumping oscillons in an excitable reaction-diffusion system

Oscillons, i.e., immobile spatially localized but temporally oscillating structures, are the subject of intense study since their discovery in Faraday wave experiments. However, oscillons can also disappear and reappear at a shifted spatial location, becoming jumping oscillons (JOs). We explain here the origin of this behavior in a three-variable reaction-diffusion system via numerical continuation and bifurcation theory, and show that JOs are created via a modulational instability of excitable traveling pulses (TPs). We also reveal the presence of bound states of JOs and TPs and patches of such states (including jumping periodic patterns) and determine their stability. This rich multiplicity of spatiotemporal states lends itself to information and storage handling.

Breatherlike solitons extracted from the Peregrine rogue wave

Based on the Peregrine solution (PS) of the nonlinear Schrödinger (NLS) equation, the evolution of rational fraction pulses surrounded by zero background is investigated. These pulses display the behavior of a breatherlike solitons. We study the generation and evolution of such solitons extracted, by means of the spectral-filtering method, from the PS in the model of the optical fiber with realistic values of coefficients accounting for the anomalous dispersion, Kerr nonlinearity, and higher-order effects. The results demonstrate that the breathing solitons stably propagate in the fibers. Their robustness against small random perturbations applied to the initial background is demonstrated too.

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system

The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.

Periodic and chaotic solitons in a semiconductor laser with saturable absorber

In a semiconductor laser with saturable absorber, solitons may spontaneously drift and/or oscillate. We study three different regimes characterized by strong intensity oscillations, both periodic and chaotic. We show that (i) soliton dynamics may be similar to that of passively Q-switched lasers, (ii) solitons may drift and oscillate simultaneously, and (iii) chaotic solitons may coexist with stationary ones and with the laser off solution.

Fundamental- and first-order localized states in a cubic-quintic reaction-diffusion system

This article analyzes the properties of rotationally symmetric self-localized solutions with different radial quantum numbers in the simplest one- and two-component reaction-diffusion systems. The consideration is made in one and two dimensions with the focus on the fundamental and first higher-order solutions showing zero and one intersections of the radial profile with zero. It is demonstrated that the solution with one intersection does not exist for the case of the quadratic-cubic nonlinearity, while the cubic-quintic extension of the models does allow existence. I show additionally that the cubic-quintic reaction diffusion system supports the existence and stability of the states with zero quantum numbers, as well as their antistates, state-antistate pairs, and clusters, which can be interpreted as the states with nonzero azimuthal quantum number.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Long-range interaction and synchronization of oscillating dissipative solitons

We study the interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which lead to the formation of bound states due to harmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

Self-organized pacemakers and bistability of pulses in an excitable medium

Pattern formation in an excitable medium described by a three-component reaction-diffusion system is investigated. Our focus is on stable self-organized pacemakers which give rise to spatially extended target patterns. Bistability of pulse solutions in the excitable regime is also reported, and interactions of the different pulses with each other and the pacemaker are studied. Self-organized pacemakers are created by a suitable perturbation from the steady state or through interaction of pulses. Bound states of one-dimensional pacemakers and phase flips are also observed.

Synergistic effect of two inhibitors on one activator in a reaction-diffusion system

We investigate a three-component reaction-diffusion system that describes the interaction of one activator and two inhibitors where one inhibitor acts as a traveling pulse generator of the activator and the other acts as a lateral inhibition localizer. It is numerically shown that the synergistic effect of these two inhibitors on one activator induces several spatiotemporal patterns such as destabilization and nonannihilation of traveling pulses and the occurrence and splitting of traveling spots. By using singular perturbation procedures, the stability of radially symmetric equilibrium solutions is discussed. Furthermore, we discuss how such dynamics are caused under the synergistic effect of two inhibitors.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.