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.

~ Transform Demonstration of Dark Soliton Solutions Found by Inverse Scattering

One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.

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.

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.

Quantum reflection of a Bose-Einstein condensate with a dark soliton from a step potential

We study dynamical behaviors of a Bose-Einstein condensate(BEC)containing a dark soliton reflected from potential wells and potential barriers,respectively.The orientation angle of the dark soliton and the width of the potential change play key roles on the reflection probability Rs.Variation of the reflection probability with respect to the orientation angleθof the dark soliton can be well described by a cosine function Rs~cos[λ(θ-π/2)],whereλis a parameter determined by the width of the potential change.There are two characteristic lengths which determine the reflection properties.The dependence of the reflection probability on the width of the potential change shows distinct characters for potential wells and potential barriers.The length of the dark soliton determines the sensitive width of potential wells,whereas for potential barriers,the decay length of the matter wave in the region of the barrier qualifies the sensitive width of the barrier.The time evolution of the density profiles of the system during the reflection process is studied to disclose the different behaviors of matter waves in the region of the potential variation.

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.

Dark soliton in one-dimensional Bose-Einstein condensate under a periodic perturbation of trap

The perturbation of a confining trap leads to the collective oscillation of a Bose-Einstein condensate, thereby the propagation of a dark soliton in the condensate is affected. In this study, periodic perturbation is employed to match the soliton oscillation. We find that the soliton dynamics depends sensitively on the coupling between the moving direction of the trap and that of the soliton. The soliton energy/depth evolves periodically, and a relevant shift in the soliton trajectory occurs as compared with the unperturbed case. Overall, the soliton oscillation frequency changes little even if the perturbation amplitude and frequency vary.

Effects of localized impurity on a dark soliton in a Bose-Einstein condensate with an external magnetic trap

The dynamics of a dark soliton has been investigated in a Bose Einstein condensate with an external magnetic trap, and the effects of localized impurity on the dynamics are discussed by the variational approach based on the renormalized integrals of motion. The reciprocal movement of the dark soliton is discussed by performing a standard linear analysis, and it is found that the effects of the localized impurity depend strictly on the positive or negative value of the impurity strength corresponding to the repulsive or attractive impurity. The numerical results confirm the theoretical analysis, and show that the effects also depend on the effective nonlinear coefficient and the harmonic frequency.

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.

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.

Bright and dark soliton solutions in growing Bose-Einstein condensates

This paper develops the Hirota method carefully for applying into the growing model of quasi-one-dimensional Bose Einstein condensations with attractive and repulsive interaction, respectively. After a tedious calculation it obtains the exact bright and dark soliton solutions analytically. It shows that the growing model has the important effect on the soliton amplitude and the time-dependent potential only contributes to the phase and phase velocity. A detailed analysis for the asymptotic behaviour of two-soliton solutions shows that the collision of two soliton is elastic.

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.

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.

Analysis of dark soliton generation in the microcavity with mode-interaction

Mode-interaction plays an important role in the dark soliton generation in the microcavity. It is beneficial to the excitation of dark solitons, but also facilitates a variety of dark soliton states. Based on the non-normalized Lugiato-Lefever equation, the evolution of dark soliton in the microcavity with mode-interaction is investigated. By means of mode-interaction, the initial continuous wave(CW) field evolves into a dark soliton gradually, and the spectrum expands from a single mode to a broadband comb. After changing the mode-interaction parameters, the original modes which result in dual circular dark solitons inside the microcavity, are separated from the resonant mode by 2 free spectral ranges(FSR). When the initial field is another feasible pattern of weak white Gaussian noise, the large frequency detuning leads to the amplification of the optical power in the microcavity, and the mode-interaction becomes stronger. Then, multiple dark solitons, which correspond to the spectra with multi-FSR, can be excited by selecting appropriate mode-interaction parameters. In addition, by turning the mode-interaction parameters, the dark soliton number can be regulated, and the comb tooth interval in the spectrum also changes accordingly. Theoretical analysis results are significant for studying the dark soliton in the microcavity with mode-interaction.

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.

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.

Propagation of dark soliton interacting with domain wall in two immiscible Bose–Einstein condensates

Reflection and transmission are two behaviors of wave propagating to an interface. The immiscible binary mixtures of Bose–Einstein condensates can form the symmetry-breaking state, in which the domain wall on the center can serve as the interface. In this study, we explore in detail the propagation of a dark soliton interacting with the domain wall in the harmonic trap. We find that the low-energy dark soliton is easy to form the transmission and the high-energy dark soliton trends to reflect from the domain wall. Both reflection and transmission of dark soliton on the domain wall induce the sound radiation. But the sound radiation in the reflection derives from the collective oscillation of condensates, and it in the transmission comes not only from the collective oscillation, but also from the condensate filling in the dark soliton.

Asymptotic analysis of multi-valley dark soliton solutions in defocusing coupled Hirota equations

We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing the uniform Darboux transformation,in addition to proposing a sufficient condition for the existence of the above dark soliton solutions.Furthermore,the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior;however,collisions for double-valley dark solitons are generally inelastic.In light of this,we further propose a sufficient condition for the elastic collisions of double-valley dark soliton solutions.Our results offer valuable insights into the dynamics of dark soliton solutions in the defocusing coupled Hirota equation and can contribute to the advancement of studies in nonlinear optics.

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.