Soliton Amplitude Research Articles

Bright solitons in fractional coupler with spatially periodical modulated nonlinearity

We study bright solitons in the fractional coupler with a spatially periodical modulated nonlinearity. The results show that the linear coupling constant κ, Lévy index α, chemical potential μ and nonlinear intensity g have a significant influence on the amplitude, width and stability of fundamental solitons, dipole solitons and tripole solitons. We investigate the stability of bright solitons by linear stability analysis and the real-time evolution method. It is found that the bright solitons tend to be stable with the increase of linear coupling constant, Lévy index, chemical potential, and nonlinear intensity in the corresponding parameter interval. The results also show that the distance between adjacent peaks is four times of the nonlinear lattice period for dipole solitons and six times for tripole solitons. We also find that the number of soliton peaks is smaller, and the stable region is wider in the fractional coupler with the spatially periodical modulated nonlinear lattice.

The mechanism that drives electrostatic solitary waves to propagate in the Earth's magnetosphere and solar wind

AbstractThe origin of the solitary waves in the Earth's magnetosphere and the solar wind, particularly close to the magnetic reconnection (MR) regions, is still an open question. For that purpose, we used the fluid model to obtain the mechanism behind the formation of solitary waves in space plasmas with a variety of components. Our findings reveal that at the separatrix of the MR in the Earth's magnetotail, the counter‐ejected electron beam leads to a broadening of the pulse amplitude. While increasing the electron beam positive velocity causes a bit increase in the soliton amplitude. Moreover, we observed that the soliton profile is very sensitive to the plasma parameters like density and temperature. The calculated electric field using the current model and the measured electric field in the Earth's magnetosphere and solar wind is highlighted.

Bilinear form and soliton solutions for a higher order wave equation

Under investigation in this paper is a higher order wave equation of the Korteweg–de Vries type, which describes the propagation of more complex wave in the shallow water. Bilinear form is derived based on the binary Bell polynomials. One-soliton solutions are constructed via the Hirota method. Two-soliton solutions are tried to be constructed, but it degenerates to the one-soliton solutions. Influences of the coefficients α and β on the soliton velocity and amplitude are analyzed through the characteristic line.

Bifurcation, bilinear forms, conservation laws and soliton solutions of the temporal-second-order KdV equation

The temporal-second-order KdV equation, which describes the propagation of two wave modes with different phase velocities and same dispersion relation, nonlinearity and dispersion parameters are investigated. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of the phase velocities [Formula: see text] and [Formula: see text] and the parameters of nonlinearity [Formula: see text] and [Formula: see text] on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the temporal-second-order KdV equation.

New unexpected soliton solutions to the generalized (2 + 1) Schrödinger equation with its four-mixing waves

The propagation of solitons in birefringent fiber is one of the phenomena that has an important role in all modern technological means of communication. The generalized [Formula: see text] nonlinear Schrödinger equation (GNLSE) with its four-mixing waves (FMW) is the famous model that describes the propagation of solitons in birefringent fiber perfectly. In fact, the FMW governed effectively the performance of the resultant solitons amplitude. Hereby, we will study this model to extract new unexpected optical soliton solutions to this model via various three techniques. The three famous methods that are a candidate for this purpose are the extended direct algebraic method (EDAM), the extended simple equation method (ESEM) and the solitary wave ansatz method (SWAM). The three techniques are implemented successively for the suggested model to establish the optical solutions of the suggested model successfully. The optical soliton solutions that are achieved by these proposed techniques give surprise expectations that weren’t achieved previously by any other authors who used other techniques.

Controllable storage and retrieval of optical solitons in triple quantum dot molecules by inter-dot tunneling coupling effect

We study the storage and retrieval of the optical field in an aligned asymmetric triple quantum dot molecule with both sides inter-dot tunneling coupling effect. It is shown that the optical solitons can be stored and retrieved by manipulating two inter-dot tunneling coupling effect, different from light memory in the ultra-cold atom system. Furthermore, the amplitude of the optical soliton emerges a trend of first decreasing and then increasing with the increasing strength of the two inter-dot tunneling coupling. It is possibility to improve the fidelity of the storage and retrieval of the optical information in semiconductor quantum dots devices.

Optical and analytical soliton solutions to higher order non-Kerr nonlinear Schrödinger dynamical model

This article studies the propagation of the short pulse optical model governed by higher order nonlinear Schrödinger equation (HNLSE) with non-Kerr nonlinearity. The model is used to describe the propagation of ultrashort photons in highly nonlinear media. Upon establishing the general framework, we discuss the statics and dynamics of HNLSE by employing an extended modified auxiliary equation mapping (EMAEM) architectonic to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Obtained optical soliton solutions are analysed graphically to represent features like as width, amplitude, and structure of solitons. We will also discuss our governing model for M-shaped, Homoclinic breathers, multiwave, kink cross rational, and interaction of M-shaped with 1 and 2 kink solutions with the help of various ansatz transformations.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas

We propose a scheme to realize a parity-time (PT) symmetric nonlinear system in a coherent atomic gas via electromagnetically induced transparency. We show that it is possible to construct an optical potential with PT symmetry due to the interplay among the Kerr nonlinearity stemmed from the atom-photon interaction, the linear potential induced by a far-detuned Stark laser field, and the optical gain originated from an incoherent pumping. Since the real part of the PT-symmetric potential depends only on the intensity of the probe field, the potential is nonlinear and its PT-symmetric properties are determined by the input laser intensity of the probe field. Moreover, we obtain the fundamental soliton solutions of the system and attain their stability region in the system parameter space. The dependence of the exceptional point (EP) location on the soliton maximum amplitude is also illustrated. The research results reported here open a new avenue for understanding the unique properties of PT symmetry of a nonlinear system. They are also promising for designing novel optical devices applicable in optical information processing and transmission.

Manipulation of Kerr cavity solitons based on projected super-position technique

We report numerical simulation on the Kerr cavity solitons (CSs) manipulation based on projected super-position technique. The CSs are generated in a fiber-based Kerr resonator. The positions and amplitudes of these CSs can be adjusted by setting intra-cavity parameters. By introducing a polarization controller (PC) and an inline polarization beam splitter (PBS) into the system, the CSs can be projected, superposed, and manipulated flexibly to operate with controllable soliton positions and amplitudes. Moreover, depending on the time separation between two polarization components, the number of solitons after projection would change based on different type of CSs.

Pattern Formation and Aggregation in Ensembles of Solitons in Quasi One-Dimensional Electronic Systems

Broken symmetries of quasi one-dimensional electronic systems give rise to microscopic solitons taking roles of carriers of the charge or spin. The double degeneracy gives rise to solitons as kinks of the scalar order parameter A; the continuous degeneracy for the complex order parameter Aexp(iθ) gives rise to phase vortices, amplitudes solitons, and their combinations. These degrees of freedom can be controlled or accessed independently via either the spin polarization or the charge doping. The long-range ordering in dimensions above one imposes super-long-range confinement forces upon the solitons, leading to a sequence of phase transitions in their ensembles. The higher-temperature T transition enforces the confinement of solitons into topologically bound complexes: pairs of kinks or the amplitude solitons dressed by exotic half-integer vortices. At a second lower T transition, the solitons aggregate into rods of bi-kinks or into walls of amplitude solitons terminated by rings of half-integer vortices. With lowering T, the walls multiply, passing sequentially across the sample. Here, we summarize results of a numerical modeling for different symmetries, for charged and neutral soliton, in two and three dimensions. The efficient Monte Carlo algorithm, preserving the number of solitons, was employed which substantially facilitates the calculations, allowing to extend them to the three-dimensional case and to include the long-range Coulomb interactions.

Completely localized solitons and their stabilities in magnetized dusty plasma of trapped ions

We numerically and theoretically investigated the completely localized solitons, obtained by the Petviashvili method, and their dynamical stabilities in a magnetized dusty plasma with trapped ions. The results suggest that its amplitudes are proportional to the square of its speed and inversely proportional to the square of the nonlinear interaction strength, which are also confirmed analytically. The dependence of the soliton amplitudes on various physical parameters is investigated systematically. Numerical results indicate that the localized solitons are always dynamically stable. When two localized solitons collide, their amplitudes and phase are nearly invariant. However, if a stable localized soliton collides with an unstable line soliton, the latter will evolve into a series of completely localized solitons.

Dust–ion acoustic solitary waves in a collisionless magnetized five components plasma

Abstract We have derived a Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation to study the nonlinear behavior of dust–ion acoustic waves in a collisionless magnetized five components dusty plasma consisting of warm adiabatic ions, nonthermal hot electrons, isothermal cold electrons, nonthermal positrons and static negatively charged dust particulates. It is found that the coefficient of the nonlinear term of the KdV-ZK equation vanishes along different family of curves in different compositional parameter planes. In this situation, to describe the nonlinear behavior of dust–ion acoustic waves, we have derived a modified KdV-ZK (MKdV-ZK) equation. When the coefficients of the nonlinear terms of both KdV-ZK and MKdV-ZK equations are simultaneously equal to zero, then we have derived a further modified KdV-ZK (FMKdV-ZK) equation which effectively describes the nonlinear behavior of dust–ion acoustic waves. Analytically and numerically, we have investigated the solitary wave solutions of different evolution equations propagating obliquely to the direction of the external static uniform magnetic field. We have seen that the amplitude of the KdV soliton strictly increases with increasing β e, whereas the amplitude of the MKdV soliton strictly decreases with increasing β e, where β e is the nonthermal parameter associated with the hot electron species. Also, there exists a critical value β r ( c ) ${\beta }_{\text{r}}^{(\text{c})}$ of β e such that the FMKdV soliton exists within the interval β r ( c ) < β e ≤ 4 7 ${\beta }_{\text{r}}^{(\text{c})}< {\beta }_{\text{e}}\le \frac{4}{7}$ , whereas the FMKdV soliton does not exist within the interval 0 < β e < β r ( c ) $0< {\beta }_{\text{e}}< {\beta }_{\text{r}}^{(\text{c})}$ . We have also discussed the effect of different parameters of the system on solitary waves obtained from the different evolution equations.

Singular soliton, shock-wave, breather-stripe soliton, hybrid solutions and numerical simulations for a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system in fluid mechanics

In this paper, a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system is investigated in fluid mechanics via the symbolic computation. With the help of the Hirota method, we derive some singular soliton, shock-wave, breather-stripe soliton and hybrid solutions. Based on the finite difference method, we get some numerical one-soliton solutions. We graphically show the singular and shock-wave solutions, and observe that the singular one-soliton solutions are explosive and unstable, but the shock-wave solutions are non-singular and stable. We observe that the breather-stripe soliton moves along the negative direction of the y axis, where y is a variable, and the amplitude and shape of the breather-stripe soliton remain invariant during the propagation. We graphically demonstrate the interaction among a rogue wave, a periodic wave and a pair of the stripe solitons: the rogue wave arises from the one stripe soliton; the rogue wave interacts with the periodic wave, the rogue wave splits into two waves and then the two waves merge into a wave; the rogue wave fuses with the other stripe soliton. We graphically present the numerical one-soliton solutions which agree with the analytic one-soliton solutions.

Quantum squeezing of slow-light dark solitons via electromagnetically induced transparency

We consider the quantum effect of slow light dark soliton (SLDS) in a cold atomic gas with defocuing Kerr nonlinearity via electromagnetically induced transparency (EIT). We calculate the quantum fluctuations of the SLDS by solving the relevant non-Hermitian eigenvalue problem describing the quantum fluctuations, and find that only one zero mode is allowed. This is different from the quantum fluctuations of bright solitons, where two independent zero modes occur. We rigorously prove that the eigenmodes, which consist of continuous modes and the zero mode, are bi-orthogonal and constitute a complete bi-orthonormalized basis, useful for the calculation on the quantum fluctuations of the SLDS. We demonstrate that, due to the large Kerr nonlinearity contributed from the EIT effect, a significant quantum squeezing of the SLDS can be realized; the squeezing efficiency can be manipulated by the Kerr nonlinearity and the soliton's amplitude, which can be much higher than that of bright solitons. Our work contributes to efforts for developing quantum nonlinear optics and non-Hermitian Physics, and for possible applications in quantum information processing and precision measurements.

Nonlinear Fourier transform assisted high-order soliton characterization

Nonlinear Fourier transform (NFT), based on the nonlinear Schrödinger equation, is implemented for the description of soliton propagation, and in particular focused on propagation of high-order solitons. In nonlinear frequency domain, a high-order soliton has multiple eigenvalues depending on the soliton amplitude and pulse-width. During the propagation along the standard single mode fiber (SSMF), their eigenvalues remain constant, while the corresponding discrete spectrum rotates along with the SSMF transmission. Consequently, we can distinguish the soliton order based on its eigenvalues. Meanwhile, the discrete spectrum rotation period is consistent with the temporal evolution period of the high-order solitons. The discrete spectrum contains nearly 99.99% energy of a soliton pulse. After inverse-NFT on discrete spectrum, soliton pulse can be reconstructed, illustrating that the eigenvalues can be used to characterize soliton pulse with good accuracy. This work shows that soliton characteristics can be well described in the nonlinear frequency domain. Moreover, as a significant supplement to the existing means of characterizing soliton pulses, NFT is expected to be another fundamental optical processing method besides an oscilloscope (measuring pulse time domain information) and a spectrometer (measuring pulse frequency domain information).

Comparative analysis of bore propagation over long distances using conventional linear and KdV-based nonlinear Fourier transform

In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The experimental flume is extended numerically to an effective relative length of x/h=3000, which allows all far-field solitons to emerge from the undular bore in the simulation data. We apply the periodic KdV-based nonlinear Fourier transform (KdV-NFT) to the time series taken at different numerical gauges and compare the results with those of the conventional Fourier transform. We find that the periodic KdV-NFT reliably predicts the number and the amplitudes of all far-field solitons from the near-field data long before the solitons start to emerge from the bore, even though the propagation is only approximated by the KdV. It is the first time that the predictions of the KdV-NFT are demonstrated over such long distances in a realistic set-up. In contrast, the conventional linear FT is unable to reveal the hidden solitons in the bore. We repeat our analyses using space instead of time series to investigate whether the space or time version of the KdV provides better predictions. Finally, we show how stepwise superposition of the determined solitons, including the nonlinear interactions between individual solitons, returns the analysed initial bore data.

N-Fold Darboux Transformation and Soliton Solutions for The relativistic Toda Lattice Equation

Under investigation in this paper is the relativistic Toda lattice (RTL) equation which is a deformation of the Toda lattice (TL) equation and may simulate many physical phenomena. The N -fold Darboux transformation (DT) of the RTL equation is constructed in terms of determinants. Compared with the usual 1-fold DT, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. Based on the N-fold DT, we obtain the N-fold explicit solutions from initial solutions. The figures of one-, two-, three-and four-soliton solutions with proper parameters are presented to illustrate the propagation of solitary waves, and elastic interactions between or among the two-, three- and four-soliton solutions are discussed: solitonic shapes and amplitudes have not changed after the interaction. What is more, the relationship between the structures of solutions and the parameters is generated with N = 1, from which we find that the 1-fold solutions may be one-soliton solutions or periodic solutions. Results in this paper might be helpful for understanding and interpreting some physical phenomena.

Dissipative solitons of the nonlinear fractional Schrödinger equation with PT-symmetric potential

We study the fractional dissipative solitons with nonlinear loss and linear gain supported by a special class of PT-symmetric potential in focused nonlinear Kerr medium. The effective width, existence and stability of fractional solitons are discussed. In such a potential, the stability of solitons changes at lower energy. Once the stability of symmetric solitons is destroyed, asymmetric solitons and symmetric solitons coexist. With the increase of linear gain coefficient, the effective width decreases first and then increases. With the increase of Lévy index, the amplitude of solitons decreases. In addition, the change of nonlinear loss coefficient has almost no effect on the soliton contour.

New Optical Solitons with Variational Principle and Collective Variable Strategy for Cold Bosons in Zig‐Zag Optical Lattices

The theme of this piece of research is to investigate the collective variable (CV) as well as semi‐inverse techniques to explore a significant model of cold bosonic atoms in a zig‐zag optical lattice. The system is reduced to an important equation by utilizing the continuum approximation and explains the soliton’s dynamics in sense of pulse variables. These parameters are amplitude, temporal position, chirp, width, frequency, and phase which are termed as collective variables (CVs). The proposed methods are more straightforward, succinct, accurate, and simple to calculate. Furthermore, to employ the computational counterfeit on the system of six ordinary differential equations that denote all the CVs incorporated in the supposed ansatz, a well‐established computational method which is the Runge–Kutta scheme of order four is applied. The CV method is exerted to resolve the evolution of pulse parameters with the propagation distance and graphical illustrations which are also given. Moreover, figures reveal the fascinating periodic oscillations of frequency, width, amplitude, and chirp of soliton. Solitons and their numerical behavior to interpret fluctuations in CVs are presented for several values of super‐Gaussian pulse parameters. Also, the results for the semi‐inverse method which are bright solitons are provided in the form of 2D, 3D, and density plots for the distinct values of the fractal parameter to understand their physical significance. This scheme is effective in finding variational principles of various nonlinear evolution equations. Some compelling characteristics pertaining to the current scrutiny are also deduced.