Gray Solitons Research Articles

Dynamics and pattern formation of ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable interactions

We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.

Black and gray solitons in holographic superfluids at zero temperature

We construct gray soliton configurations, which move at constant speeds, in holographic superfluids for the first time. Since there should be no dissipation for a moving soliton to exist, we use the simplest holographic superfluid model at zero temperature, considering both the standard and alternative quantizations. For comparison purpose, we first investigate black solitons in the zero temperature holographic superfluids, which are static configurations. Then we focus on the numerical construction of gray solitons under both quantizations, which interpolate between the (static) black solitons and sound waves (moving at the speed of sound). Interestingly, under the standard quantization, a peculiar oscillation of the soliton configurations is observed, very much resembling the Friedel oscillation in fermionic superfluids at the BCS regime. Some implications and other aspects of the soliton configurations are also discussed.

Gray and Dark Soliton Behavior and Population Under a Symmetric and Asymmetric Potential Trap

We numerically study the impact of Gaussian barrier height and width on the gray solitons population in a symmetric and asymmetric potential trap. The gray solitons are created in a double-well potential by the density engineering method. Two identical Bose-Einstein condensate fragments are confined and made to collide by switching off the Gaussian barrier in a double-well potential. We find that the gray solitons population can be manipulated by Gaussian barrier height and width. We also study the gray solitons population dependence on the coupling strength. Moreover, we also study the impact of an asymmetry present in the double-well potential. We observe that such an asymmetry always swings the point of collision of the gray solitons. Later, a stationary dark soliton is created by the phase imprinting method and we observe that the initial asymmetry in the double-well potential trap sets the dark soliton into oscillation.

Coupling of optical spatial solitons in photorefractive multiple quantum well planar waveguide

A comprehensive theory predicting the existence of both incoherently coupled and coherently coupled grey and dark optical spatial solitons in multiple quantum well planar (MQW) waveguides is presented. In addition, we also show the existence of dark Manakov soliton pair and the bright-dark Manakov soliton pair in such Multiple Quantum well waveguides for the first time. In the case of a coherently coupled soliton pair, the mutual phase difference, the overlap of the optical mode on the waveguide, the external bias field and beam intensity ratio have a profound effect on the FWHM of the soliton pairs. As a general case, a theory for the interesting and hitherto uninvestigated case of coherently coupled multicomponent solitons has been presented and analyzed for the first time. The effect of the mutual phase difference, the overlap of the optical mode on the waveguide(overlap parameter), the external bias field and beam intensity ratio on the FWHM is studied for these coherently coupled multicomponent solitons also. The Manakov soliton solutions are found analytically and they exist in the limit of low total intensity of the beams. We find that the dark-dark and bright-dark Manakov soliton pairs exist in such MQW waveguides whereas bright-bright Manakov soliton pair does not exist. The Manakov solitons’ spatial profile depends upon the overlap parameter, the intensity of the components, and the external electric field.

Dynamical Critical Behavior of an Attractive Bose-Einstein Condensate Phase Transition.

When matter undergoes a continuous phase transition on a finite timescale, the Kibble-Zurek mechanism predicts universal scaling behavior with respect to structure formation. The scaling is dependent on the universality class and is irrelevant to the details of the system. Here, we examine this phenomenon by controlling the timescale of the phase transition to a Bose-Einstein condensate using sympathetic cooling of an ultracold Bose thermal cloud with tunable interactions in an elongated trap. The phase transition results in a diverse number of bright solitons and gray solitons in the condensate that undergo attractive and repulsive interactions, respectively. The power law dependence of the average soliton number on the timescale of the phase transition is measured for each interaction and compared. The results support the Kibble-Zurek mechanism, in that the scaling behavior is determined by universality and does not rely on the interaction properties.

Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a higher-order nonlinear Schrödinger system for the simultaneous propagation of two ultrashort optical pulses in an optical fiber. Based on the Lax pair, with respect to the two-component electric fields, infinitely-many conservation laws and one/N-fold binary Darboux transformations are derived, where N=2,3,…. Solitons are discussed: (1) Nondegenerate one dark–dark soliton, which is black or gray in each component, is obtained. ϵ has no relation with the soliton amplitude in each component, but has a linear correlation with the soliton velocities, where ϵ, σ1 and σ2 are the constant coefficients in the system; (2) Overtaking and head-on interactions between the two dark–dark solitons as well as the bound state are depicted. With the decreasing value of σ1, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes decrease. With the decreasing value of σ2, the gray solitons’ amplitudes increase, but the black solitons’ amplitudes do not change; (3) Interaction among the three overtaking solitons and interaction between a bound state and one soliton are displayed. We find that the interactions between the two solitons and among the three solitons are elastic.

W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and potentials modulated in time and space

ABSTRACTThis paper explores the higher-order nonlinear Schrodinger equation with dual power-law nonlinearities and external potential that can depend on space and time coordinates. We constructed exact soliton-like solutions with arbitrary power-law nonlinearities using the similarity transformation approach. We obtained a family of soliton-like solutions: the so-called W-shaped, dark and the grey solitons. These solutions are at first, found by considering just the time modulated nonlinearities, where the frame variable coincides with the characteristic coordinate ξ. Then the nonlinearities coefficients and potential are space–time modulated with a bounded function . In both cases, a numerical study based on appropriate choices of parameters is achieved. For a fixed absolute value of nonlinearities, we examine how different soliton profiles are influenced when increasing the power-law index n. These variations of parameters display some substantial changes in solitary wave profiles; paving the way for many applications.

Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. I. Cubic-quintic systems

We consider an envelope equation with space-time symmetry for describing scalar waves in systems with spatiotemporal dispersion and a generic saturable nonlinearity. Exact bright and gray solitons are derived by direct integration methods and coordinate transformations, with the results for cubic-quintic systems [see companion article— Phys. Rev. A 98, 053842 (2018)] recovered in the limit of weak saturation. Classic predictions from a nonlinear Schrodinger formulation of the propagation problem are shown to emerge asymptotically as subsets of the more general spatiotemporal solutions. The robustness of the new solitons against perturbations to the local pulse shape is then tested by deploying integral stability criteria, symmetry principles, and numerical analysis.

Incoherently coupled two-color vector dark solitons in self-defocusing media.

We investigate the stability of incoherently coupled two-color vector dark and grey solitons in a nonlinear self-defocusing media. We employ the variational technique to derive approximate solutions of the two-color vector dark solitons. The relations between the soliton parameters and the representation of the transverse velocities of the vector solitons are analytically obtained. We also demonstrate numerically the propagation dynamics of the two-color vector solitons, which shows that the two-color vector dark and grey solitons with different wavelengths are always unstable.

Optical Solitons and Stability Analysis in Ring-Cavity Fiber System with Carbon Nanotube as Saturable Absorber

This paper addresses the ring-cavity fiber laser system. A class of gray and black soliton solutions of the model are reported by adopting an appropriate envelope ansatz. Further more, the modulation instability (MI) of the equation is studied using the linear-stability analysis (LSA) technique and the MI gain spectrum is derived. Some physical interpretations and analysis of the results obtained are also presented.

Dynamics of optical solitons, multipliers and conservation laws to the nonlinear schrödinger equation in (2+1)-dimensions with non-Kerr law nonlinearity

ABSTRACTThis work studies the (2 + 1)-dimensional nonlinear Schrödinger equation which arises in optical fibre. Grey and black optical solitons of the model are reported using a suitable complex envelope ansatz solution. The integration lead to some certain conditions which must be satisfied for the solitons to exist. On applying the Chupin Liu's theorem to the grey and black optical solitons, we construct new sets of combined optical soliton solutions of the model. Moreover, classification of conservation laws (Cls) of the model is implemented using the multipliers approach. This is achieved by constructing a set of first-order multipliers of a system of nonlinear partial differential equations acquired by transforming the model into real and imaginary components are derived, which are subsequently used to construct the Cls.

Many-body dissipative flow of a confined scalar Bose-Einstein condensate driven by a Gaussian impurity

The many-body dissipative flow induced by a mobile aussian impurity harmonically oscillating within a cigar-shaped Bose-Einstein condensate is investigated. For very {small and large driving frequencies} the superfluid phase is preserved. Dissipation is identified, for intermediate driving frequencies, by the non-zero value of the drag force whose abrupt increase {signals the spontaneous downstream emission of an array of gray solitons. After each emission event, typically each of the solitary waves formed decays and splits into two daughter gray solitary waves that are found to be robust propagating in the bosonic background for large evolution times.} In particular, a smooth transition towards dissipation is observed, with the {\it critical} velocity for solitary wave formation depending on both the characteristics of the obstacle, namely its driving frequency and width as well as on the interaction strength. The variance of a sample of single-shot simulations indicates the fragmented nature of the system; here it is found to increase during evolution for driving frequencies where the coherent structure formation becomes significant. Finally, we demonstrate that for fairly large particle numbers in-situ single-shot images directly capture the gray soliton's decay and splitting.

Thermalization, condensate growth, and defect formation in an out-of-equilibrium Bose gas

We experimentally and numerically investigate thermalization processes of a trapped $^{87}$Rb Bose gas, initially prepared in a non-equilibrium state through partial Bragg diffraction of a Bose-Einstein condensate (BEC). The system evolves in a Gaussian potential, where we observe the destruction of the BEC due to collisions, and subsequent growth of a new condensed fraction in an oscillating reference frame. Furthermore, we occasionally observe the presence of defects, which we identify as gray solitons. We simulate the evolution of our system using the truncated Wigner method and compare the outcomes with our experimental results.

Dark solitons manipulation using optical event horizon.

We demonstrate that the optical event horizon can provide an effective technique to actively control the propagation properties of a dark soliton with another weak probe wave. Careful power adjustment of the probe wave enables the black soliton converted into a gray one with varying grayness through the nonlinear interaction, corresponding to a nearly adiabatic variation of the soliton's speed. The sign of the phase angle for the newly formed gray soliton at optical event horizon is significantly dependent on the frequency of the launched probe wave. Linear-stability analysis of dark solitons under the perturbation of a weak probe wave is performed to clarify the intrinsic mechanism of the nonlinear interaction. The probe wave manipulated collisional dynamics between both dark solitons are investigated as an analogue of the combined white-hole and black-hole horizons which provides some insights into exploring the transition between integrable and non-integrable systems.

Optimal control of dark solitons

We consider optimal control methods for dark solitons and their higher-order states based on the completely integrable nonautonomous nonlinear Schrödinger equation (NLSE) models with varying gravitational-like and harmonic oscillator potentials, and in the presence of gain or absorption. In view of the fact that the time-dependence of the dark soliton phase (chirp) represents a major difference between bright and dark solitons, our main goal here is to gain a deeper understanding at a fundamental level based on the Lax pairs of how this time-dependent phase appears in the framework of the nonautonomous NLSE models with varying dispersion and nonlinearity. Specifically, we consider dynamics of dark solitons and their higher-order states in a variety of different optical scenarios. The main features of dark solitons are supplied by numerical calculations performed both for the exactly integrable and nonintegrable cases. We reveal nontrivial dynamics of black and gray solitons under the optimal control conditions. Since the experimental realization of dark solitons is possible only if a finite background pedestal is used, we consider dynamics of dark solitons and their higher-order states superimposed on the ground state of nonlinear harmonic oscillator. Direct numerical experiments performed in the framework of the Gross-Pitaevskii (GP) model in a broad range of parameters reveal many fundamental features of the dark solitons dynamics in confining harmonic oscillator potential, including the background BEC cloud spreading near the Feshbach resonance (when repulsive nonlinearity has a dispersive form). One of the most important effects is related to the “saturation” of the maximum value of self-compression near the Feshbach resonance. This effect exists both for the central black soliton and for the gray solitons formed during the higher-order dark soliton oscillations in the parabolic trap. The soliton becomes black at the turning points in the harmonic potential and gray at the trap center. This makes possible to observe the nontrivial dynamics of the periodic transformation of the black soliton to the gray one.

Grayness-dependent emission of dispersive waves from dark solitons in optical fibers.

We report the experimental observation of dispersive wave emission from gray solitons propagating in the normal dispersion region of an optical fiber. Besides observing for the first time, to the best of our knowledge, the emission of a dispersive wave from an isolated dark soliton, we show that the dispersive wave frequency and amplitude strongly depend on soliton grayness. This process can be explained by the higher-order dispersion contribution into the phase-matching condition and the grayness of the soliton. Numerical simulations and theoretical predictions are in good agreement with the experiments.

Gray optical soliton, linear stability analysis and conservation laws via multipliers to the cubic nonlinear Schrödinger equation

This paper addresses the cubic nonlinear Schrödinger equation with a bounded potential (CNLSE) which describes optical solitary waves propagation properties in optical fiber. A gray optical soliton solution of this equation is retrieved for the first time by adopting an appropriate solitary wave ansatz which play a vital role in understanding various physical phenomena in nonlinear systems. The integration lead to a constraint condition on the solitary wave parameters which must hold for the soliton to exist. We studied the conservation laws (Cls) of the CNLSE by analyzing a system of partial differential equations (PDEs) obtained by transforming the equation into real and imaginary components. The multiplier approach is employed to retrieve the conservation laws. Moreover, the modulation instability (MI) analysis of the model is studied by employing the linear-stability analysis and the MI gain spectrum is got. Physical interpretations of the acquired results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CNLSE.

Coupling effects for separate spatial solitons in a biased series centrosymmetric photorefractive crystal circuit considering grey solitons

The existence and coupling of grey–grey, grey–bright and grey–dark separate spatial solitons in a biased series centrosymmetric photorefractive crystal circuit is investigated for the first time. The numerical solution of the separate spatial solitons is presented. The coupling between the two separate spatial solitons is analysed for all three cases of separate coupled solitons, namely grey–grey, grey–bright, and grey–dark. Changing the intensity of the soliton in one crystal affects the soliton in both crystals due to flow of the light induced current through the circuit. The effect of the background intensity of each crystal on both the spatial solitons is investigated. Also, the effect of changing the temperature of one crystal affects the soliton in both crystals due to the coupling effect. The soliton width dependence on the temperature is different for each crystal.

Black and gray soliton interactions and cascade compression in the variable coefficient nonlinear Schrödinger equation

We present the theoretical study of dark soliton dynamics in inhomogeneous fiber system by means of a variable coefficient nonlinear Schrödinger equation (Vc-NLSE). The dark soliton interaction in the dispersion-managed system with gain/loss and cascade compression of solitons by the nonlinear tunneling phenomena are studied in detail. Numerically, we study the dark soliton propagation in the continuous wave background, and the obtained soliton solution is found to be robust against perturbations. The elastic collision behavior of the dark solitons are observed by means of asymptotic analysis. For better insight, we study the impact of the blackness factor in the phase profile and also on the interaction dynamics of the different dark soliton settings, such as black and gray soliton. Additionally, we extend the study to the investigation of tunneling of the dark soliton across the barrier/well with a particular emphasize on the role of blackness parameter in the nonlinear tunneling phenomena, and demonstrated a cascaded compression of soliton in the inhomogeneous system.

Weakly decaying solutions of nonlinear Schrödinger equation in the plane

We show that the nonlinear Schrödinger equation in 2 + 1 dimensions possesses a class of regular and rationally decaying solutions associated to interacting solitons. The interesting dynamics of the associated pulses is studied in detail and related to homothetic Lagrange configurations of certain N- body problems. These solutions correspond to the discrete spectrum of the Lax pair associated operator. A natural characterization of this spectrum is given. We show that a certain subset of solutions correspond to rogue waves, localized along curves in the plane. Other configurations like grey solitons, cnoidal waves and general N- lumps solutions are also described.