Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schr?dinger(NLS) equation and Boussinesq equation are two very important integrable equations.They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.     

Small Amplitude Solitons in the Higher-Order Nonlinear Schroedinger Equation in an Optical Fibre

By taking advantage of the approximate approach of small amplitude soliton, we study the higher-order nonlinear Schroeinger equation in an optical fibre. Our results show that the bright and dark solitons of small amplitude can appear on the background of a continuous wave in normal dispersion regime or in anomalous dispersion regime simultaneously due to the higher-order effects. Interesting connection between the higher-order nonlinear Schroedinger equation and the Korteweg-de Vries equation is also demonstrated.     

Kuznetsov–Ma soliton and Akhmediev breather of higher-order nonlinear Schrdinger equation

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrdinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability(MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov–Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrdinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.     

Stability of dark soliton solutions of the quintic complex Ginzburg--Landau equation inthe case of normal dispersion

Dark soliton solutions of the one-dimensional complex Ginzburg--Landau equation (CGLE) are analysed for the case of normal group-velocity dispersion. The CGLE can be transformed to the nonlinear Schr\"{o}dinger equation (NLSE) with perturbation terms under some practical conditions. The main properties of dark solitons are analysed by applying the direct perturbation theory of the NLSE. The results obtained may be helpful for the research on the optical soliton transmission system.     

Nonautonomous solitons in the continuous wave background of the variable-coefficient higher-order nonlinear Schro¨dinger equation

We reduce the variable-coefficient higher-order nonlinear Schrdinger equation (VCHNLSE) into the constantcoefficient (CC) one. Based on the reduction transformation and solutions of CCHNLSE, we obtain analytical soliton solutions embedded in the continuous wave background for the VCHNLSE. Then the excitation in advancement and sustainment of soliton arrays, and postponed disappearance and sustainment of the bright soliton embedded in the background are discussed in an exponential system.     

Dark Solitons for the Defocusing Cubic Nonlinear Schr?dinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schr¨odinger(NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincar′e map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.     

Soliton Solutions of Discrete Complex Ginzburg-Landau Equation via Extended Hyperbolic Function Approach

In this paper, we extend the hyperbolic function approach for constructing the exact solutions of nonlinear differential-difference equation (NDDE) in a unified way. Applying the extended approach and with the aid of Maple,we have studied the discrete complex Ginzburg-Landau equation (dCGLE). As a result, we find a set of exact solutions which include bright and dark soliton solutions.     

Exact Soliton Solutions for the (2+1)-Dimensional Coupled Higher-Order Nonlinear Schr¨odinger Equations in Birefringent Optical-Fiber Communication

In birefringent optical fibers, the propagation of femtosecond soliton pulses is described by coupled higherorder nonlinear Schrdinger equations. In this paper, we will investigate the bright and dark soliton solutions of(2+1)-dimensional coupled higher-order nonlinear Schrdinger equations, with the aid of symbolic computation and the Hirota method. On the basis of soliton solutions, we test and discuss the interactions graphically between the solitons in the x-z, x-t, and z-t planes.     

Multi-Solitons and Combined Solitons with Self-Similar Behaviors in a Birefringent Fiber with Higher-Order Effects

A coupled variable-coefficient higher-order nonlinear Schr(o|¨)dinger equation in biretringent fiber is studied,and analytical multi-soliton,combined bright and dark soliton,W-shaped and M-shaped soliton solutions are obtained.Nonlinear tunnelling of these combined solitons in dispersion barrier and dispersion well on an exponential background is discussed,and the decaying or increasing,even lossless tunnelling behaviors of combined solitons are decided by the decaying or increasing parameter.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion(CTE) method is applied to the(2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution,and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlev′e truncated expansion method.And we investigate interactive properties of solitons and periodic waves.     

Evolution of dark solitons in the presence of Raman gain and self-steepening effect

Based on the equation satisfied by optical pulse that is a slowly varying function, the higher-order nonlinear Schr o¨dinger equation(NLSE) including Raman gain and self-steepening effect is deduced in detail, and a new Raman gain function is defined. By using the split-step Fourier method, the influence of the combined effect between Raman gain and self-steepening on the propagation characteristic of dark solitons is simulated in the isotropic fiber. The results show that gray solitons can be symmetrically formed by high order dark soliton, however self-steepening effect will inhibit the formation mechanism through the phenomenon that gray solitons are produced only in the trailing edge of the central black soliton. Meanwhile, the Raman gain changes the propagation characteristic of optical soliton and inhibits the self-steepening effect, resulting in the broadening of pulse width and the decreasing of pulse offset.     

Spatial Solitons in 2D Graded-Index Waveguides with Different Distributed Transverse Diffractions

We discuss the nonlinear Schr6dinger equation with variable coefficients in 21） graded-index waveguides with different distributed transverse diffractions and obtain exact bright and dark soliton solutions. Based on these solutions, we mainly investigate the dynamical behaviors of solitons in three different diffraction decreasing waveguides with the hyperbolic, Gaussian and Logarithmic profiles. Results indicate that for the same parameters, the amplitude of bright solitons in the Logarithmic profile and the amplitude of dark solitons in the Gaussian profile are biggest respectively, and the amplitude in the hyperbolic profile is smallest, while the width of solitons has the opposite case.     

STEADY SOLUTIONS OF THE OPTICAL SOLITON EQUATION WITH A NONLINEAR RESPONSE DELAY TERM

The exact solution of the optical soliton equation with a nonlinear response delay term has been obtained by using the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found for a special case.     

Dynamics of three nonisospectral nonlinear Schrdinger equations

Dynamics of three nonisospectral nonlinear Schrdinger equations(NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrdinger equation(NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover,we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes,velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.     

Exact solitary wave solutions of a nonlinear Schrdinger equation model with saturable-like nonlinearities governing modulated waves in a discrete electrical lattice

In this paper, we introduce and propose exact and explicit analytical solutions to a novel model of the nonlinear Schr¨odinger(NLS) equation. This model is derived as the equation governing the dynamics of modulated cutoff waves in a discrete nonlinear electrical lattice. It is characterized by the addition of two terms that involve time derivatives to the classical equation. Through those terms, our model is also tantamount to a generalized NLS equation with saturable;which suggests that the discrete electrical transmission lines can potentially be used to experimentally investigate wave propagation in media that are modeled by such type of nonlinearity. We demonstrate that the new terms can enlarge considerably the forms of the solutions as compared to similar NLS-type equations. Sine–Gordon expansion-method is used to derive numerous kink, antikink, dark, and bright soliton solutions.     

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.     

Exact self-similar solitary waves and collisions in nonlinear optical media

This paper analyses bright and dark spatial self-similar waves propagation and collision in graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities. It finds an appropriate transformation for the first time such that the nonlinear Schrodinger equation （NLSE） with varying coefficients transform into standard NLSE. It obtains one-solitonlike, two-solitonlike and multi-solitonlike self-similar wave solutions by using the transformation. Furthermore, it analyses the features of the self-similar waves and their collisions.     

Localized Properties of Rogue Wave for a Higher-Order Nonlinear Schr?dinger Equation

In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higherorder nonlinear Schr¨odinger equation(HONLS) by the Darboux transformation and confirm the decomposition rule of the rogue wave solutions up to fourth-order. These solutions have two parameters α and β which denote the contribution of the higher-order terms(dispersions and nonlinear effects) included in the HONLS equation. Two localized properties, i.e.,length and width of the first-order rogue wave solution are expressed by above two parameters, which show analytically a remarkable influence of higher-order terms on the rogue wave. Moreover, profiles of the higher-order rogue wave solutions demonstrate graphically a strong compression effect along t-direction given by higher-order terms.     

Asymptotic analysis of multi-valley dark soliton solutions in defocusing coupled Hirota equations

We construct uniform expressions of such dark soliton solutions encompassing both single-valley and double-valley dark solitons for the defocusing coupled Hirota equation with high-order nonlinear effects utilizing the uniform Darboux transformation, in addition to proposing a sufficient condition for the existence of the above dark soliton solutions. Furthermore, the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior; howeve...     

Propagation of bright femtosecond pulses in a nonlinear optical fibre with the third- and fourth-order dispersions

We solve the generalized nonlinear Schr\"{o}dinger equation describing the propagation of femtosecond pulses in a nonlinear optical fibre with higher-order dispersions by using the direct approach to perturbation for bright solitons, and discuss the combined effects of the third- and fourth-order dispersions on velocity, temporal intensity distribution and peak intensity of femtosecond pulses. It is noticeable that the combined effects of the third- and fourth-order dispersions on an initial propagated soliton can partially compensate each other, which seems to be significant for the stability controlling of soliton propagation features.