Coupled Nonlinear Schrodinger Equations Research Articles

Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

In this article, truncated M-fractional coupled nonlinear Schrodinger equation (NLSE) with quadratic–cubic nonlinearity is under observation. The studied model is composed of chromatic dispersion, magneto-optic parameter and inter-modal dispersion. The NLSE is the most significant physical model to explain the fluctuations of optical soliton proliferation. The NLSEs have become more popular because of the clarity with which they explain a wide range of complex physical phenomena and the depth with which they display dynamical patterns via localized wave solutions. Optical soliton propagation in magneto-optic is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. The optical solitons are secured in the forms of bright, dark, singular and combo solitons. In addition, hyperbolic, periodic and exponential function solutions have been recovered. The modified Sardar subequation and enhanced modified extended tanh-expansion approaches recently developed integration tools are adopted in this study for securing the solutions. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The effect of parameters have been observed by allotting suitable values and sketching the different shapes of the graphs.

Novel solution structures localized in fractional integrable coupled nonlinear Schrödinger equations

Fractional calculus highlights its importance and has been expanded to many fields. This paper focuses on fractional integrable systems and their novel structural solutions to comply with the forefront development of nonlinear waves. Specifically, a system of fractional integrable coupled nonlinear Schrödinger (CNLS) equations are first derived from the associated linear spectral problem equipped with space-time conformable fractional derivatives of different orders. Then, by extending Darboux transformation (DT) and its generalized version to the derived fractional CNLS equations, we obtained fractional single-soliton solutions and double- and triple-semirational solutions that obey the dual power laws of independent variables. Under the dominant influences of space-time fractional orders, some novel solution structures are finally revealed based on the obtained solutions. In terms of characteristics, these spatial structures exhibit inclination, symmetry and spikes. Observing from the perspective of propagation, some of these obtained solutions have time-varying velocities and widths. From the perspective of spatiotemporal structures, some solutions form collapsed quadrangular pyramids. All of these novel solution structures cannot be supported by the integer-order CNLS equations.

Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas

An asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct (carrier) wavenumbers (k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}) and amplitudes (Ψ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _1$$\\end{document} and Ψ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _2$$\\end{document}) are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-distributed electron background in one dimension. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the two carrier wavenumbers. These coefficients present no symmetry whatsoever, in the general case (of different wavenumbers). The possibility for coupled envelope (vector soliton) solutions to occur has been investigated. Although the CNLS equations are asymmetric and non-integrable, in principle, the system admits various types of vector soliton solutions, physically representing nonlinear, localized electrostatic plasma modes, whose areas of existence is calculated on the wavenumbers’ parameter plane. The possibility for either bright (B) or dark (D) type excitations for either of the (2) waves provides four (4) combinations for the envelope pair (BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the soliton parameters (maximum amplitude, width) are also calculated for each type of vector soliton solution, in its respective area of existence. The dependence of the vector soliton characteristics on the (two) carrier wavenumbers and on the spectral index κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} characterizing the electron distribution has been explored. In certain cases, the (envelope) amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among copropagating wavepackets. As κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} decreases from large values, modulational instability occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of vector solitons on the parameter plane(s) also varies significantly with decreasing κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}, and in fact dramatically for κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} between 3 and 2. Deviation from the Maxwell-Boltzmann picture therefore seems to favor modulational instability as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

Coupled nonlinear Schrödinger (CNLS) equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

The nonlinear dynamics of two co-propagating electrostatic wavepackets in a one-dimensional non-magnetized plasma fluid model is considered, from first principles. The coupled waves are characterized by different (carrier) wavenumbers and amplitudes. A plasma consisting of non-thermalized (κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-distributed) electrons evolving against a cold (stationary) ion background is considered. The original model is reduced, by means of a multiple-scale perturbation method, to a pair of coupled nonlinear Schrödinger (CNLS) equations for the dynamics of the wavepacket envelopes. For arbitrary wavenumbers, the resulting CNLS equations exhibit no known symmetry and thus intrinsically differ from the Manakov system, in general. Exact analytical expressions have been derived for the dispersion, self-modulation (nonlinearity) and cross-modulation (coupling) coefficients involved in the CNLS equations, as functions of the wavenumbers (k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document}, k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}) and of the spectral index κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} characterizing the electron profile. An analytical investigation has thus been carried out of the modulational instability (MI) properties of this pair of wavepackets, focusing on the role of the intrinsic (variable) parameters. Modulational instability is shown to occur in most parts of the parameter space. The instability window(s) and the corresponding growth rate are calculated numerically in a number of case studies. Two-wave interaction favors MI by extending its range of occurrence and by enhancing its growth rate. Growth rate patterns obtained for different κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} index (values) suggest that deviation from thermal (Maxwellian) equilibrium, for low κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} values, leads to enhance MI of the interacting wave pair. To the best of our knowledge, the dynamics of two co-propagating wavepackets in a plasma described by a fluid model with κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-distributed electrons is investigated thoroughly with respect to their MI properties as a function of κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} for the first time, in the framework of an asymmetric CNLS system whose coefficients present no obvious symmetries for arbitrary k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}. Although we have focused on electrostatic wavepacket propagation in non-thermal (non-Maxwellian) plasma, the results of this study are generic and may be used as basis to model energy localization in nonlinear optics, in hydrodynamics or in dispersive media with Kerr-type nonlinearities where MI is relevant.

Novel Liouville integrable Hamiltonian models with six components and three signs

This paper aims at constructing a six-component integrable hierarchy associated with a matrix spatial spectral problem with six potentials and three signs. The zero curvature formulation and the trace identity are used to generate integrable models and their Hamiltonian structures, respectively. Two expository examples of integrable models of lower orders are six-component integrable coupled nonlinear Schrödinger (NLS) equations and modified Korteweg–de Vries (mKdV) equations. The motivation of this study is to explore typical integrable coupled NLS equations and mKdV equations, and the innovative idea and main advance is to introduce a specific matrix spectral problem involving three signs to construct integrable coupled equations.

Modulational electrostatic wave–wave interactions in plasma fluids modeled by asymmetric coupled nonlinear Schrödinger (CNLS) equations

The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid evolving against a thermalized (Maxwell–Boltzmann distributed) electron background. A multiple-scale perturbation method is employed to reduce the original model equations to a pair of coupled nonlinear Schrödinger (CNLS) equations governing the dynamics of the wavepacket amplitudes (envelopes). The CNLS equations are in general asymmetric for arbitrary carrier wavenumbers. Similar CNLS systems have been derived in the past in various physical contexts, and were found to support soliton, breather, and rogue wave solutions, among others. A detailed stability analysis reveals that modulational instability (MI) is possible in a wide range of values in the parameter space. The instability window and the corresponding growth rate are determined, considering different case studies, and their dependence on the carrier and the perturbation wavenumber is investigated from first principles. Wave–wave coupling is shown to favor MI occurrence by extending its range of occurrence and by enhancing its growth rate. Our findings generalize previously known results usually associated with symmetric NLS equations in nonlinear optics, though taking into account the difference between the different envelope wavenumbers and thus group velocities.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

A split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger equations

In this paper, we propose a split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger (CNLS) equations. The space fractional CNLS equations can be split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly. The Riesz space fractional derivative is approximated by a Fourier pseudo-spectral method. The unconditional stability, convergence, discrete charge and multi-symplectic preserving properties for the proposed method are investigated. Then, the proposed method is extended for solving the two-dimensional problem. Finally, some numerical experiments are performed to confirm our theoretical analysis and illustrate the efficiency of the proposed scheme.

Theoretical investigations of weakly- and strongly-coupled multi-core fibers for the applications of optical submarine communications under power and fiber count limits.

The practical cable design for optical submarine communications has a limited fiber pair count due to the mechanical considerations of cable weight and size. Consequently, multi-core fibers (MCFs) could exhibit higher capacity than conventional single-mode fibers (SMFs) thanks to space division multiplexing (SDM). That is because the power supply to a submarine cable is fed by the voltage difference between shores. Under the power-limited condition, SDM improves the cable capacity by using more paths which outperforms the SMF link whose capacity approximately complies with a logarithmic relationship to optical power. At the same time, fiber nonlinearity can be alleviated by the reduced power density of transmitted light in MCFs, due to the increased spatial diversity and mode coupling among coupled cores. In this work, we theoretically investigate the potentials of MCFs including weakly-coupled multicore fiber (WC-MCF) and strongly-coupled multicore fiber (SC-MCF) as the propagation media for submarine communications across the Atlantic and the Pacific. To fairly compare the performances of SMFs- and MCFs-based submarine cables, the Gaussian noise (GN) model for SDM links is employed to optimize the systematic settings including spatial multiplicity and single span length. Then, we develop an SDM and wavelength division multiplexing (WDM) fiber transmission model based on coupled nonlinear Schrodinger equations (CNSE) to investigate the optical filed coupling effect in MCFs-based cables. The developed transmission model has been self-examined by measuring the inter-core crosstalk (IC-XT) and spatial mode dispersion (SMD), referring to the set values. As indicated by the theoretical analysis, the WC-MCFs cable exhibits a larger capacity than the SMFs cable, when the fiber pair count is limited below 32. Moreover, the SC-MCFs cable outperforms the WC-MCFs cable thanks to the reduced fiber nonlinearity due to the random mode coupling and the assistance of multiple-input and multiple-output digital signal processing (MIMO-DSP). At last, the marginal influences of IC-XT, SMD, and insertion loss of Fan-in and Fan-out couplers are also analyzed for the MCFs cable.

Structure and properties of soliton molecules in a single-mode dual core fiber with Kerr nonlinearity

We consider the case where two different frequency optical solitons are incident in a single-mode dual core fiber with Kerr nonlinearity. In this situation, the propagation dynamics of the two optical solitons are controlled by a set of coupled nonlinear Schrödinger equations(CNLSE). In this paper, the one- and two-soliton solutions of the CNLSE are obtained by Horita’s bilinear method, and the differences in constraints between one- and two-soliton solutions are discussed. Subsequently, when analyzing the characteristics of soliton transmission, it is found that the two-solitons will form the soliton molecules under certain conditions. This paper may have implications for the design of the stable mode-locked fiber laser and all-optical switches.

Some anomalous exact solutions for the four-component coupled nonlinear Schrödinger equations on complex wave backgrounds

The coupled nonlinear Schrödinger (CNLS) equations are the important models for the study of the multicomponent Bose-Einstein condensates (BECs). In this paper, we study the four-components CNLS equations via Darboux transformation and obtain the N-soliton solutions with zero seed and non-zero seed solutions(q_i=0 or q_i=e^{-2it}). The 1-soliton solution and 2-soliton solution are calculated on complex wave backgrounds, the dark-bright-bright-bright soliton solutions and dark-dark-bright-bright soliton solutions are constructed. We can obtain a new class of dark-bright-bright-bright soliton solutions, which admit one-valley dark soliton in component q_1 and triple-hump bright solitons in the other three components. The collision properties between dark-dark-bright-bright solitons are considered, and the vector solitons are expected to be much more abundant than those of previously reported vector soliton collisions.

Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain

In this paper, we considered a discrete coupled nonlinear Schrödinger equations (CNLSEs) which describes the propagation of solitonic waves in a gyrotropic molecular chain (GMC) of left and right circularly polarized intramolecular vibrations. From the linear analysis, we shown the forward and backward waves for left and right-handed modes. We underlined the effects of the gyrotropy term and effective mass (EM) on both modulation instability (MI) gain and modulated wave (MW) pattern. It reveals for strong enough values of these parameters that the generation of new sides lobes and MI bands. For numerical simulation, we shown out the propagation and interaction of the MW bright-soliton with high energy on the peak. It results from this investigation that gyrotropy term and EM behave as being energy sources in GMC. Finally, the reported outcomes can be used during the transfer of energy in molecular chain.

A novel investigation of exact solutions of the coupled nonlinear Schrodinger equations arising in ocean engineering, plasma waves, and nonlinear optics

We have analyzed the two coupled nonlinear Schrödinger equations (CNLSE) in the current work. This model has applications in high birefringence fibers. To generate different types of analytical solutions, including exponential, periodic, and soliton-type, we operated generalized Kudryashov, modified Kudryashov and exponential rational function procedures. Thanks to this implementation, we contributed to ocean engineering, plasma waves, nonlinear birefringence phenomena, and pulse compression features. In addition, in our article, we have included 2D and 3D graphics of solutions to help understand the physical meaning of solutions. The article highlights the method’s ability to provide various solutions to various physical problems.

Inelastic soliton interactions for nonlinear directional couplers in optical metamaterials with Kerr nonlinearity modulation stability

A directional coupler is made up of two parallel monolayer wave guides filled with high nonlinear optical metamaterial. The model equations of the directional twin-core couplers, which are coupled nonlinear Schrödinger equations (CNLSEs), were proposed in the literature. These equations have received the attention of a variety of works. The results found, therein, show mainly dark, bright or singular soliton solutions. The objective of this work is to search for different soliton geometric structures, which show novel patterns. To this issue a novel transformation, with complex amplitude, is introduced. It allows to inspect the waves produced via soliton periodic wave collision. The collision is elastic whenever the waves generated are, everywhere, smooth. Otherwise it is inelastic. Here, the unified method is used to find the traveling wave solutions of the CNLSEs. The solutions obtained are evaluated numerically and they are used to display the amplitude and the real part of the optical field in figures. These figures exhibit super-lattices, zig-zag soliton with rhombus modulation and graded solitons with blow up. These behaviors occur due to inelastic collisions. The contour plots show lattice waves with gap (region of blow up) or without gap. Also, ripples that occur currently in optical cables are observed. Furthermore, it is found that modulation instability (MI) is triggered for positive nonlinear phase shift coefficients or in the case when the coupling coefficients exceed a critical value. As far as MI holds, this leads to traveling waves blow up and gaps.

Modulational instability and rogue waves in one-dimensional nonlinear acoustic metamaterials: case of diatomic model

In this paper, by means of the expanded Taylor series and Lindstedt-Poincaré perturbation methods, the coupled nonlinear Schrödinger equations (CNLSE) modeling the propagation of acoustic waves in acoustic metamaterial is obtained. Using these equations, the Modulational Instability (MI) phenomenon is observed in disturbance mode. Manakov integrable system is derived with suitable parameters and we shown that the Rogue Waves (RWs) can propagate diatomic acoustic metamaterials.

Mixed interaction solutions for the coupled nonlinear Schrödinger equations

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple.

Survey of third- and fourth-order dispersions including ellipticity angle in birefringent fibers on W-shaped soliton solutions and modulation instability analysis

In this work, the modified Sardar sub-equation method is used to construct optical soliton solutions to the coupled nonlinear Schrodinger equation (CNLSE) having third-order and fourth-order dispersions describing short-pulse propagation in two-core fibers. Several solutions are determined including W-shaped bright, dark soliton solutions, singular soliton solutions, periodic function solutions and the combined complex soliton solutions. It should be noted that this integration scheme is very powerful mathematical tools for obtaining exact optical soliton solutions of nonlinear evolution equations. Under suitable values for the physical parameters, some representative wave structures are graphically displayed. In addition, the linear stability technic is used to analyze the modulation gain spectra in the birefringence associated with an ellipticity angle. In our knowledge, these obtained results are new in the context of nonlinear birefringent optical fibers.

Scattering of The Vector Soliton in Coupled Nonlinear Schrödinger Equation with Gaussian Potential

Nonlinear Schrodinger equation (NLSE) is the fundamental equation which describes the wave field envelope dynamics in a nonlinear and dispersive medium. However, if the fields have many components, one should consider the Coupled Nonlinear Schrodinger equation (CNLSE). We considered the interactions of orthogonally polarized and equal-amplitude vector solitons with two polarization directions. In this paper, we focused on the effect of Gaussian potential on the scattering of the vector soliton in CNLSE. The scattering process was investigated by the variational approximation method and direct numerical solution of CNLSE. Analytically, we analyzed the dynamics of the width and center of mass position of a soliton by the variational approximation method. Soliton may be reflected from each other or transmitted through or trapped. Initially, uncoupled solitons may form the coupled state if the kinetic energy of solitons less than the potential of attractive interaction between solitons but when its’ velocity above the critical velocity, the soliton will pass through each other easily. Meanwhile, a direct numerical simulation of CNLSE had been run to check the accuracy of the approximation. The result of the variational model gives a slightly similar pattern with direct numerical simulation of CNLSE by fixing the parameters for both solutions with the same value. The interaction of the vector soliton with Gaussian potential depends on the initial velocity and amplitude of the soliton and the strength of the external potential.

Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case

In this paper, we study the solution behavior of two coupled non–linear Schrodinger equations (CNLS) in the critical case, where one equation includes gain, while the other includes losses. Next, we present two numerical methods for solving the CNLS equations, for which we have made a comparison. These numerical experiments permit to illustrate other theoretical results proven by the authors [ 11 ]. We also obtain several numerical results for different non–linearities and investigate on the value of the blow up time relatively to some parameters.

Numerical analysis and simulations for coupled nonlinear Schrödinger equations based on lattice Boltzmann method

In this paper, we will extend the lattice Boltzmann model (LBM) to deal with the complex coupled nonlinear Schrödinger equations. A D1Q5 nonstandard complex lattice Boltzmann model (CLBM) with amending term for the coupled nonlinear Schrödinger equations (CNLSEs) is proposed. Complex equilibrium distribution functions and amending terms are selected, and the CNLSEs can be recovered by using the Chapman–Enskog multi-scale analysis and Taylor expansion technique. Some initial boundary value problems of the coupled system are simulated to verify the effectiveness and stability of our model.