Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity

In this Letter, we investigate the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters

Bessel solitary wave solutions to a two-dimensional strongly nonlocal nonlinear Schrödinger equation with distributed coefficients are obtained. Bessel solitary wave solutions have unique characteristics compared with Gaussian solitary wave solutions, Laguerre-Gaussian solitary wave solutions, and Hermite-Gaussian solitary wave solutions. The generalized two-dimensional nonlocal nonlinear Schrödinger equation with distributed coefficients is investigated for the first time to our knowledge.     

Complex Wave Excitations in Generalized Broer-Kaup System

Starting from an improved projective method and a linear variable separation approach, new families of variable separation solutions （including solltary wave solutlons, periodic wave solutions and rational function solutions） with arbitrary functions [or the （2＋ 1）-dimensional general/zed Broer-Kaup （GBK） system are derived. Usually, in terms of solitary wave solutions and/or rational function solutions, one can find abundant important localized excitations. However, based on the derived periodic wave solution in this paper, we reveal some complex wave excitations in the （2＋1）-dimensional GBK system, which describe solitons moving on a periodic wave background. Some interesting evolutional properties for these solitary waves propagating on the periodic wave bactground are also briefly discussed.     

Bifurcations and new exact travelling wave solutions for the bidirectional wave equations

By using the method of dynamical system, the bidirectional wave equations are considered. Based on this method, all kinds of phase portraits of the reduced travelling wave system in the parametric space are given. All possible bounded travelling wave solutions such as dark soliton solutions, bright soliton solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, numerical simulations are conducted for dark soliton solutions, bright soliton solutions and periodic travelling wave solutions to the bidirectional wave equations. The results presented in this paper improve the related previous studies.     

Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to(1+1)-Dimensional Dispersive Long Wave Equation

In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.     

Travelling Wave Solutions to a Special Type of Nonlinear Evolution Equation

A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of "rank". The key idea of this method is to make use of the arbitrariness of the manifold in Painleve analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.     

Bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation

In this paper, we study the bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation by using the method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space. Then we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, compactions and kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions and periodic wave solutions are implicitly given,while the expressions of kink-like and antikink-like wave solutions are explicitly shown. The dynamics of these new traveling wave solutions will greatly enrich the previews results and further help us understand the physical structures and analyze the propagation of the nonlinear wave.     

The exact solutions to （2＋1）-dimensional nonlinear Schroedinger equation

By using the extended F-expansion method,the exact solutions,including periodic wave solutions expressed by Jaeobi elliptic functions,for (2 1)-dimensional nonlinear Schroedinger equation are derived.In the limit cases,the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Exponential--fraction trial function method to the 5th-order mKdV equation

This paper obtains some solutions of the 5th-order mKdV equation by using the exponential--fraction trial function method, such as solitary wave solutions, shock wave solutions and the hopping wave solutions. It successfully shows that this method may be valid for solving other nonlinear partial differential equations.     

The exact solutions to (2+1)-dimensional nonlinear Schrǒdinger equation

By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrǒdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

The Peridic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.     

New solitons and periodic wave solutions for the dispersive long wave equations

We make use of Jacobi elliptic functions to construct periodic wave solutions for the dispersive long wave equations. In the limit cases the multiple soliton solutions are also obtained. The properties of some periodic and soliton solutions for the dispersive long wave equations are shown by some figures. As a result, we can successfully obtain the solitary wave solutions that can be found by the previous work.     

Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.     

The Periodic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobi elliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

Applications of F-expansion method to the coupled KdV system

An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.     

Extended F-Expansion Method and Periodic Wave Solutionsfor Klein-Gordon-Schrödinger Equations

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrödinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

The periodic wave solutions for the generalized Nizhnik－Novikov－Veselov equation

The periodic wave solutions expressed by Jacobi elliptic functions for the generalized Nizhnik－Novikov－Veselov equation are obtained using the F-expansion method proposed recently. In the limiting cases, the solitary wave solutions and other types of travelling wave solutions for the system are obtained.     

Direct Approach to Construct the Periodic Wave Solutions for Two Nonlinear Evolution Equations

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna-Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Furthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.