Stabilization of dark solitons in the cubic ginzburg-landau equation

Nozaki-Bekki solitons in semiconductor lasers

Optical frequency-comb sources, which emit perfectly periodic and coherent waveforms of light1, have recently rapidly progressed towards chip-scale integrated solutions. Among them, two classes are particularly significant-semiconductor Fabry-Perót lasers2-6 and passive ring Kerr microresonators7-9. Here we merge the two technologies in a ring semiconductor laser10,11 and demonstrate a paradigm for the formation of free-running solitons, called Nozaki-Bekki solitons. These dissipative waveforms emerge in a family of travelling localized dark pulses, known within the complex Ginzburg-Landau equation12-14. We show that Nozaki-Bekki solitons are structurally stable in a ring laser and form spontaneously with tuning of the laser bias, eliminating the need for an external optical pump. By combining conclusive experimental findings and a complementary elaborate theoretical model, we reveal the salient characteristics of these solitons and provide guidelines for their generation. Beyond the fundamental soliton circulating inside the ring laser, we demonstrate multisoliton states as well, verifying their localized nature and offering an insight into formation of soliton crystals15. Our results consolidate a monolithic electrically driven platform for direct soliton generation and open the door for a research field at the junction of laser multimode dynamics and Kerr parametric processes.

Fermi-Pasta-Ulam-Tsingou recurrence of periodic anomalous waves in the complex Ginzburg-Landau and in the Lugiato-Lefever equations

The complex Ginzburg-Landau (CGL) equation, an envelope model relevant in the description of several natural phenomena like binary-fluid convection and second-order phase transitions, and the Lugiato-Lefever (LL) equation, describing the dynamics of optical fields in pumped lossy cavities, can be viewed as nonintegrable generalizations of the nonlinear Schrödinger (NLS) equation, including diffusion, linear and nonlinear loss or gain terms, and external forcing. In this paper we treat the nonintegrable terms of both equations as small perturbations of the integrable focusing NLS equation, and we study the Cauchy problem of the CGL and LL equations corresponding to periodic initial perturbations of the unstable NLS background solution, in the simplest case of a single unstable mode. Using the approach developed in a recent paper by the authors with P. G. Grinevich [Phys. Rev. E 101, 032204 (2020)10.1103/PhysRevE.101.032204], based on the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic models describing quantitatively how the solution evolves, after a suitable transient, into a Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of anomalous waves (AWs) described by slowly varying lower dimensional patterns (attractors) in the (x,t) plane, characterized by Δx=L/2 or Δx=0 in the case in which loss or gain, respectively, effects prevail, where Δx is the x-shift of the position of the AW during the recurrence and L is the period. We also obtain, in the CGL case, the analytic condition for which loss and gain exactly balance, stabilizing the ideal FPUT recurrence of periodic NLS AWs; such a stabilization is not possible in the LL case due to the external forcing. These processes are described, to leading order, in terms of elementary functions of the initial data in the CGL case, and in terms of elementary and special functions of the initial data in the LL case.

On the influence of additive and multiplicative noise on holes in dissipative systems

We investigate the influence of noise on deterministically stable holes in the cubic-quintic complex Ginzburg-Landau equation. Inspired by experimental possibilities, we specifically study two types of noise: additive noise delta-correlated in space and spatially homogeneous multiplicative noise on the formation of π-holes and 2π-holes. Our results include the following main features. For large enough additive noise, we always find a transition to the noisy version of the spatially homogeneous finite amplitude solution, while for sufficiently large multiplicative noise, a collapse occurs to the zero amplitude solution. The latter type of behavior, while unexpected deterministically, can be traced back to a characteristic feature of multiplicative noise; the zero solution acts as the analogue of an absorbing boundary: once trapped at zero, the system cannot escape. For 2π-holes, which exist deterministically over a fairly small range of values of subcriticality, one can induce a transition to a π-hole (for additive noise) or to a noise-sustained pulse (for multiplicative noise). This observation opens the possibility of noise-induced switching back and forth from and to 2π-holes.

Stabilization of a scroll ring by a cylindrical Neumann boundary

We study the interaction of phase singularities with homogeneous Neumann boundaries in one, two, and three spatial dimensions for the complex Ginzburg-Landau equation. The existence of a boundary-induced drift attractor, well known for spiral waves in two spatial dimensions, is demonstrated for scroll waves in three spatial dimensions. We find that a cylindrical Neumann boundary can lock a scroll ring, thus preventing the collapse of its closed filament.

Dark solitons in the presence of higher-order effects

Dark soliton propagation is studied in the presence of higher-order effects, including third-order dispersion, self-steepening, linear/nonlinear gain/loss, and Raman scattering. It is found that for certain values of the parameters a stable evolution can exist for both the soliton and the relative continuous-wave background. Using a newly developed perturbation theory we show that the perturbing effects give rise to a shelf that accompanies the soliton in its propagation. Although, the stable solitons are not affected by the shelf it remains an integral part of the dynamics otherwise not considered so far in studies of higher-order nonlinear Schrödinger models.

Dissipative dark soliton in a complex plasma

The observation of a dark soliton in a three-dimensional complex plasma containing monodisperse microparticles is presented. We perform our experiments using neon gas in the bulk plasma of an rf discharge. A gas temperature gradient of 500K/m is applied to balance gravity and to levitate the particles in the bulk plasma. The wave is excited by a short voltage pulse on the electrodes of the radio frequency discharge chamber. It is found that the wave propagates with constant speed. The propagation time of the dark soliton is approximately 20 times longer than the damping time.

Solitary pulses in linearly coupled Ginzburg-Landau equations

This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau (CGL) equations. In the simplest case, the system features linear gain, cubic nonlinearity (possibly combined with cubic loss), and group-velocity dispersion (GVD) in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive (lossy) one. The model gives rise to exact analytical solutions for stationary solitary pulses (SPs). The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons (holes). In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic (second-harmonic-generating) nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers.