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Under investigation in this paper is a discrete variable-coefficient Ablowitz–Ladik equation, which has certain applications in the electrical and optical systems. Employing the Hirota method and symbolic computation, we obtain the dark one- and two-soliton solutions under a variable-coefficient constraint. Linear-, parabolic-, periodic- and s-shaped dark one solitons are observed: We find that the space-time-modulated inhomogeneous frequency shift only affects the velocity of the dark soliton, the coefficient of tunnel coupling between the sites only affects the amplitude of the dark soliton, the time-modulated effective gain/loss term has no effect on either the dark soliton’s velocity or amplitude, and the velocity of the dark soliton decreases as the lattice spacing increases with the amplitude unchanged. Via the asymptotic analysis, we prove that the interactions between the dark two solitons are elastic on the soliton solutions. Overtaking interactions between the linear- and parabolic-shaped dark t...

In this paper, we investigate a (2+1)-dimensional Bogoyavlenskii–Kadontsev–Petviashili equation in a fluid, plasma or ferromagnetic thin film. Through the Bell polynomials, Hirota method and symbolic computation, the one- and two-kink-soliton solutions are derived. Backlund transformation, Lax pair and conservation laws are presented. Elastic collisions including the oblique, parallel, unidirectional and bidirectional collisions between the two-kink solitons are discussed. In addition, the relation between the velocities and wave numbers of the two-kink solitons are analysed. When wave numbers bj>0,νjx, the velocities in the x axis, increase with wave numbers aj increasing. With bj increasing, νjx increase when aj4-3bj2>0, while decrease when aj4-3bj2

(2+1)-dimensional non-linear optical waves through the coherently excited resonant medium doped with the erbium atoms can be described by a (2+1)-dimensional non-linear Schrodinger equation coupled with the self-induced transparency equations. For such a system, via the Hirota method and symbolic computation, linear forms, one-, two- and N-soliton solutions are obtained. Asymptotic analysis is conducted and suggests that the interaction between the two solitons is elastic. Bright solitons are obtained for the fields E and P, while the dark ones for the field N, with E as the electric field, P as the polarization in the resonant medium induced by the electric field, and N as the population inversion profile of the dopant atoms. Head-on interaction between the bidirectional two solitons and overtaking interaction between the unidirectional two solitons are seen. Influence of the averaged natural frequency ω on the solitons are studied: (1) ω can affect the velocities of all the solitons; (2) Amplitu...

Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential, which describes the dynamics of the (2+1)-dimensional Bose–Einstein condensate. Employing the Hirota method and symbolic computation, we obtain the dark one-soliton, two-soliton, three-soliton, breather-wave and rouge-wave solutions, respectively. We graphically study the dark solitons with the time-varying harmonic potential and scaled scattering length. Parallel and period solitons are observed. We obtain that when the external trapping potential increases with time, amplitudes of the dark solitons increase and widths of those solitons become narrower; when the external trapping potential is a periodic function, amplitudes and widths of the dark solitons periodically change. Decrease in the scaled scattering length leads to the narrower solitons’ widths, but does not affect the solitons’ amplitudes. Breather waves and rouge waves are also displayed: Rouge waves emerge when the period ...

In this paper, we investigate the rogue waves for an integrable coupled nonlinear Schrodinger (CNLS) system with the self-phase modulation, cross-phase modulation and four-wave mixing term, which can describe the propagation of optical waves in a multi-mode fibre. We construct a generalized Darboux transformation (GDT) for the CNLS system and find a gauge transformation which converts the Lax pair into the constant-coefficient differential equations. Solving those equations, we can obtain the vector solutions of the Lax pair. Using the GDT, we derive an iterative formula for the nth-order rogue-wave solutions for the CNLS system. We derive the first- and second-order rogue-wave solutions for the CNLS system and analyse the profiles for the rogue waves with respect to the self-phase modulation term a, cross-phase modulation term c and four-wave mixing term b, respectively. The rogue waves become thinner with the increase in the value for the real part of b and that the effect of a or c on the rogue waves i...

The coupled equations for the incoherent optical spatial solitons in a nonlocal nonlinear medium is studied analytically. With the soliton solutions hereby obtained via the symbolic computation, the optical-soliton motion in the nonlocal nonlinear medium is studied: |ψ| is inversely related to κ, ω, and n0, while Δn is positively related to ω and n0, but Δn is independent of κ, with ψ as the slowly varying amplitude of the beam, Δn as the refractive index change, κ as the beam intensity distribution, ω as the frequency of the propagating beam, and n0 as the unperturbed refractive index. Head-on and overtaking interactions are observed, and head-on interaction is transformed into an overtaking one with ω increasing. Bound-state interaction is displayed, and with n0 increasing, the period of ψ decreases, while that of Δn increases. Considering the external forces in the nonlocal nonlinear medium, we explore the chaotic motions in the nonlinear nonlocal medium, including effects of the external forces on suc...

Under investigation in this paper is a set of the (2+1)-dimensional nonlinear Schrodinger–Maxwell–Bloch (NLS–MB) equations, which describes the optical pulse propagation in an erbium-doped fibre. Employing the Hirota method and symbolic computation, we obtain the one- two- and N-soliton solutions. We prove that the interactions between the two solitons are elastic through the asymptotic analysis on the soliton solutions. Figures are plotted to show the one soliton and the interaction between the two solitons. We find that the decrease of the frequency shift from the resonance can make the angle between the propagation directions and the amplitudes of the two MB solitons bigger, implying that the change of the amplitudes of the two MB solitons is closely related to the propagation directions of the two NLS solitons.

In this paper, a stochastic higher-order nonlinear Schrodinger equation, which describes the wave propagation in the optical fiber with stochastic dispersion and nonlinearity, is investigated analytically. Via the symbolic computation and white noise functional approach, the stochastic dark one- and two-soliton solutions are obtained, and effects of the Gaussian white noise on the stochastic dark one and two solitons are discussed. For the stochastic dark one soliton, velocity and phase change randomly because of the Gaussian white noise, but the energy, shape and amplitude keep unchanged during the soliton propagation. For the stochastic dark two solitons, effect of the Gaussian white noise leads to the inversion of the velocity directions, while the velocities have the same varying trend so that the interaction appears that the stochastic dark two solitons keep parallel.

In this paper, we study the generalized coupled nonlinear Schrodinger equations with the cubic-quintic nonlinearity and arbitrary coupling parameters which describe the effects of the quintic nonlinearity on the ultrashort optical soliton pulse propagation in the non-Kerr media. Dark–dark N- and bright–dark two-soliton solutions are derived with symbolic computation. The bilinear Backlund transformation and the corresponding one-soliton solution are also given. Interactions of the dark–dark solitons including the repulsive and coherent interactions are investigated analytically and graphically, which are different from those in previous studies. The relationship between the amplitude and the velocity of the dark soliton is studied, especially the soliton with the small amplitude catches up with the one with the large amplitude. Head-on and overtaking interactions of dark–dark solitons are presented. Interactions between a stationary dark soliton and another dark one are shown. The asymptotic analysis show...

A type of mode-locked fiber laser, which can generate the bound solitons in the anomalous group-velocity dispersion regime, is suggested in this paper via symbolic computation. A transformation is given to convert the coupled nonlinear Schrodinger equations to a simpler form. Analytic two-, three- and N-soliton solutions for the coupled nonlinear Schrodinger equations are obtained. Based on those solutions, formation of the bound solitons is analyzed, and methods for the soliton control are presented. The soliton intensity and collision period can be controlled with the changes in the nonlinearity and group-velocity dispersion of optical fibers. The slight phase shift has a great influence on the bound states of solitons after the soliton collision. To support the results, numerical simulations of three-soliton propagation are performed. Finally, the modulational instability of bound solitons is analyzed.

In this letter, under investigation is a nonlinear Schrodinger equation with varying dispersion, nonlinearity and loss for the propagation of ultra-short optical pulses in the normal dispersion regime of optical fibers. By virtue of the modified Hirota's method and symbolic computation, the analytic two-soliton solution is explicitly obtained. Both the bright and dark solitons are observed in the normal dispersion regime of optical fibers with dispersion management. An asymptotic analysis to verify the elastic collision between solitons is performed and the stability of the soliton solutions is investigated. Besides, a new bright solitonic generator for generating high-power and narrow bandwidth pulses is advised. Furthermore, possible applicable soliton control techniques which might be used for the design of optical switch and dispersion-managed systems are proposed.

In this paper, for the propagation of the ultra-short optical pulses in the normal dispersion regime of inhomogeneous optical fibers, the variable-coefficient higher-order nonlinear Schrodinger equation is investigated. A bilinear form and analytic soliton solutions are obtained with the help of the modified Hirota method and symbolic computation. Through choosing the different dispersion profiles of the inhomogeneous optical fibers, relevant properties of the soliton solution are graphically discussed. Parabolic-type evolution of the soliton is observed. Additionally, periodic and s-shaped solitons are derived, respectively. Finally, a possibly applicable compression technique for the dark soliton is proposed. The results might be of potential application to soliton control, soliton compression, signal amplification and dispersion management.

Transmission control using dispersion decreasing fibers with potential applications to the design of high-speed optical devices and long distance transmission of pulses is investigated based on solving the variable-coefficient nonlinear Schrodinger equation. By using Hirota's method, the analytic fundamental soliton solution of this model is obtained. In addition, according to this solution, the relevant properties and features of physical and optical interest are illustrated. Furthermore, the profiles of dispersion decreasing fibers are found to be able to control the pulse speed under certain conditions.

The soliton-effect pulse compression of ultrashort solitons in a dispersion-shifted fiber (DSF) is investigated based on solving the higher-order nonlinear Schrodinger equation with the effects of third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). By using Hirota's bilinear method with a set of parametric conditions, the analytic one-, two- and three-soliton solutions of this model are obtained. According to those solutions, the higher-order soliton is shown to be compressed in the DSF for the pulse with width in the range of a few picoseconds or less. An appealing feature of the soliton-effect pulse compression is that, in contrast to the second-order soliton compression due to the combined effects of negative TOD and SRS, the third-order soliton can significantly enhance the soliton compression in the DSF with small values of the group-velocity dispersion (GVD) at the operating wavelength.

A variable-coefficient nonlinear Schrodinger model is hereby investigated, which arises in the optical communication systems. A family of the exact analytic dark-soliton-like solutions is presented with computerized symbolic computation, which may be of value for future studies on the solitonic features in optical communication systems.