Vector Dark Solitons Research Articles

A binary Darboux transformation for multi-component nonlinear Schrödinger equations and dark vector soliton solutions

By taking the plane wave potentials as the seed solutions, we harness a binary Darboux transformation to generate dark vector soliton solutions for multi-component nonlinear Schrödinger equations. We introduce a generalized Darboux matrix such that the eigenvalues could equal the adjoint eigenvalues. The method which is purely algebraic could be useful and convenient, particularly in the construction of dark soliton solutions of integrable systems.

Fundamental and second-order dark soliton solutions of two- and three-component Manakov equationsin the defocusing regime.

We present exact multiparameter families of soliton solutions for two- and three-component Manakov equationsin the defocusing regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatiotemporal dynamics. The complexity increases in the case of three-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (nonoscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.

Nonlocal two-color vector dark solitons

In this paper, vector dark solitons with two colors are demonstrated in nonlocal nonlinear Kerr medium. Using Lagrangian method and rectangular profile of nonlocal response of the media, solutions of the solitons are obtained analytically. The stability of the solitons is also studied numerically, which confirms the analytical results. Nonlocality can suppress instability induced by the difference of diffraction coefficients, leading to formation of stationary two-color vector dark solitons which are always unstable in local nonlinear media.

Vector solutions of the coupled discrete conformable fractional nonlinear Schrödinger equations

By considering the conformable fractional derivative to modify the discrete Riccati equation mapping method, we obtain rich vector exact solutions for the coupled discrete conformable fractional nonlinear Schrödinger equations, including the vector bright solitons, the vector dark solitons and the vector trigonometric function solutions. The parameter range existing these vector exact solutions is also given. The influence of parameters related to the group velocity and the effect of fractional derivatives on the amplitude modulation of wave is investigated.

Strong and weak interactions of rational vector rogue waves and solitons to anyn-component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate anyn-component fourth-order nonlinear Schrödinger (n-FONLS) system with non-zero backgrounds containing then-Hirota equation and then-Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of then-FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of then-FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of then-FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

Darboux transformation and dark vector soliton solutions for complex mKdV systems

A Darboux transformation and general vector dark soliton solutions are constructed for multi-component complex modified Korteweg–de Vries (mKdV) system. Dark soliton solutions exist when nonlinearities are either defocusing or mixed focusing and defocusing. Single-dark–dark–dark and two-dark–dark–dark soliton solutions of three component complex mKdV system are explicitly presented, whose properties and dynamics are illustrated. It is also shown during the interaction between those solitons, energies in different components completely transmit through. Moreover, we perform an asymptotic analysis for the n -dark soliton solutions rigorously. Our results can be applicable to the study of dark solitons in different physical fields.

Coexistence of dark vector soliton Kerr combs in normal dispersion resonators

We investigate the formation of dark vector dissipative solitons in the presence of nonlinear polarization mode coupling in optical resonators subject to a coherent optical injection in the normal dispersion regime. This simple device is described by coupled Lugiato-Lefever equations. The stabilization of dark dissipative solitons is attributed to a front locking mechanism in the bistable regime as shown in a recent communication [B. Kostet et al., OSA Continuum 4, 1564 (2021)]. Here, we focus on a tristable homogeneous steady state regime. We show that two branches of dark dissipative solitons can coexist for a fixed value of the system parameters. These coexisting solutions possess different polarization states and different peak powers in the microresonator. We characterize their formation by drawing their bifurcation diagrams in regimes far from any modulational instability. It is shown that both branches of localized structures exhibit a heteroclinic collapsed snaking type of behavior. The coexistence of two vectorial branches of dark localized states is not possible without taking into account polarization degrees of freedom.

Self-starting triple-wavelength vector dark soliton with a bismuth-doped fiber saturable absorber.

We report on the generation of a triple-wavelength vector dark soliton in an all-fiber ring cavity of erbium-doped fiber laser mode-locked with a bismuth-doped fiber saturable absorber. The formation of the triple-wavelength vector dark soliton is due to the cross-phase coupling derived from the cavity birefringence. The mode-locked laser operated at a 1.89 MHz repetition rate with a 335 ns pulse width, and its robustness is confirmed.

Vector solitons for the (2 + 1)‐dimensional coupled nonlinear Schrödinger system in the Kerr nonlinear optical fiber

AbstractInvestigated in this paper is the (2 + 1)‐dimensional coupled nonlinear Schrödinger system, which describes the propagation of the optical soliton through the Kerr nonlinear fiber in which each component diffracts in two transverse dimensions. By virtue of the Kadomtsev‐Petviashvili hierarchy reduction, we obtain the bright, dark and mixed vector soliton solutions in terms of the Gramian. We investigate the vector solitons graphically: (1) Bright‐dark one solitons with one of the components being gray or black are shown. Bouncing, beating, oscillating and breathing effects which appear during the interactions between the bright‐dark two solitons are discussed, and those interactions are shape‐preserving; (2) Degenerate and non‐degenerate dark‐dark one solitons are presented. Interaction of the dark‐dark two solitons is discussed and the two solitons keep their amplitudes and velocities invariant except for some phase shifts during the interaction; (3) Bound‐state and complexes of the bright‐bright solitons are presented.

Modulation of dark vector bisolitons

In the manuscript, we report the theoretical simulation results about modulating dark vector bisolitons in 1064 nm wavelength band, when input soliton state is polarization-locking or group-velocity-locking. Varieties of pulse shapes and corresponding optical spectra in orthogonally polarized directions can be achieved by changing orthogonally polarized modes’ pulse parameters. When input soliton state is polarization-locking, one dip always occurs for orthogonally polarized modes regardless of pulse parameters’ changes. While for input soliton state of group-velocity-locking, “1 + 2″/“2 + 1″ pseudo-high-order dark vector soliton, vector solitons with chirp-like structures can be achieved. The theoretical simulation results give reference for studying the modulation of vector multi-solitons and vector bisolitons at other wavelengths.

Pulse shaping of bright-dark vector soliton pair**Project supported by the National Key Research and Development Program of China (Grant No. 2018YFB0504500) and the National Natural Science Foundation of China (Grant No. 51672177).

We simulate pulse shaping of bright–dark vector soliton pair in an optical fiber system. Through changing input pulse parameters (amplitude ratio, projection angle, time delay, and phase difference), different kinds of pulse shapes and spectra can be generated. For input bright–dark vector soliton pair with the same central wavelength, “2+1”- and “2+2”-type pseudo-high-order bright–dark vector soliton pairs are achieved. While for the case of different central wavelengths, bright–dark vector soliton pairs with multiple pulse peaks/dips are demonstrated with appropriate pulse parameter setting.

Interference properties of two-component matter wave solitons

Wave properties of solitons in a two-component Bose–Einstein condensate are investigated in detail. We demonstrate that dark solitons in one of components admit interference and tunneling behavior, in sharp contrast to the scalar dark solitons and vector dark solitons. Analytic analyses of interference properties show that spatial interference patterns are determined by the relative velocity of solitons, while temporal interference patterns depend on the velocities and widths of two solitons, differing from the interference properties of scalar bright solitons. Especially, for an attractive interactions system, we show that interference effects between the two dark solitons can induce some short-time density humps (whose densities are higher than background density). Moreover, the maximum hump value is remarkably sensitive to the variation of the solitons’ parameters. For a repulsive interactions system, the temporal-spatial interference periods of dark–bright solitons have lower limits. Numerical simulation results suggest that interference patterns for the dark–bright solitons are more robust against noises than bright–dark solitons. These explicit interference properties can be used to measure the velocities and widths of solitons. It is expected that these interference behaviors can be observed experimentally and can be used to design matter wave soliton interferometer in vector systems.

Manipulation of polarization- and group-velocity-locked dark vector solitons

In this manuscript, we simulate the manipulation of polarization- and group-velocity-locked dark vector solitons. Through changing pulse parameters of incident dark vector solitons, different kinds of pulse shapes could be generated. For incident fundamental dark polarization-locked vector soliton (PLVS), “2 + 2” pseudo-high-order dark PLVS and bright-dark PLVS could be generated. While for incident fundamental dark group-velocity-locked vector soliton (GVLVS), dark vector solitons with multiple side dips could be achieved depending on input pulse parameters.

Phase dynamics of inhomogeneous Manakov vector solitons

We report the exact phase dynamics of Manakov bright and dark vector solitons in an inhomogeneous optical system by means of a variable coefficient coupled nonlinear Schrödinger equation. To investigate the phase dynamics, we have modified the Manakov system with a relation between two modes of propagation, that are obtained by the Hirota bilinear method. The importance of the phase study in soliton interaction is revealed by asymptotic analysis of two-soliton solutions. In contrast with the Manakov bright soliton, the time-dependent dark vector soliton exhibits a gradual phase shift due to the blackness factor. The various inhomogeneous effects on the soliton phase are investigated, with a particular emphasis on nonlinear tunneling. The intensity and corresponding phase of the tunneling soliton either forms a peak or valley and retains its shape after tunneling. Unlike the bright counterpart, the gain or loss term significantly affects the phase of the dark soliton. Apart from the study of soliton intensity, the phase profile of bright and dark vector solitons and its dynamical features are also explored. As the study is not limited to intensity description, the present study could serve as a reference for the future studies on multisolitons phase dynamics in photonics and related fields.

Vector dark solitons in a single mode fibre laser

We report on the first experimental observation of vector dark solitons in an erbium-doped single mode fiber ring laser. We show experimentally that two orthogonal linearly polarized dark solitons could couple together incoherently through the cross polarization phase modulation and form a bound state of dark solitons. In particular, the formed vector dark solitons are stable even under existence of high frequency periodic background intensity modulation. Numerical simulations have well confirmed the existence of vector dark solitons in single mode fiber lasers and their stability under periodic background intensity modulation.©2018 Optical Society of America OCIS codes: (190.4370) nonlinear optics, fibers; (190.5530) pulse propagation and temporal solitons.

Magnetized vector solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate with Zeeman coupling

The property of matter-wave vector solitons in a spin-1 Bose-Einstein condensate with spin-orbit and Zeeman couplings is investigated by multiscale perturbation method. The excitation spectrum and the corresponding state vectors of the system are obtained analytically, and they can be adjusted by the parameters of the system. The bright and dark vector solitons are formulated by reducing the three-component coupled Gross-Pitaeviskii equations to a standard nonlinear Schrödinger equation, which has the solutions of the bright and dark solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients. The moving vector solitons are demonstrated by adjusting specific momentum near the energy minimum. Finally, the magnetized features of the vector solitons are discussed by the spin polarization of the system.

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.

Three coupled slow temporal vector optical solitons in a cold seven-level atomic system

A theoretical scheme is proposed to investigate the formation of three coupled slow temporal vector optical solitons in a lifetime-broadened seven-state atomic medium. We demonstrate that the three polarized components of a low intensity linear-polarized pulsed probe field can evolve into three coupled slow temporal vector optical solitons due to the fact that group-velocity dispersion can be well compensated by the self- and cross-phase-modulation effects. Moreover, this atomic system could support the existence of various types of three coupled slow temporal vector solitons, such as bright–bright–bright, bright–dark–bright, dark–dark–dark, and bright–dark–dark vector solitons.

Dark and antidark solitons for the defocusing coupled Sasa–Satsuma system by the Darboux transformation

In this article, we construct the N-fold Darboux transformation for the defocusing coupled Sasa–Satsuma system which describes the simultaneous propagation of two nonlinear waves in optical fibers with higher order effects. With the non-zero constant background as a seed, we derive the dark and antidark soliton solutions from the once-iterated formula. We find that this coupled system can exhibit the dark–dark, dark–antidark and antidark–dark vector solitons.

The Nth-order Darboux transformation, vector dark solitons and breathers for the coupled defocusing Hirota system in a birefringent nonlinear fiber

Under investigation in this paper is the coupled defocusing Hirota system, which describes the propagation of ultra-short pulses in a birefringent nonlinear fiber. With respect to the amplitudes of the pulse envelopes, we construct the Nth-order Darboux transformation, which is different from those in the existing literatures, where N is a positive integer. The first- and second-order solutions are obtained via the Darboux transformation. Pulse envelopes including the gray-gray/antidark-gray/dark-bright solitons, vector Akhmediev breathers, temporal cavity solitons and (time, space)-periodic breathers are acquired. As the relative width of the spectrum that arises due to the quasi monochromocity decreases, we find that: the velocity and width of the dark-bright solitons increase; the width of the temporal cavity soliton decreases; the temporal period of the vector (time, space)-periodic breather becomes longer while its spacial period becomes shorter. Relative width of the spectrum that arises due to the quasi monochromocity does not affect the amplitudes of the pulse envelopes obtained. Elastic interactions between the breathers and solitons, and inelastic interactions between the two gray-gray solitons are obtained.