We introduce a model which integrates the complex Ginzburg-Landau (CGL) equation in two dimensions (2D) with the linear-cubic-quintic combination of loss and gain terms, self-defocusing nonlinearity, and a periodic potential. In this system, stable 2D \textit{dissipative gap solitons} (DGSs) are constructed, both fundamental and vortical ones. The soliton families belong to the first finite bandgap of the system's linear spectrum. The solutions are obtained in a numerical form and also by means of an analytical approximation, which combines the variational description of the shape of the fundamental and vortical solitons and the balance equation for their total power. The analytical results agree with numerical findings. The model may be implemented as a laser medium in a bulk self-defocusing optical waveguide equipped with a transverse 2D grating, the predicted DGSs representing spatial solitons in this setting.
In this paper we study the behavior of dissipative solitons in systems with high order nonlinear dissipation and show how they cannot survive under the effect of trapping potentials both of rigid wall type or asymptotically increasing ones. This provides an striking example of a soliton which cannot be trapped and only survives to the action of a weak potential.
We show dissipative spatial solitons in nonlinear optical micro-resonators in which the refractive index is laterally modulated. In addition to "normal" and "staggered" dissipative solitons, similar to those in spatially modulated conservative systems, a narrow "mid-band" soliton is shown, having no counterparts in conservative systems.
Dissipative solitons are non-dispersive localized waves that exhibit intriguing dynamics. In optical systems, the discovery of new soliton dynamics and how to accurately manipulate dissipative solitons are still largely unexplored and challenging in the ultrafast fiber laser community. Herein, we unveil a new type of dissipative soliton behavior in a net-normal-dispersion bidirectional ultrafast fiber laser. That is, the bidirectional dissipative solitons will always reveal similar spectral and temporal characteristics through common gain and loss modulation, even if the transient instability …
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…
We present the results of asymptotic and numerical analysis of dissipative Kerr solitons in whispering gallery mode microresonators influenced by higher order dispersive terms leading to the appearance of a dispersive wave (Cherenkov radiation). Combining direct perturbation method with the method of moments we find expressions for the frequency, strength, spectral width of the dispersive wave and soliton velocity. Mutual influence of the soliton and dispersive wave was studied. The formation of the dispersive wave leads to a shift of the soliton spectrum maximum from the pump frequency (spect…
We consider two-component nonlinear dissipative spatially extended systems of reaction-cross-diffusion type. Previously, such systems were shown to support "quasi-soliton" pulses, which have fixed stable structure but can reflect from boundaries and penetrate each other. Presently we demonstrate a different type of quasi-solitons, with a phenomenology resembling that of the envelope solitons in Nonlinear Schroedinger equation: spatiotemporal oscillations with a smooth envelope, with the velocity of the oscillations different from the velocity of the envelope.
The effect of an externally applied force upon dynamics of dissipative solitons is analyzed in the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation supplemented by a linear potential term. The potential accounts for the external force manipulations and consists of three symmetrically arranged potential wells whose depth is considered to be variable along the longitudinal coordinate. It is found out that under an influence of such potential a transition between different soliton waveforms coexisted under the same physical conditions can be achieved. A low-dimensio…
We study the dynamics of localized pulses in the complex cubic-quintic Ginzburg-Landau (GL) equation with strong nonlinearity management. The generalized complex GL equation, averaged over rapid modulations of the nonlinearity, is derived. We employ the variational approach to this averaged equation to derive a dynamical system for the dissipative soliton parameters. By using numerical simulations of this dynamical system and the full GP equation, we show that nonlinearity-managed dissipative solitons exist in the model. The parameter regions for stabilization of exploding soliton are obtained…
We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a `simple' and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions it becomes characterized by normal diffusion.
We analyze the dynamics of dissipative solitons in silicon on insulator waveguides embedded in a gain medium. The optical propagation is modeled through a cubic Ginzburg-Landau equation for the field envelope coupled with an ordinary differential equation accounting for the generation of free carriers owing to two-photon absorption. Our numerical simulations clearly indicate that dissipative solitons accelerate due to the carrier-induced index change and experience a considerable blue-shift, which is mainly hampered by the gain dispersion of the active material. Numerical results are fully exp…
Dissipative Kerr solitons arising from parametric gain in ring microresonators are usually described within a classical mean-field framework. Here, we develop a quantum-mechanical model of dissipative Kerr solitons in terms of the truncated Wigner method, which accounts for quantum effects to lowest order. We show that the soliton experiences a finite lifetime due to quantum fluctuations originating from losses. Reading the results in terms of the theory of open quantum systems, allows to estimate the Liouvillian spectrum of the system. It is characterized by a set of eigenvalues with finite i…
We study the influence of higher-order effects such as third order dispersion (TOD), fourth order dispersion (FOD), quintic nonlinearity (QN), self steepening (SS) and second order nonlinear dispersion (SOND) on the dynamics of dissipative soliton (DS) in metamaterials. Considering each higher-order effect as a perturbation to the system and following Lagrangian variational method, we demonstrate stable dynamics of DS as a result of the interplay between different higher-order effects. We also perform numerical analysis to confirm the analytical results.
Numerical simulations of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal five entirely novel classes of pulse or solitary waves solutions, viz. pulsating, creeping, snaking, erupting, and chaotical solitons. Here, we develop a theoretical framework for analyzing the full spatio-temporal structure of one class of dissipative solution (snaking soliton) of the CCQGLE using the variational approximation technique and the dynamical systems theory. The qualitative…
We report a new effect of a cascade replication of dissipative solitons from a single one. It is discussed in the framework of a common model based on the one-dimensional cubic-quintic complex Ginzburg-Landau equation in which an additional linear term is introduced to account the perturbation from a particular potential of externally applied force. The effect manifestation is demonstrated on the light beams propagating through a planar waveguide. The waveguide consists of a nonlinear layer able to guide dissipative solitons, and a magneto-optic substrate. In the waveguide an externally applie…
New resonant emission of dispersive waves by oscillating solitary structures is considered analytically and numerically. The resonance condition for the radiation is derived and it is demonstrated that the predicted resonances match the spectral lines observed in numerical simulations perfectly. The complex recoil of the radiation on the soliton dynamics is discussed.
Nonlinear properties of a multi-layer stack of graphene sheets are studied. It is predicted that such a structure may support dissipative plasmon-solitons generated and supported by an external laser radiation. Novel nonlinear equations describing spatial dynamics of the nonlinear plasmons driven by a plane wave in the Otto configuration are derived and the existence of single and multi-hump dissipative solitons in the graphene structure is predicted.
We show that the balance between localized gain and nonlinear cubic dissipation in the twodimensional nonlinear Schrodinger equation allows for existence of stable two-dimensional localized modes which we identify as solitons. Such modes exist only when the gain is strong enough and the energy flow exceeds certain threshold value. The observed solitons neither undergo diffractive spreading nor collapse. Above the critical value of the gain the symmetry breaking occurs and asymmetric dissipative solitons emerge.
Approximate analytical chirped solitary pulse (chirped dissipative soliton) solutions of the one-dimensional complex cubic-quintic nonlinear Ginzburg-Landau equation are obtained. These solutions are stable and highly-accurate under condition of domination of a normal dispersion over a spectral dissipation. The parametric space of the solitons is three-dimensional, that makes theirs to be easily traceable within a whole range of the equation parameters. Scaling properties of the chirped dissipative solitons are highly interesting for applications in the field of high-energy ultrafast laser phy…
We report dissipative surface solitons forming at the interface between a semi-infinite lattice and a homogeneous Kerr medium. The solitons exist due to balance between amplification in the near-surface lattice channel and two-photon absorption. The stable dissipative surface solitons exist in both focusing and defocusing media, when propagation constants of corresponding states fall into a total semi-infinite and or into one of total finite gaps of the spectrum (i.e. in a domain where propagation of linear waves is inhibited for the both media). In a general situation, the surface solitons fo…
We analyze the stability and dynamics of dissipative Kerr solitons in the presence of a parabolic potential. This potential stabilizes oscillatory and chaotic regimes, favoring the generation of static DKSs. Furthermore, the potential induces the emergence of new dissipative structures, such as asymmetric breathers and chimera-like states. Based on a mode decomposition of these states, we unveil the underlying modal interactions.
The driven-dissipative photonic dimer comprised of two evanescently coupled high-Q microresonators is a fundamental element of multimode soliton lattices. It has demonstrated a variety of emergent nonlinear phenomena including supermode soliton generation, symmetry breaking, and soliton hopping. In this article, we present another aspect of dissipative soliton generation in coupled resonators, revealing the advantages of this system over conventional single resonator platforms. Namely, we show that the accessibility of solitons drastically varies for symmetric and antisymmetric supermode famil…
In this work, we introduce a method for stabilizing spatiotemporal solitons. These solitons correspond to light bullets in multimode optical fiber lasers, energy-scalable waveguide oscillators and amplifiers, localized coherent patterns in Bose-Einstein condensates, etc. We show that a three-dimensional confinement potential, formed by a spatial transverse (radial) parabolic graded refractive index and dissipation profile, combined with quadratic temporal phase modulation, may permit the generation of stable spatiotemporal dissipative solitons. This corresponds to combining phase mode-locking …
The paper is devoted to the dynamics of dissipative gap solitons in the periodically corrugated optical waveguides whose spectrum of linear excitations contains a mode that can be referred to as a quasi-Bound State in the Continuum. These systems can support a large variety of stable bright and dark dissipative solitons that can interact with each other and with the inhomogeneities of the pump. One of the focus points of this work is the influence of slow variations of the pump on the behavior of the solitons. It is shown that for the fixed sets of parameters the effect of pump inhomogeneities…
We propose two models for the creation of stable dissipative solitons in optical media with the $χ^{(2)}$ (quadratic) nonlinearity. To compensate spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized "hot spot", carrying the linear gain, is added, acting either on the FF component, or on the SH one. In both systems, we use numerical methods to find families of dissipative $χ^{(2)}$ solitons pinned to the "hot spot". The shape of the existence and stability domains may be rather complex. An existence boundary for th…
We introduce a mechanism of stable spatiotemporal soliton formation in a multimode fiber laser. This is based on spatially graded dissipation, leading to distributed Kerr-lens mode-locking. Our analysis involves solutions of a generalized dissipative Gross-Pitaevskii equation. This equation has a broad range of applications in nonlinear physics, including nonlinear optics, spatiotemporal patterns formation, plasma dynamics, and Bose-Einstein condensates. We demonstrate that careful control of dissipative and non-dissipative physical mechanisms results in the self-emergence of stable (2+1)-dime…
We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The passed wave either retains the form of a soliton (though of different parameters) or becomes a bi-soliton. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapour saturation/condensation area. Some rough e…
We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with finite dissipation. The modelling included the case of a finite dissipative layer similar to a wave passing through the air-glass-air as well as a wave passing from a non-dissipative layer into a dissipative one (similar to the passage of light from air to water). The dissipation predictably results in reducing the soliton amplitude/velocity, but some new effects occur in the case of finite barrier on the soliton path: after the wave leaves the dissipative barrier it retains a soliton f…
The paper addresses the bistability caused by spontaneous symmetry breaking bifurcation in a one-dimensional periodically corrugated nonlinear waveguide pumped by coherent light at normal incidence. The formation and the stability of the switching waves connecting the states of different symmetries are studied numerically. It is shown that the switching waves can form stable resting and moving bound states (dissipative solitons). The protocols of the creation of discussed nonlinear localized waves are suggested and verified by numerical simulations.
A system of first-order differential-difference equations with time lag describes the formation of density waves, called as quasi-solitons for dissipative systems in this paper. For co-moving density waves, the system reduces to some exactly solvable lattice models. We construct a shock-wave solution as well as one-quasi-soliton solution, and argue that there are pseudo-conserved quantities which characterize the formation of the co-moving waves. The simplest non-trivial one is given to discuss the presence of a cascade phenomena in relaxation process toward the pattern formation.
We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite dissipative layer as well as a wave passing from a dissipative layer into a non-dissipative one and vice versa. New effects are presented in the case of numerically finite barrier on the soliton path: first, if the form of dissipation distribution has a form of a frozen soliton, the wave that leaves the dissipative barrier becomes a bi-soliton and a reflection wave arises as a comparatively small and quasi-harmonic oscillation. Se…
We study how nonlinear delayed-feedback in the Ikeda model can induce solitary impulses, i.e. dissipative solitons. The states are clearly identified in a virtual space-time representation of the equations with delay, and we find that conditions for their appearance is bistability of a nonlinear function and negative character of the delayed feedback. Both dark and bright solitons are identified in numerical simulations and physical electronic experiment showing an excellent qualitative correspondence and proving thereby the robustness of the phenomenon. Along with single spiking solitons, a v…
We consider settings providing the existence of stable two-dimensional (2D) dissipative solitons with zero and nonzero vorticity in optical media with the quadratic ($χ^{(2)}$) nonlinearity. To compensate the spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized amplifying region ("hot spot",HS), carrying the linear gain, is included, acting onto either the FF component or SH one. In both cases, the Gaussian radial gain profile supports stable fundamental dissipative solitons pinned to the HS. The structure of exis…
We report the observation of different localized structures coexisting for the same parameter values in an extended system. The experimental findings are carried out in a nonlinear optical interferometer, and are fully confirmed by numerical simulations. The existence of each kind of localized structure is put in relation to a corresponding delocalized pattern observed. Quantitative evaluation of the range of pump parameter allowing bistability between localized structures is given. The dependence of the phenomena on the other relevant parameters is discussed.
An unstable particle in quantum mechanics can be stabilized by frequent measurements, known as the quantum Zeno effect. A soliton with dissipation behaves like an unstable particle. Similar to the quantum Zeno effect, here we show that the soliton can be stabilized by modulating periodically dispersion, nonlinearity, or the external harmonic potential available in BEC. This can be obtained by analyzing a Painleve integrability condition, which results from the rigorous Painleve analysis of the generalized nonautonomous nonlinear Schrodinger equation. The result has a profound implication to th…
We introduce a model for spatiotemporal modelocking in multimode fiber lasers, which is based on the (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation (cGLE) with conservative and dissipative nonlinearities and a 2-dimensional transverse trapping potential. Systematic numerical analysis reveals a variety of stable nonlinear modes, including stable fundamental solitons and breathers, as well as solitary vortices with winding number $n=1$, while vortices with $n=2$ are unstable, splitting into persistently rotating bound states of two unitary vortices. A characteristic feature of …
We study nonlinear modes in one-dimensional arrays of doped graphene nanodisks with Kerr-type nonlinear response in the presence of an external electric field. We present the theoretical model describing the evolution of the disks' polarizations, taking into account intrinsic graphene losses and dipole-dipole coupling between the graphene nanodisks. We reveal that this nonlinear system can support discrete dissipative scalar solitons of both longitudinal and transverse polarizations, as well as vector solitons composed of two mutually coupled polarization components. We demonstrate the formati…
We consider an optical resonator containing a photonic crystal fiber and driven coherently by an injected beam. This device is described by a generalized Lugiato-Lefever equation with fourth order dispersion. We use an asymptotic approach to derive interaction equations governing the slow time evolution of the coordinates of two interacting dissipative solitons. We show that Cherenkov radiation induced by positive fourth-order dispersion leads to a strong increase of the interaction force between the solitons. As a consequence, large number of equidistant soliton bound states in the phase spac…
We investigate the existence and stability of dissipative soliton solution in a system described by complex Ginzburg-Landau (CGL) equation with asymmetric complex potential, which is obtained from original parity reflection - time reversal ($\mathcal{PT}$) symmetric Rosen-Morse potential. In this study, stability of solution is examined by numerical analysis to show that solitons are stable for some parameter ranges for both self-focusing and self-defocusing nonlinear modes. Dynamical properties such as evolution and transverse energy flow for both modes are also analyzed. Obtained results are…
Optical frequency combs in microresonators (microcombs) have a wide range of applications in science and technology, due to its compact size and access to considerably larger comb spacing. Despite recent successes, the problems of self-starting, high mode efficiency as well as high output power have not been fully addressed for conventional soliton microcombs. Recent demonstration of laser cavity soliton microcombs by nesting a microresonator into a fiber cavity, shows great potential to solve the problems. Here we comprehensively study the dissipative soliton generation and interaction dynami…
We show the existence of stable two- and three-dimensional vortex solitons carrying multiple, spatially separated, single-charge topological dislocations nested around a vortex-ring core. Such new nonlinear states are supported by elliptical gain landscapes in focusing nonlinear media with two-photon absorption. The separation between the phase dislocations is dictated mostly by the geometry of gain landscape and it only slightly changes upon variation of the gain or absorption strength.
Clouds of cold neutral atoms driven by a coherent light beam in a ring cavity exhibit self-structured states transversely with respect to the beam axis due to optomechanical forces and the back action of the atomic structures on the beam. Below the instability threshold for extended hexagonal structures, localized soliton-like excitations can be stable. These constitute peaks or holes of atom density, depending on the linear susceptibility of the cloud. Complex rotating and spiralling motion of coupled atom-light solitons, and hence atomic transport, can be achieved via phase gradients in the …
We demonstrate that necklace-shaped arrays of localized spatial beams can merge into stable fundamental or vortex solitons in a generic model of laser cavities, based on the two-dimensional complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The outcome of the fusion is controlled by the number of beads in the initial necklace, 2N, and its topological charge, M. We predict and confirm by systematic simulations that the vorticity of the emerging soliton is the absolute value of difference of N and M. Threshold characteristics of the fusion are found and explained too. If the i…
A plethora of applications have recently motivated extensive efforts on the generation of low noise Kerr solitons and coherent frequency combs in various platforms ranging from fiber to whispering gallery and integrated microscale resonators. However, the Kerr (cubic) nonlinearity is inherently weak, and in contrast, strong quadratic nonlinearity in optical resonators is expected to provide an alternative means for soliton formation with promising potential. Here, we demonstrate the formation of a dissipative quadratic soliton via non-stationary optical parametric amplification in the presence…
We propose a system of coupled microring resonators for the generation frequency combs and dissipative Kerr solitons in silicon at telecommunication frequencies. By taking advantage of structural slow-light, the effective non-linearity of the material is enhanced, thus relaxing the requirement of ultra-high quality factors that currently poses a major obstacle to the realization of silicon comb devices. We demonstrate a variety of frequency comb solutions characterized by threshold power in the 10-milliwatt range and a small footprint of $0.1$ mm$^2$, and study their robustness to structural d…
We examine spatial localisation and dynamical stability of Bose-Einstein condensates of exciton-polaritons in microcavities under the condition of off-resonant spatially inhomogeneous optical pumping both with and without a harmonic trapping potential. We employ the open-dissipative Gross-Pitaevskii model for describing an incoherently pumped polariton condensate coupled to an exciton reservoir, and reveal that spatial localisation of the steady-state condensate occurs due to the effective self-trapping created by the polariton flows supported by the spatially inhomogeneous pump, regardless of…
We report the existence of vectorial dark dissipative solitons in optical cavities subject to a coherently injected beam. We assume that the resonator is operating in a normal dispersion regime far from any modulational instability. We show that the vectorial front locking mechanism allows for the stabilisation of dark dissipative structures. These structures differ by their temporal duration and their state of polarization. We characterize them by constructing their heteroclinic snaking bifurcation diagram showing evidence of multistability within a finite range of the control parameter.
We consider the model of fiber-laser cavities near the zero-dispersion point, based on the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity, including the third-order dispersion (TOD) term. It is well known that this model supports stable dissipative solitons. We demonstrate that the same model gives rise to several families of robust bound states of the solitons, which exists only in the presence of the TOD. There are both stationary and dynamical bound states, with oscillating separation between the bound solitons. Stationary states are multistable, corresponding to diffe…
We study Kerr frequency combs and dissipative Kerr solitons in silicon photonic crystal coupled-cavity waveguides (CCW) with globally optimized dispersion at telecom wavelengths. The corresponding threshold for comb generation is found to explicitly depend on the main CCW figures of merit, namely, mode volume, normal mode quality factor and slow-light group index. Our analysis is carried out by solving the non-linear dynamics of the CCW Bloch modes in presence of Kerr non-linearity and two-photon absorption. Our results open the way to CCW comb generation via dispersion engineering and slow-li…
When a laser cavity supports the propagation of several ultrashort pulses, these pulses interact and can form compact bound states called soliton molecules. Soliton molecules are fascinating objects of nonlinear science, which present striking analogies with their matter molecules counterparts. The soliton pair, composed of two identical pulses, constitutes the chief soliton molecule of fundamental interest. The relative timing and phase between the two propagating pulses are the most salient internal degrees of freedom of the soliton molecule. These two internal degrees of freedom allow self-…
We report on the experimental observation of period doubling of multiple dissipative-soliton-resonance (DSR) pulses in an all-normal-dispersion fiber laser based on a nonlinear amplifying loop mirror. By increasing the pump power, in addition to the typically linearly pulse broadening under fixed pulse peak power, the jump from a single DSR pulse to multiple DSR pulses was observed. During this process, the period doubling appears, that is, the DSR pulses can exhibit the characteristics of period doubling in a multi-pulse state. The typical DSR performance of linearly pulse duration change ver…
