The exact solutions to (2 1)-dimensional nonlinear Schrdinger equation

1　IntroductionConsiderthe ( 2 1 ) dimensionalnonlinearSchr dingerequation (NLSE)iψt =ψxy γ2 vψ ( 1 )vx =2 ( ｜ψ｜2 ) y ( 2 )whereγ2 =± 1 .Eqs.( 1 )and ( 2 )playimportantroleinnonlinearopticalphysicalfield (see ,e .g .[1 ]) .InRef.[2 ],ZakharovappliedinversescatteringmethodtoderivethesolitonsolutionsforEqs.( 1 )and ( 2 ) ,andNsolitonsolutionscanbefoundinRef.[3].InRef.[4],RadhaandlakshmananhasinvestigatedthePainlev啨ｐｒｏｐｅｒｔｉｅｓｆｏｒＥｑｓ.( 1 )and ( 2 ) .Thehiddensymme…     

A series of new double periodic solutions to a (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov equation

By means of a new general ans?tz and with the aid of symbolic computation, a new algebraic method named Jacobi elliptic function rational expansion is devised to uniformly construct a series of new double periodic solutions to (2+1)-dimensional asymmetric Nizhnik－Novikov－Veselov (ANNV) equation in terms of rational Jacobi elliptic function.     

Wronskian and Grammian solutions for the （3＋ 1）-dimensional Jimbo-Miwa equation

<正>In this paper,based on Hirota’s bilinear method,the Wronskian and Grammian techniques,as well as several properties of the determinant,a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented.They guarantee that the Wronskian determinant and the Grammian determinant solve the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form.Then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions.At last,with the aid of Maple,some of these special exact solutions are shown graphically.     

Evolution property of soliton solutions for the Whitham－Broer－Kaup equation and variant Boussinesq equation

Using the standard Painlevé analysis approach, the (1+1)-dimensional Whitham－Broer－Kaup (WBK) and variant Boussinesq equations are solved. Some significant and exact solutions are given. We investigate the behaviour of the interactions between the multi-soliton-kink-type solution for the WBK equation and the multi-solitonic solutions and find the interactions are not elastic. The fission of solutions for the WBK equation and the fusions of those for the variant Boussinesq equation may occur after their interactions.     

Variable separation solutions and new solitary wave structures to the (1+1)-dimensional equations of long-wave－short-wave resonant interaction

A variable separation approach is proposed and extended to the (1+1)-dimensional physical system. The variable separation solutions of (1+1)-dimensional equations of long-wave－short-wave resonant interaction are obtained. Some special type of solutions such as soliton solution, non-propagating solitary wave solution, propagating solitary wave solution, oscillating solitary wave solution are found by selecting the arbitrary function appropriately.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

New exact periodic solutions to （2＋1）-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.     

A new method of new exact solutions and solitary wave-like solutions for the generalized variable coefficients Kadomtsev——Petviashvili equation

Using the solution of general Korteweg--de Vries (KdV) equation, the solutions of the generalized variable coefficient Kadomtsev--Petviashvili (KP) equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.     

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrdinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.     

Analytical solutions and rogue waves in （3＋1）-dimensional nonlinear SchrSdinger equation

Analytical solutions in terms of rational-like functions are presented for a(3+1)-dimensional nonlinear Schrdinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz.Several free functions of time t are involved to generate abundant wave structures.Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.     

(2+1)维Boussinesq方程的Backlund变换与精确解

借助于符号计算软件Maple，对方程的种子解作适当的未知函数替换，然后利用Backlund 变 换通过具体的符号演算获得了(2+1)维Boussinesq方程的一系列精确解.这些解包括类孤子解 和有理解,其中有的解中含有任意函数，当任意函数取特殊函数时，这些解具有丰富的结构 ，有些结构可能对物理现象的研究是有意义的. 关键词： (2+1)维Boussinesq方程 Backlund 变换 精确解 类孤子解     

(2+1) 维Broer-Kau-Kupershmidt方程一系列新的精确解

借助于符号计算软件Maple，通过一种构造非线性偏微分方程(组)更一般形式精确解的直接方法即改进的代数方法，求解(2+1) 维 Broer-Kau-Kupershmidt方程，得到该方程的一系列新的精确解，包括多项式解、指数解、有理解、三角函数解、双曲函数解、Jacobi 和 Weierstrass 椭圆函数双周期解. 关键词： 代数方法 (2+1) 维 Broer-Kau-Kupershmidt 方程 精确解 行波解     

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.     

(2+1)维色散长波方程的新精确解及其复合波激发

利用Riccati方程展开法和变量分离法,得到了(2+1)维色散长波方程的变量分离解.根据得到的孤波解,构造出该方程新颖的复合波局域结构,研究了复合波随时间的演化.     

Lump,lumpoff and predictable rogue wave solutions to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation

This paper mainly uses Hirota bilinear form to investigate the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. We obtain the general lump solutions and discuss its positiveness, the propagation path, amplitude and position at any time. Based on the general lump solutions, lumpoff solutions which a combination of lump solitons and stripe solitons, are also triumphantly acquired. Similarly, according to the general lump solutions, we are also consider a particular rogue wave by introducing a pair of stripe solitons, and research its predictability which include the time of the rogue wave appearance, position at time, propagation path and the maximum value of wave height. Finally, some figures are given to explain the movement mechanism of these solutions.     

A General Mapping Approach and New Travelling Wave Solutions to (2+1)-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.     

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.     

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.