Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equation with an External Potential

An improved homogeneous balance principle and an F-expansiontechnique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schrödinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented.     

New Optical Solitons in High-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

By using the generalized tanh-function method, we find bright and dark solitary wave solutions to an extended nonlinear Schrödinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. At the same time, we also obtained other types of exact solutions.     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrödinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrödinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrödinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrödinger equation is given.     

Analytical Periodic Wave and Soliton Solutions of the Generalized Nonautonomous Nonlinear Schrödinger Equation in Harmonic and Optical Lattice Potentials

We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schrödinger equation with time- and space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain (loss) coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.     

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrödinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T₀ to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X₀.     

Spatiotemporal self-similar waves for the (3 + 1)-dimensional inhomogeneous cubic-quintic nonlinear medium

Spatiotemporal self-similar waves of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation, describing propagation of optical pulses in a cubic-quintic nonlinear medium with inhomogeneous dispersion and gain, are derived. A one-to-one correspondence between such self-similar waves and solutions of the constant-coefficient cubic-quintic nonlinear Schrödinger equation exists when two certain compatibility conditions are satisfied. Under these conditions, we discuss dynamical behaviors of self-similar waves in dispersion decreasing fiber.     

Exact Solutions of Five Complex Nonlinear Schrödinger Equations by Semi-Inverse Variational Principle

In this paper, we establish exact solutions for five complex nonlinear Schrödinger equations. The semi-inverse variational principle (SVP) is used to construct exact soliton solutions of five complex nonlinear Schrödinger equations. Many new families of exact soliton solutions of five complex nonlinear Schrödinger equations are successfully obtained.     

Optical solitons for the decoupled nonlinear Schrödinger equation using Jacobi elliptic approach

Most of the important aspects of soliton propagation through optical fibers for transcontinental and transoceanic long distances can best be described using the nonlinear Schrödinger equation. Optical solitons are electromagnetic waves that span in nonlinear dispersive media and permit the stress and intensity to stay unaltered as a result of the delicate balance between dispersion and nonlinearity effects. However, this study exploited the Jacobi elliptic method and obtained different soliton solutions of the decoupled nonlinear Schrödinger equation with ease. Discussions about the obtained solutions were made with the aid of some 3D graphs.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

色散渐变光纤中相移控制研究

相移控制能有效的避免孤子间的相互作用.通过利用符号计算和双线性方法,本文解析研究用于描述色散渐变光纤传输特点的非线性Schr?dinger方程,并得到该方程的双孤子解.基于所得到的双孤子解,通过研究发现,当色散渐变光纤中的群速度色散呈Gauss型变化时,可以利用该类光纤实现孤子相移控制,从而避免孤子相互作用,提高光通信系统中信号传输质量.此外,本文还将讨论色散渐变光纤中各类参数对相移控制的影响.本文结论还有助于逻辑门和全光开光的研究.     

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.     

Breathers and Rogue Waves Derived from an Extended Multi-dimensional N-Coupled Higher-Order Nonlinear Schrödinger Equation in Optical Communication Systems

In this paper, an extended multi-dimensional N-coupled higher-order nonlinear Schrödinger equation (NCHNLSE), which can describe the propagation of the ultrashort pulses in wavelength division multiplexing (WDM) systems, is investigated. By the bilinear method, we construct the breather solutions for the extended (1+1), (2+1) and (3+1)-dimensional N-CHNLSE. The rogue waves are derived as a limiting form of breathers with the aid of symbolic computation. The effect of group velocity dispersion (GVD), third-order dispersion (TOD) and nonlinearity on breathers and rogue waves solutions are discussed in the optical communication systems.     

New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schrödinger Equation

Some new exact travelling wave and period solutions of discrete nonlinear Schrödinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.     

Rogue waves of the sixth-order nonlinear Schrödinger equation on a periodic background

In this paper, we construct the rogue wave solutions of the sixth-order nonlinear Schrödinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.     

Mixed Local-Nonlocal Vector Schrödinger Equations and Their Breather Solutions

To the best of our knowledge, all nonlinearities in the known nonlinear integrable systems are either local or nonlocal. A natural problem is whether there exist some nonlinear integrable systems with both local and nonlocal nonlinearities, and how to solve this kinds of spectral nonlinear integrable systems with both local and nonlocal nonlinearities. Recently, some novel mixed local-nonlocal vector Schrödinger equations are presented, which are different from the single local and nonlocal coupled Schrödinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrödinger equations with a spectral problem. Starting from a special Lax pairs, the mixed localnonlocal vector Schrödinger equations are constructed. We obtain the one- and two- and N-soliton solution formulas of the mixed local-nonlocal vector Schrödinger equations with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-solitons are exhibited, the overtaking elastic interactions among the two-breather solitons are considered. We find that unlike the local and nonlocal cases, the mixed local-nonlocal vector Schrödinger equations have some novel results. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

Direct Perturbation Method for Derivative Nonlinear Schrödinger Equation

We extend Lou's direct perturbation method for solving the nonlinear Schrödinger equation to the case of the derivative nonlinear Schrödinger equation (DNLSE). By applying this method, different types of perturbation solutions are obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.     

The Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation with Variable Coefficients

The cubic-quintic nonlinear Schrödinger equation (CQNLS) plays important parts in the optical fiber and the nuclear hydrodynamics. By using the homogeneous balance principle, the bell type, kink type, algebraic solitary waves, and trigonometric traveling waves for the cubic-quintic nonlinear Schrödinger equation with variable coefficients (vCQNLS) are derived with the aid of a set of subsidiary high-order ordinary differential equations (sub-equations for short). The method used in this paper might help one to derive the exact solutions for the other high-order nonlinear evolution equations, and shows the new application of the homogeneous balance principle.     

An Integrable Decomposition of Defocusing Nonlinear Schrödinger Equation

The method of nonlinearization of spectral problems is developed to thedefocusing nonlinear Schrödinger equation. As an application, an integrable decomposition of the defocusing nonlinear Schrödinger equation is presented.     

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.     

Nonautonomous Solitons in the(3+1)-Dimensional Inhomogeneous Cubic-Quintic Nonlinear Medium

We have constructed explicit nonautonomous soliton solutions of the generalized nonlinear Schrdinger equation in the(3+1)-dimensional inhomogeneous cubic-quintic nonlinear medium.The gain parameter has no effects on the motion of the soliton's phase or their velocities,and it affects just the evolution of their peaks.As two examples,we discuss the propagation of nonautonomous solitons in the periodic distributed amplification system and the exponential dispersion decreasing system.Results show that the presence of the chirp not only makes the intensity of solitons weaken more promptly,but also broadens their width.