Head-on collision of ring dark solitons in Bose-Einstein condensates

The ring dark solitons and their head-on collisions in a Bose-Einstein condensates with thin disc-shaped potential are studied. It is shown that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by the two cylindrical KdV equations in the respective reference frames. By using the extended Poincaré-Lighthill-Kuo perturbation method, the analytical phase shifts following the head-on collisions between two ring dark solitary waves are derived. It is shown that the phase shifts decrease with the radial coordinate r according to the r-1/3 law and depend on the initial soliton amplitude and radius.

Experimental Observation of Bright and Dark Solitons Mode-Locked with Zirconia-Based Erbium-Doped Fiber Laser

We demonstrate the generation of dark and bright solitons with our homemade zirconia-based erbium-doped fiber and graphene oxide（GO） saturable absorber in anomalous dispersion region.The GO is fabricated using an abridged Hummer＇s method,which is combined with polyethylene oxide to produce a composite film.The film is sandwiched between two optical ferrules and embedded in the laser cavity to enhance its birefringence and nonlinearity.The self-starting bright soliton is easily generated at pump power of 78 mW with the whole length cavity of 14.7 m.The laser produces the bright pulse train with repetition rate,pulse width,pulse energy and central wavelength being 13.9 MHz,0.6 ps,2.74 p J and 1577.46 nm,respectively.Then,by adding the 10 m of single mode fiber into the laser cavity,dark soliton pulse is produced.For the formation of dark pulse train,the measured repetition rate,pulse width,pulse energy and central wavelength are 8.3 MHz,20 ns and 4.98 p J and1596.82 nm,respectively.Both pulses operate in the anomalous region.

Dissipative dynamics of dark solitons in elongated Bose-Einstein condensates

The dissipative dynamic stability is investigated of dark solitons in elongated Bose-Einstein condensates that can be described by the Gross-Pitaevskii equation including an additional term. Based on the direct perturbation theory for the nonlinear SchrSdinger equation, the dependence of the soliton velocity on time is explicitly given, and the shape of dark solitons remaining unchanged under the dissipative condition is confirmed theoretically for the first time. It is found that the dynamically stable dark solitons turn out to be thermodynamically unstable.

Dark Solitons in FPU Lattice Chain

Based on multiple scales method, we study the nonlinear properties of a new Fermi-Pasta-Ulam lattice model analytically. It is found that the lattice chain exhibits a novel nonlinear elementary excitation, i.e. a dark soliton. Moreover, the modulation depth of dark soliton is increasing as the anharmonic parameter increases.

Quantum Squeezing of Dark Solitons in Optical Fibers

The quantum theory of dark soliton propagation in fibers is studied based on the linearization approximation. Then the uncertainties in photon number, phase, position （time） and momentum of quantized dark solitons are calculated. Finally, the squeezing of the dark soliton is investigated by using homodyne detection and compared with bright soliton case.

Evolution of dark solitons in the presence of Raman gain and self-steepening effect

Based on the equation satisfied by optical pulse that is a slowly varying function, the higher-order nonlinear Schr o¨dinger equation(NLSE) including Raman gain and self-steepening effect is deduced in detail, and a new Raman gain function is defined. By using the split-step Fourier method, the influence of the combined effect between Raman gain and self-steepening on the propagation characteristic of dark solitons is simulated in the isotropic fiber. The results show that gray solitons can be symmetrically formed by high order dark soliton, however self-steepening effect will inhibit the formation mechanism through the phenomenon that gray solitons are produced only in the trailing edge of the central black soliton. Meanwhile, the Raman gain changes the propagation characteristic of optical soliton and inhibits the self-steepening effect, resulting in the broadening of pulse width and the decreasing of pulse offset.

Dark and multi-dark solitons in the three-component nonlinear Schrodinger equations on the general nonzero background

We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrodinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrodinger systems, and require more in-depth studies in the future.

Effects of periodic optical lattice potential on dark solitons in a Bose Einstein condensate

This paper investigates the dynamics of dark solitons in a Bose-Einstein condensate with a magnetic trap and an optical lattice （OL） trap, and analyses the effects of the periodic OL potential on the dynamics by applying the variational approach based on the renormalized integrals of motion. The results show that the dark soliton becomes only a standing-wave and free propagation of the dark soliton is not possible when the periodic length of the OL potential is approximately equal to the effective width of the dark soliton. When the periodic length is very small or very large, the effects of the OL potential on the dark soliton will be sharply reduced. Finally, the numerical results confirm these theoretical findings.

Propagation characteristics of parallel dark solitons in silicon-on-insulator waveguide

The propagation characteristic of two identical and parallel dark solitons in a silicon-on-insulator(SOI)waveguide is simulated numerically using the split-step Fourier method.The parallel dark solitons imposed by the initial chirp are investigated mainly by changing their power,their relative time delay.The simulation shows that the time delay deforms the parallel dark soliton pulse,forming a bright-like soliton in the transmission process and making the transmission quality down.By increasing the power of one dark soliton,the energy of the other dark soliton can be increased,and larger increase in a soliton’s power leads to larger increase in the energy of the other.When the initial chirp is introduced into one of the dark solitons,higher energy consumption is observed.In particular,positive chirps resulting in pulse broadening width while negative chirps narrowing,with an obvious compression effect on the other dark soliton.Finally,large negative chirps are found to have a profound impact on parallel and nonparallel dark solitons.

Bound states of spatial optical dark solitons in nonlocal media

It is shown that multiple dark solitons can form bound states in a series of balance distances in nonlocal bulk media. Dark solitons can either attract or repel each other depending on their separated distance. The stability of such bound states are studied numerically. There exist unstable degenerate bound states decaying in different ways and having different lifetimes.

Dark solitons in WS_2 erbium-doped fiber lasers

Tungsten disulfide(WS_2) is a type of anisotropic-layered compound and has broadband saturable absorption features as saturable absorbers(SAs). With WS_2-based SAs, dark solitons in erbium-doped fiber(EDF) lasers are first obtained. For the generated dark solitons, the center wavelength is measured to be 1530 nm, and the repetition rate is about 116.5 MHz. A series of optical spectra is exhibited. The electrical signal-to-noise ratio is better than 94 d B.Results in this paper demonstrate that WS_2-based SAs are the promising SAs for generating dark solitons in EDF lasers.

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrodinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schr?dinger equation,since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices.Based on the Lax pair,the binary Darboux transformation is constructed under certain constraints,thus the multi-dark soliton solutions are presented.Soliton propagation and collision are graphically discussed with the group-velocity dispersion,third-and fourth-order dispersions,which can affect the solitons’velocities but have no effect on the shapes.Elastic collisions between the two dark solitons and among the three dark solitons are displayed,while the elasticity cannot be influenced by the above three coefficients.

Dynamics and formation of vortices collapsed from ring dark solitons in a two-dimensional spin–orbit coupled Bose–Einstein condensate

We consider the dynamics and formation of vortices from ring dark solitons in a two-dimensional Bose–Einstein condensate with the Rashba spin–orbit coupling based on the time-dependent coupled Gross–Pitaevskii equation.Compared with previous results,the system exhibits complex dynamical behaviors in the presence of the spin–orbit coupling.With the modulation of the spin–orbit coupling,not only the lifetime of ring dark solitons is greatly prolonged,but also their attenuation kinetics is significantly affected.For two shallow ring dark solitons with the equal strength of the spin–orbit coupling,the radius of ring dark solitons increases to a maximum value over time and then shrinks into a minimum value.Due to the effect of the snake instability,ring dark solitons split into a series of ring-like clusters of vortex pairs,which perform complex oscillations.This indicates that the system is strongly dependent on the presence of the spin–orbit coupling.Furthermore,the effect of different initial modulation depths on the dynamics of ring dark solitons is investigated.

Symplectic schemes and symmetric schemes for nonlinear Schr¨odinger equation in the case of dark solitons motion

In this paper,symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrodinger Equation(NLSE)in case of dark soliton motion.Firstly,by Ablowitz–Ladik model(A–L model),the NLSE is discretized into a non-canonical Hamiltonian system.Then,different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system.Therefore,the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE.The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect,and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons,preserving the invariants and the approximations of conserved quantities.Moreover,it is obvious that coordinate transformations with more symmetry have a better simulation effect.

Dynamics of Nonautonomous Dark Solitons

We solve a generalized nonautonomous nonlinear Schrodinger equation analytically by performing the Hirota＇s bilinearization method. The precise expression of a parameter e, which provides a compatibility condition and dark soliton management, is obtained. Comparing with nonautonomous bright soliton, we find that the gain parameter affects both the background and the valley of dark soliton （∈2 ≠ 1） while it has no effects on the wave central position. Moreover, the precise expressions of a nonautonomous black soliton＇s （∈2 = 1） width, background and the trajectory of its wave central, which describe the dynamic behavior of soliton＇s evolution, are investigated analytically. Finally, the stability of the dark soliton solution is demonstrated numerically. It is shown that the main characteristic of the dark solitons keeps unchanged under a slight perturbation in the compatibility condition.

Solitons and Bifurcations for the Generalized Tzitzéica Type Equation in Nonlinear Fiber Optics

Solitons and bifurcations for the generalized Tzitzéica type equation are studied by using the theory of dynamical systems and Hamilton function. With the help of Maple and bifurcation theory of differential equations, the bifurcation parameter conditions and all the bifurcation phase portraits are obtained. Because the same energy value of the Hamiltonian function is corresponding to the same orbit, thus the periodic wave solutions, bright soliton and dark soliton solutions are defined.

Coupled Optical Solitons for Pulse Propagation in Multi-Level Media

We investigate the shape-preserving propagation of N optical pulses in an （N ＋ 1）-level medium. We solve Maxwell-Schroedinger equations exactly and provide several types of explicit coupled soliton solutions, which are temporally amplitude- and group-velocity-matched multi-mode slow-optical pulses of the system.

Effect of dark soliton on the spectral evolution of bright soliton in a silicon-on-insulator waveguide

The spectral evolution of bright soliton in a silicon-on-insulator optical waveguide is numerically simulated using the split-step Fourier method.The power and input chirp of the dark soliton and the second-order dispersion are varied to investigate the effect of dark soliton on the spectrum of bright soliton.The simulations prove that the dark soliton modifies the spectral shape of the bright soliton.Further,the variation in the power of dark soliton affects the splitting of bright soliton.Furthermore,the chirped dark soliton can improve the spectral width and flatness.The variation in the dispersion of dark soliton modifies the phase matching of the bright soliton and the dispersive wave emission,thereby affecting the spectral evolution.

Small Amplitude Solitons in Bose-Einstein Condensates with External Perturbation

By developing a small amplitude soliton approximation method, we study analytically weak nonlinear excitations in cigar-shaped condensates with repulsive interatomic interaction under consideration of external perturbation potential. It is shown that matter wave solitons may exist and travel over a long distance without attenuation and change in shape by properly adjusting the strength of interatomic interaction to compensate for the effect of external perturbation potential.

Self—Steepening and Thired—Order Dispersion Induced Optical Solitons in Fiber

Usually,one considers only the group velocity dispersion(GVD)-and self-phase modulation(SPM)-induced solitons in optic soliton communication while other higher order effects such as the third-order dispersion(TOD),self-steepening(SS),and stimulated Raman scattering are considered only perturbatively,In this paper,we study the existence of the TOD-and SS-induced soliton solutions.The existence conditions of the TOD-and SS-induced bright and dark solitons are quite different from those of the GVD-and SPM-induced solitons.