Evolution of the exact spatiotemporal periodic wave and soliton solutions of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients

In this paper the spatiotemporal evolution of the periodic wave is investigated analytically when the laser passes through the inhomogeneous nonlinear medium. Firstly, the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients is solved analytically by an improved homogeneous balance principle and F-expansion technique. A number of exact periodic traveling wave and spatiotemporal soliton solutions are obtained. Then, their propagation characteristics are analyzed in detail. It is found that the evolutions of propagation of spatiotemporal soliton and periodic wave solutions are regular when the diffraction and dispersion coefficients are the identical distributed coefficients, but the evolutions of propagation of these solutions are irregular with other coefficients.     

非线性Schrodinger方程的Jacobi椭圆函数包络解

通过函数变换和扩展Jacobi椭圆函数展开法,利用吴消元法,借助符号运算软件Maple,得到非线性Schringer方程丰富的包络形式精确解,特别是由两个Jacobi椭圆函数表示的精确解.当模数m→1或m→0时,一部分解退化为双曲函数或三角函数表示的解,F-展开法和扩展的F-展开法得到的精确解是本文结果的特例.     

Manakov型非线性Schr

简化了扩展的Jacobi椭圆函数展开法,亦即对修正的Jacobi椭圆函数展开法进行了扩展.把这种方法应用于Manak0v型非线性Schrodinger方程,得到了Jacobi椭圆函数包络解.在一定条件下,这些解退化成相应的包络冲击波解和包络孤立波解.     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

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.     

变系数非线性方程的Jacobi椭圆函数展开解

把Jacobi椭圆函数展开法扩展并应用到求解变系数的非线性演化方程,比较方便地得到新的解析解 关键词： Jacobi椭圆函数 变系数非线性方程 类椭圆余弦波解 类孤子解     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

非线性发展方程的丰富的Jacobi椭圆函数解

通过把十二个Jacobi椭圆函数分类成四组，提出了新的广泛的Jacobi椭圆函数展开法，利用这一方法求得了非线性发展方程的丰富的Jacobi椭圆函数双周期解.当模数m→0或1时，这些解退化为相应的三角函数解或孤立波解和冲击波解. 关键词： 非线性发展方程 Jacobi椭圆函数 双周期解 行波解     

非线性Schr(o)inger方程的Jacobi椭圆函数包络解

通过函数变换和扩展Jacobi椭圆函数展开法,利用吴消元法,借助符号运算软件Maple,得到非线性Schr(o)inger方程丰富的包络形式精确解,特别是由两个Jacobi椭圆函数表示的精确解.当模数m→1或m→0时,一部分解退化为双曲函数或三角函数表示的解,F-展开法和扩展的F-展开法得到的精确解是本文结果的特例.     

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.     

非线性波动方程的Jacobi椭圆函数包络周期解

应用Jacobi椭圆函数展开法求得了一类非线性波方程的包络周期解,而且用这种方法得到的周期解在一定条件下可以退化为包络冲击波解或包络孤立波解 关键词： Jacobi椭圆函数 非线性方程 包络周期解 包络孤立波解     

Optical solitons and modulation instability analysis of the (1+1)-dimensional coupled nonlinear Schrödinger equation

This study successfully reveals the dark, singular solitons, periodic wave and singular periodic wave solutions of the (1+1)-dimensional coupled nonlinear Schrödinger equation by using the extended rational sine-cosine and rational sinh-cosh methods. The modulation instability analysis of the governing model is presented. By using the suitable values of the parameters involved, the 2-, 3-dimensional and the contour graphs of some of the reported solutions are plotted.     

Breather and double-pole solutions of the derivative nonlinear Schrödinger equation from optical fibers

The derivative nonlinear Schrödinger (DNLS) equation, which governs the propagation of the femtosecond optical pulse in a monomodal optical fiber, is analytically studied in this Letter. Breather and double-pole solutions are derived from the two-soliton solution with the choice of parameters. It is found that the parameters in the DNLS equation cannot only control the phase and propagation direction of the breather and double pole, but also influence the interaction period of the breather. Elastic collisions between a breather and a dark/anti-dark soliton are studied by the qualitative analysis and graphical illustration. The stability of the breather and double-pole solutions is also analyzed.     

Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation

The (1+2)-dimensional chiral nonlinear Schrödinger equation (2D-CNLSE) as a nonlinear evolution equation is considered and studied in a detailed manner. To this end, a complex transform is firstly adopted to arrive at the real and imaginary parts of the model, and then, the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE. The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.     

非线性薛定谔方程的新多级包络周期解

基于Lamé方程和新的Lamé函数,应用摄动方法和Jacobi椭圆函数展开法求解了非线性薛定谔方程,获得多种新的多级准确解。这些解对应着不同的形式的包络周期解。这些解在极限条件下可以退化为各种形式的包络孤波解。     

Exact analytical solutions for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients

Analytical solutions are reported for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients using the similarity transformation and tri-function method, which involve three free functions of spaces to generate abundant wave structures. Three types of free functions are chosen to exhibit the corresponding nonlinear wave propagations.     

Wobbling phenomena with logarithmic law nonlinear Schrödinger equations for incoherent spatial Gaussons

We apply the equivalence particle principle to several nonlinear Schrödinger equations (NLSEs) that model the propagation of a spatial beam with logarithmic law nonlinearity. Using this principle, expressions for acceleration, spatial frequency, spatial period and other variables for a spatial soliton can be derived from the solution of the homogenous NLSE with logarithmic law nonlinearity. These results agree well with numerical simulations of the perturbed NLSE. We show that if the expression of the acceleration is bounded this means the spatial soliton propagates with a swing effect.