Identical Solitons Research Articles

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

Long-range interaction between solitons in strongly nonlocal nonlinear media with oscillatory response

The long-range interaction between two identical solitons in strongly nonlocal nonlinear media with the sine-oscillation response function is studied. It is found that the two solitons may attract each other, repel each other, or keep in parallel in the initial stage. The behavior of the soliton interactions depends periodically on the separation between solitons; it is essentially controlled by the superposition of the periodic light-induced nonlinear refractive index. The different versions of the soliton interactions can be interchanged by adjusting the separation between solitons.

Observation of complex multimode soliton molecules in spatiotemporal mode-locked Er-doped fiber laser

In this letter, we have reported experimental observation of diverse structural multimode (MM) soliton molecules in an all fiber spatiotemporal mode-locked (STML) laser. By adjusting the intracavity devices, the STML operation state at ∼1560 nm with the identical soliton separation is achieved, including soliton triplet, soliton quartet, and soliton quintet. Additionally, the STML soliton molecule with compound structure can also be obtained, where two different pulse separations exist in the soliton molecule complex, for the inner soliton with the unequal or identical intensity. The transverse mode distributions for different soliton molecule complexes differ from each other, indicating the STML structure of the MM solitons. Our experimental results will extend our further understanding of the spatiotemporal nonlinear dynamics in the MM lasers.

Ultrafast Photonics of Ternary RexNb(1-x)S2 in Fiber Lasers.

Two-dimensional (2D) transition metal chalcogenides (TMCs) become more attractive upon addition of a third element owing to their unique structure and remarkable physical and chemical properties, which endow these materials with considerable potential for applications in nanoscale devices. In this work, a RexNb(1-x)S2-based saturable absorber (SA) device for ultrafast photonics applications is studied. The device is assembled by placing RexNb(1-x)S2 nanosheets with a thickness of 1-3 nm onto a microfiber to increase their compatibility with an all-fiber laser cavity. The prepared RexNb(1-x)S2-based device exhibits a modulation depth of 24.3%, a saturation intensity of 10.1 MW/cm2, and a nonsaturable loss of 28.5%. Furthermore, the RexNb(1-x)S2-based device is used to generate ultrashort pulses in an erbium-doped fiber (EDF) laser cavity. At a pump power of 260 mW, the EDF laser operates in a conventional soliton mode-locked region. The pulse width is 285 fs, and the repetition frequency is 61.993 MHz. In particular, the bound-state soliton mode-locking operation is successfully obtained in a pump power range of 300-900 mW. The bound-state pulses are formed by doubling identical solitons with a temporal interval of 0.8 ps. The output power is as high as 47.9 mW, and the repetition frequency is 123.61 MHz. These results indicate that the proposed RexNb(1-x)S2-based SAs have comparable properties to currently used 2D SAs and provide a basis for their application in the field of ultrafast photonics.

Conventional solitons and bound-state solitons in an erbium-doped fiber laser mode-locked by TiSe2-based saturable absorber

Conventional solitons (CSs) as well as bound-state solitons in a passively mode-locked erbium-doped fiber (EDF) laser based on 1 T-phase titanium diselenide (1 T-TiSe2) saturable absorber (SA) have been systematically demonstrated for the first time. The mode locker is assembled by sandwiching the 1 T-TiSe2 film between two fiber ferrules to improve compatibility with the all-fiber-integrated ring cavity configuration. The modulation depth, saturation intensity and nonsaturable loss of the prepared 1 T-TiSe2 SA are 14.36%, 1.33 MW cm−2 and 9.44%, respectively. The system is switchable between two states: CS and bound-state CS, by carefully adjusting the orientations of the polarization controller (PC). In the CS mode-locked regime, the oscillating wavelength is centered at 1558.294 nm with a pulse duration of 1.74 ps, a pulse repetition rate of 3.23 MHz and a maximum average output power of 2.904 mW. In the bound-state CS regime, two identical solitons form the bound-state pulses with a temporal separation of 6.1 ps, and the bound-state pulses are equally distributed at a repetition rate of 3.23 MHz, corresponding to the fundamental cavity repetition rate. The experimental results further indicate that 1 T-TiSe2 SA is competitive with the existing SAs explored so far and will promote the applications of 1 T-TiSe2-based SAs in the field of ultrafast lasers.

Soliton molecules and novel smooth positons for the complex modified KdV equation

In this research, based on Darboux transformation, a molecule consisting of two identical soliton waves is firstly obtained by velocity resonance for modified KdV equation. And we also get molecules containing a plurality of solitons. Further, we study the elastic interaction between soliton molecules and typical smooth higher-order positon via semi-degenerate Darboux transformation. Last but not least, we find a new type of smooth positons called rational positons. The dynamic properties of higher-order rational positon are discussed in detail and related propositions are given in this paper. The nature of rational positons is fundamentally different from that of typical smooth positons and breather positons. The method used in this paper to get interaction solutions and rational positons can be applied to other integrable equations as well.

Propagation characteristics of parallel dark solitons in silicon-on-insulator waveguide**Project supported by the National Natural Science Foundation of China (Grant No. 61741509).

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.

Interactions of three-dimensional solitons in the cubic-quintic model.

We report results of a systematic numerical analysis of interactions between three-dimensional (3D) fundamental solitons, performed in the framework of the nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity, combining the self-focusing and defocusing terms. The 3D NLSE with the CQ terms may be realized in terms of spatiotemporal propagation of light in nonlinear optical media, and in Bose-Einstein condensates, provided that losses may be neglected. The first part of the work addresses interactions between identical fundamental solitons, with phase shift φ between them, separated by a finite distance in the free space. The outcome strongly changes with the variation of φ: in-phase solitons with φ = 0, or with sufficiently small φ, merge into a single fundamental soliton, with weak residual oscillations in it (in contrast to the merger into a strongly oscillating breather, which is exhibited by the 1D version of the same setting), while the choice of φ = π leads to fast separation between mutually repelling solitons. At intermediate values of φ, such as φ = π/2, the interaction is repulsive too, breaking the symmetry between the initially identical fundamental solitons, there appearing two solitons with different total energies (norms). The symmetry-breaking effect is qualitatively explained, similar to how it was done previously for 1D solitons. In the second part of the work, a pair of fundamental solitons trapped in a 2D potential is considered. It is demonstrated that they may form a slowly rotating robust "molecule," if initial kicks are applied to them in opposite directions, perpendicular to the line connecting their centers.

Effect of Raman scattering on soliton interactions in optical fibers

We investigate numerically the effect of Raman scattering on the interaction of two temporally separated pulses with identical spectra that propagate inside a single-mode fiber as fundamental solitons. We take into account all interpulse Raman scattering terms in the generalized nonlinear Schrodinger equation and study the interplay between the Kerr, intrapulse Raman, and interpulse Raman effects. We observe considerable differences from the well-known two-soliton interaction behavior caused by the Kerr nonlinearity. We study in detail the mechanism for a net energy transfer from the leading pulse to the trailing pulse caused by the delayed nature of the Raman response in the case of two identical in-phase solitons. Long-range interactions, where the pulses do not temporally overlap, are found to not cause any energy transfer between the pulses, but solitons are still shown to affect each others’ phase owing to the long tail of the Raman response function. We also study and compare how a phase difference between two otherwise identical solitons affects the interaction scenario and changes the collision dynamics, both in the presence and in the absence of the Raman effect. Finally, we look at the effect of changing the relative amplitude of the two interacting solitons by a small amount.

Anomalous waves in gas-liquid mixtures near gas critical point in Gardner equation approximation

The waves in a bubbled incompressible liquid with Van der Waals gas in a bubbles being near critical points is considered in a frame of Gardner equation. It is shown that both coefficients on quadratic and cubic nonlinear terms in Gardner equation change the sign near gas critical point and it results the anomalous waves: negative and limited solitons, kinks, antikinks and breathers. The dynamics and interactions of these waves was studied numerically by high accuracy Fourier methods with periodically boundary conditions. In particular it is revealed that limited solitons always arise from initial distribution with a few identical soliton's pair and stand stable in their form after numerous interactions.

Quadrupolar matter-wave soliton in two-dimensional free space

We study two-dimensional (2D) matter-wave solitons in the mean-field models formed by electric quadrupole particles with long-range quadrupole–quadrupole interaction (QQI) in 2D free space. The existence of 2D matter-wave solitons in the free space was predicted using the 2D Gross–Pitaevskii Equation (GPE). We find that the QQI solitons have a higher mass (smaller size and higher intensity) and stronger anisotropy than the dipole–dipole interaction (DDI) solitons under the same environmental parameters. Anisotropic soliton–soliton interaction between two identical QQI solitons in 2D free space is studied. Moreover, stable anisotropic dipole solitons are observed, to our knowledge, for the first time in 2D free space under anisotropic nonlocal cubic nonlinearity.

Nature of soliton interaction in fiber lasers with continuous external optical injection

It has been shown by numerical simulation that the nonlinear interaction between a laser soliton and an injected monochromatic cw results in their phase locking. As a consequence, the velocity of the soliton begins to depend on the amplitude and frequency of the injected radiation. It has been found that if the frequency of the external signal coincides with the frequency of the dispersive waves emitted by solitons in a laser cavity with lumped intracavity elements, a mechanism for controlling long-range soliton interaction occurs. This mechanism is related to the interference between the injected wave and the dispersive waves involved in the strong long-range interaction between solitons. We have demonstrated the mechanism of soliton-soliton repulsion and, as its consequence, the occurrence of harmonic passive mode locking (multipulse generation with an equidistant arrangement of identical solitons in the laser cavity).

The superposition of algebraic solitons for the modified Korteweg–de Vries equation

Many real nonlinear evolution equations exhibiting soliton properties display a special superposition principle, where an infinite array of equally spaced, identical solitons constitutes an exact periodic solution. This arrangement is studied for the modified Korteweg–de Vries equation with positive cubic nonlinearity, which possesses algebraic solitons with nonvanishing far field conditions. An infinite sum of equally spaced, identical algebraic pulses is evaluated in closed form, and leads to a complex valued solution of the nonlinear evolution equation.

Dynamics of gap solitons in one-dimensional binary lattices with saturable self-defocusing nonlinearity and alternating spacing

The existence and stability of spatial solitons in one-dimensional binary photonic lattices with alternating spacing and a saturable defocusing type of nonlinearity are investigated. Five types of nonlinear localized structures are found to exist: two in the mini-gap in the energy spectrum and others in the regular gap. It is proved that some of them are stable in certain ranges of the system parameters. Interactions between two identical localized structures propagating parallel to each other are investigated, too. It is shown that this interaction leads to formation of different localized patterns, such as solitons, breather-like modes, and breather complexes. The interaction output depends on the power and type of interacting identical solitons, the separation between them, the width of the mini-gap, and the phase relation between the tails of interacting solitons.

Two-dimensional solitons in media with stripe-shaped nonlinearity modulation

We introduce a model of media with the cubic attractive nonlinearity concentrated along a single or double stripe in the two-dimensional (2D) plane. The model can be realized in terms of nonlinear optics (in the spatial and temporal domains alike) and BEC. It is known from recent works that search for stable 2D solitons in models with a spatially localized self-attractive nonlinearity is a challenging problem. We make use of the variational approximation (VA) and numerical methods to investigate conditions for the existence and stability of solitons in the present setting. The result crucially depends on the transverse shape of the stripe: while the rectangular profile supports stable 2D solitons, its smooth Gaussian-shaped counterpart makes all the solitons unstable. This difference is explained, in particular, by the VA. The double stripe with the rectangular profile admits stable solitons of three distinct types: symmetric and asymmetric ones with a single-peak, and double-peak symmetric solitons. The shape and stability of the single-peak solitons of either type are accurately predicted by the VA. Collisions between identical stable solitons are briefly considered too, by means of direct simulations. Depending on the collision velocity, we observe excitation of intrinsic oscillations of the solitons, or their decay, or the collapse (catastrophic self-focusing).

Interactions between dispersion-managed solitons with unequal powers

We revisit the problem of intrachannel interactions between solitons in dispersion-managed nonlinear fiber links, which is a major limiting factor for the use of solitons in fiber-optic networks. By means of systematic simulations, we demonstrate the existence of a well-defined maximum of the interaction length, zint, as a function of the ratio of peak powers of interacting solitons, while in the case of identical solitons the interaction length always attains a minimum. An explanation to these features is proposed. Interaction effects in three-soliton sets are studied too. The results suggest that the transmission characteristics of soliton streams in the dispersion-managed system may be improved by using alternating values of the peak powers whose ratio corresponds to the maximum of zint.

Collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices

Abstract Systematic results of collisions between discrete spatiotemporal dissipative Ginzburg-Landau solitons in two-dimensional photonic lattices are reported. The generic outcomes are identified for (i) the collision of two identical solitons located in the corner, at the edge, and in the center of the photonic lattice, and for (ii) the collision of two non-identical corner and edge solitons located at different distances from the boundaries of the photonic lattice. Depending on the values of the kick (collision momentum) and of the nonlinear (cubic) gain, the collision scenarios include soliton merging, creation of an extra soliton, soliton bouncing, soliton spreading, and quasi-elastic (symmetric) interactions.

Observation of High-Order Polarization-Locked Vector Solitons in a Fiber Laser

We report on the experimental observation of a new type of polarization-locked vector soliton in a passively mode-locked fiber laser. The vector soliton is characterized by the fact that not only are the two orthogonally polarized soliton components phase-locked, but also one of the components has a double-humped intensity profile. Multiple phase-locked high-order vector solitons with identical soliton parameters and harmonic mode locking of the vector solitons were also obtained in the laser. Numerical simulations confirmed the existence of stable high-order vector solitons in the fiber laser.

Multisoliton ejection from an amplifying potential trap

We study theoretically the dynamics of a beam launched inside an amplifying trap potential. Raising the amplification transforms the dynamics from linear tunneling at low amplification to periodic ejection of a sequence of identical solitons (when the amplification rate exceeds the tunneling rate) and, at strong amplification, to nonperiodic multisoliton ejection.

Deviation from one dimensionality in stationary properties and collisional dynamics of matter-wave solitons

By means of analytical and numerical methods, we study how the residual three-dimensionality affects dynamics of solitons in an attractive Bose-Einstein condensate loaded into a cigar-shaped trap. Based on an effective 1D Gross-Pitaevskii equation that includes an additional quintic self-focusing term, generated by the tight transverse confinement, we find a family of exact one-soliton solutions and demonstrate stability of the entire family, despite the possibility of collapse in the 1D equation with the quintic self-focusing nonlinearity. Simulating collisions between two solitons in the same setting, we find a critical velocity, $V_{c}$, below which merger of identical in-phase solitons is observed. Dependence of $V_{c} $ on the strength of the transverse confinement and number of atoms in the solitons is predicted by means of the perturbation theory and investigated in direct simulations. Symmetry breaking in collisions of identical solitons with a nonzero phase difference is also shown in simulations and qualitatively explained by means of an analytical approximation.