Second-order Solitons Research Articles

Lead Sulfide Saturable Absorber Based Passively Mode-Locked Tm-Doped Fiber Laser

In this paper, a passively mode-locked Tm-doped fiber laser by employing lead sulfide (PbS) nanoparticles as the saturable absorber (SA) is successfully demonstrated in the 2 μm region for the first time, to the best of our knowledge. Measured by a home-made balanced twin-detector setup at 2 μm, the PbS SA was characterized by the modulation depth of 10.69%, the non-saturable loss of 74.27%, and the saturable peak intensity of 8.62 MW/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively. The laser delivered stable conventional soliton with a pulse duration of 1.24 ps and a repetition rate of 21.93 MHz. The center wavelength and 3 dB bandwidth are 1957.37 nm and 3.43 nm, respectively. Additionally, the second-order harmonic soliton pulses with a repetition rate of 43.86 MHz were also observed by further increasing the pump power. Our work reveals that PbS material is a reliable SA applied for pulse generation in the 2 μm spectral region.

Compact all-fiber 2.1-2.7 μm tunable Raman soliton source based on germania-core fiber.

Although ultrafast rare-earth-doped fiber lasers mode-locked at near-infrared and ∼3 μm wavelengths have been well developed, it is relatively difficult to achieve ultrafast fiber laser emitting in the 2.1-2.7 μm spectral gap between ∼2 μm (Tm fiber) and ∼2.8 μm (Er or Ho fluoride fiber). In this paper, we report the generation of 2.1-2.7 μm tunable femtosecond Raman solitons from a compact fusion-spliced all-fiber system using a home-made 1.96 μm ultrafast pump source and a MIR-available germania-core fiber. At first, a Tm-doped double-clad fiber amplifier is used to not only boost up the power of 1957 nm femtosecond seed laser, but also to generate the first-order soliton self-frequency shift (SSFS). The first-order Raman solitons can be tuned from 2.036 to 2.152 μm, have a pulse duration of ∼480 fs and can reach a pulse energy of 1.07 nJ. The first-order Raman solitons are further injected into a 94 mol.% germania-core fiber to excite the second-order SSFS. The second-order solitons can be tuned to longer wavelengths, i.e. from 2.157 μm up to 2.690 μm. Our work could provide an effective way to develop compact, all-fiber ultrafast MIR laser sources with the continuous wavelength tuning of 2.1-2.7 μm.

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.

The Nth-order bright and dark solitons for the higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrödinger equation, which can describe the attosecond pulses in an optical fiber. The Nth-order bright and dark soliton solutions are derived via the Hirota method. We demonstrate that the interaction between the two solitons is elastic unless the solitons propagate with the same velocities. Meanwhile, mathematical expressions of the amplitude, velocity and phase shift of each soliton are given under five cases. We reveal that two interactive bright solitons can evolve to the parallel-state solitons or bound solitons when the two solitons have the same velocities while the interaction of the dark solitons can not generate the bound solitons. Furthermore, if two branches of the second-order solitons possess the identical wave numbers and angular frequencies, the second-order solitons will degenerate into the first-order soliton.

Rogue Waves and Lump Solitons of the (3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves**Supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023, and 11471050, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for

In this paper, the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order, higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Active control of adiabatic soliton fission by external dispersive wave at optical event horizon

We show that the group-velocity-led optical event horizon (OEH) in optical fibers provides a convenient way to actively control the propagation property of higher-order solitons by a comparatively weak dispersive wave (DW) pulse. It has been found numerically that clean soliton breakup, a process by which a second-order soliton completely splits into a pair of constituent solitons with vastly different power proportions after interacting with the weak DW pulse, will occur while external DWs become polychromatic. The temporal separation between both constituent solitons can be controlled by adjusting the power of the external DW. The more energetic main soliton is advanced/trailed in time depending on the selected frequency of input DW pulse. We have developed an analytic formalism describing the external acting-force (AF) perturbation. These results provide a fundamental explanation and physical scaling of optical pulse evolution in optical fibers and can find applications in improved supercontinuum sources.

Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model

In this paper, via the generalized Darboux transformation, rational soliton solutions are derived for the parity-time-symmetric nonlocal nonlinear Schr\"{o}dinger (NLS) model with the defocusing-type nonlinearity. We find that the first-order solution can exhibit the elastic interactions of rational antidark-antidark, dark-antidark, and antidark-dark soliton pairs on a continuous wave background, but there is no phase shift for the interacting solitons. Also, we discuss the degenerate case in which only one rational dark or antidark soliton survives. Moreover, we reveal that the second-order rational solution displays the interactions between two solitons with combined-peak-valley structures in the near-field regions, but each interacting soliton vanishes or evolves into a rational dark or antidark soliton as $|z|\ra \infty$. In addition, we numerically examine the stability of the first- and second-order rational soliton solutions.

Nonautonomous solitons, breathers and rogue waves for the cubic-quintic nonlinear Schrödinger equation in an inhomogeneous optical fibre

Under investigation in this paper is a cubic-quintic nonlinear Schrödinger equation which can describe the propagation of ultrashort pulses in an inhomogeneous optical fibre. Lax pair and conservation laws are constructed from which the integrability of the equation can be verified. Through a gauge transformation, spectral problem for the equation is converted to an Ablowitz–Kaup–Newell–Segur spectral problem. Nonautonomous solitons and breathers are derived based on the Darboux transformation (DT), while nonautonomous rogue waves are obtained via the generalized DT. Influence of the group-velocity dispersion, gain-or-loss coefficient and group-velocity coefficient on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves is also discussed. The gain-or-loss coefficient influences the amplitude of nonautonomous soliton. The characteristic line and velocity for the nonautonomous soliton are dependent on the group-velocity dispersion and group-velocity coefficient, but independent of the gain-or-loss coefficient. For the second-order nonautonomous soliton, if the two spectral parameters have the same real part, the bound-state structure can be formed, and the intensity becomes amplified in the attractive procedure. There exist two types of the nonautonomous breathers. Quasi-periods for the nonautonomous breathers are given, and effects of those coefficients on the quasi-periods are also discussed: The group-velocity dispersion can influence the trajectories of the nonautonomous breathers and rogue waves, while the gain-or-loss coefficient affects the backgrounds for the nonautonomous breathers and rogue waves

Nonautonomous solitons and Wronskian solutions for the (3+1)-dimensional variable-coefficient forced Kadomtsev–Petviashvili equation in the fluid or plasma

Under investigation in this paper is a ( 3 + 1 )-dimensional variable-coefficient forced Kadomtsev–Petviashvili equation which can describe the nonautonomous solitons in such areas as fluids and plasmas. The first- and second-order nonautonomous solitons are constructed via the Hirota bilinear method. Propagation and interaction of the nonautonomous solitons are analyzed. Perturbation coefficient affects the amplitude of the nonautonomous soliton. The background where the nonautonomous soliton exists can be influenced by the external force coefficient. Breathers and resonant interaction, which are the special interaction structures for the second-order nonautonomous solitons, are also presented. Nonuniformity coefficient influences the period of the breather. For the resonant interaction, the two nonautonomous solitons merge into a single solitary wave and form three branches, the amplitudes of which are influenced by the perturbation coefficient. Solutions in terms of the Wronskian determinants are constructed and verified via the direct substitution into the bilinear form.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

Interaction of high-order solitons with external dispersive waves.

The effect of mutual interaction between second-order soliton and dispersive waves (DWs) is investigated. It is predicted analytically and confirmed numerically that DWs (both transmitted and reflected components) become polychromatic after interaction with the soliton. Collision with DWs of considerable intensity can lead to acceleration/deceleration and central frequency shift of the soliton, while still preserving the soliton's oscillating structure. Two second-order solitons with resonant DWs trapped between them can form an effective solitonic cavity with "flat" or "concave mirrors," depending on the intensity of the input.

Breather solutions of the generalized nonlinear Schrödinger equation with spatially modulated parameters and a special external potential

By applying the similarity transformation we find analytical solutions of the generalized nonlinear Schrodinger equation with spatially modulated parameters and a special external potential in the form of periodic breathing nonlinear wave packets. These breathers are built by the products of parabolic-cylinder functions and the one-soliton solutions with the continuous-wave background or the second-order soliton solutions of the standard nonlinear Schrodinger equation with constant coefficients. Some examples of such composed solutions possessing applicative potential are given, in which the breathers appear in different forms. The numerical simulations are performed to compare the numerical solution with the analytical solution and to confirm the stability of the breathers found.

The Interaction of Peregrine Solitons

We theoretically investigate the interaction of two in-phase and out-of-phase Peregrine solitons in a Kerr nonlinear medium, addressing both the cases of first- and second-order solitons. Upon adjusting the interval between the solitons, their interactions exhibit different properties. If the interval is sufficiently large, two Peregrine solitons will propagate individually and will not interact each other. However, if the interval is not very large, the Peregrine solitons will strongly interact and display varying behavior.

Generalized Darboux transformation and rational soliton solutions for Chen–Lee–Liu equation

In this paper, the Chen–Lee–Liu (C–L–L) equation is investigated by the Darboux transformation (DT) method. A specific construction of the N-fold DT for C–L–L equation is derived in a simple way. The form of the N-fold DT is a matrix polynomial and each element of the matrix can be expressed by a ratio of two determinants. Furthermore, by choosing suitable eigenvalues and eigenfunctions under the reduction conditions, we can obtain the determinant solution of the reduced C–L–L equation. Moreover, the generalized DT (gDT) for C–L–L equation is also constructed through the limiting technique. As applications of the gDT method, the first and the second-order rational solitons with vanished background (VBC) and non-vanished background (NVBC) from different seeds are calculated for C–L–L equation. We hope our results can be realized by experiments in plasma physics and optical fibers.

Observation of surface dark solitons in nonlocal nonlinear media

We investigated surface dark solitons (SDSs) at the interface between a self-defocusing nonlocal nonlinear medium and a linear medium, both theoretically and experimentally. We demonstrate that fundamental and higher-order SDSs can exist when the linear refractive index of the self-defocusing medium is much greater than that of the linear medium. The fundamental and second-order solitons are observed at the interface between air and a weakly absorbing liquid.

Rotation-induced nonlinear wavepackets in internal waves

The long time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual formation of a localised wavepacket. Here this initial value problem is considered within the context of the Ostrovsky, or the rotation-modified Korteweg-de Vries (KdV), equation and a numerical method for obtaining accurate wavepacket solutions is presented. The flow evolutions are described in the regimes of relatively-strong and relatively-weak rotational effects. When rotational effects are relatively strong a second-order soliton solution of the nonlinear Schrödinger equation accurately predicts the shape, and phase and group velocities of the numerically determined wavepackets. It is suggested that these solitons may form from a local Benjamin-Feir instability in the inertia-gravity wave-train radiated when a KdV solitary wave rapidly adjusts to the presence of strong rotation. When rotational effects are relatively weak the initial KdV solitary wave remains coherent longer, decaying only slowly due to weak radiation and modulational instability is no longer relevant. Wavepacket solutions in this regime appear to consist of a modulated KdV soliton wavetrain propagating on a slowly varying background of finite extent.

On the amplification of unchirped soliton pulses in a dispersion-decreasing fiber

In this paper, based on a variable-coefficient nonlinear Schrodinger (vcNLS) equation, amplification of the fundamental and second-order unchirped solitons in the dispersion-decreasing fiber without any external amplification device, which is different from those in the existing literatures, is studied. Via symbolic computation, soliton solutions of the vcNLS equation are obtained. For a fundamental-soliton pulse, the amplitude is amplified by the gain during the propagation, whereas the width keeps unchanged. Because of the equilibrium between the gain, nonlinearity and varying dispersion, soliton structure is not destroyed, and the amplified fundamental soliton is free from the pedestal and chirp. With the increase of the absolute value of the gain coefficient $$\alpha $$ , magnification of the fundamental-soliton amplitude is enhanced in the same propagation distance. For the second-order soliton, the width is compressed and the amplitude is amplified, because the amplification process is accompanied by the compression of the soliton. Period of the second-order soliton decreases exponentially during the propagation, and decreases with the increase of the absolute value of $$\alpha $$ in the same propagation distance.

Nonlocal surface dark solitons and their stability analysis

Due to their future applications in optical communication, nonlocal dark solitons in bulk medium and surface bright solitons have received much attention recently. However, nonlocal surface dark solitons have not been investigated till now. In this paper, 1+1 dimensional nonlocal fundamental and second-order surface dark solitons have been found numerically at the interface between thermal nonlinear medium and linear medium. The relation between the wave shape of nonlocal surface dark soliton and propagation constant and nonlocality degree is investigated. Moreover, the stability of them is analyzed theoretically. The numerical simulation results show that 1+1 dimensional nonlocal fundamental surface dark Solitons are always stable in the domain of their existence, while second-order surface dark solitons are oscillatorily unstable and the width of unstable domain depends more greatly on propagation constant than nonlocality degree of nonlocal nonlinear medium. The figure showing the propagation, with the initial input of noise added, confirms the correctness of stability analysis results.

Cascaded second-order surface plasmon solitons due to intrinsic metal nonlinearity

We theoretically show the existence of cascaded second-order surface plasmon solitons propagating at the interface between a metal and a linear dielectric. Non-local multipole nonlinearities originating from the free conduction electron plasma of the metal lead to strong interaction between co-propagating surface plasmon polariton beams at the fundamental and second-harmonic frequencies. Finite element numerical modelling for an effective two-dimensional medium explicitly demonstrates soliton formation, confirming the theoretical results. The non-diffractive regime of propagation has been demonstrated at a silica/silver interface for 5λ-wide surface plasmon polariton beams with the loss-limited propagation distance of the order of 100 μm for the 750/1550 nm wavelength pair. Plasmon–soliton formation in phase-matched conditions has been shown to be beneficial for non-diffractive surface plasmon polariton propagation.

Nonlocal dark soliton and its linear stability analysis

Due to its future use in communication area, nonlocal spatial optical soliton has been a hot research topic recently. However, because of its special border condition, little research has been done on spatial dark solitons especially on its linear stability. In this paper, a method to analyze linear stability of nonlocal spatial dark soliton is put forward, moreover a numerical simulation and analysis is done on the linear stability of (1+1)-dimensional fundamental and second-order dark soliton in thermal nonlinear medium. Numerical results show that (1+1)-dimensional fundamental nonlocal dark solitons are always stable in their entire existence domain, while second-order dark solitons are oscillatorily unstable and the width of unstable domain depends on propagation constant and nonlocality degree of thermal nonlinear medium. The propagation graphs of initial input with noise addition confirms the correctness of linear stability analysis results.