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.     

Lump and new interaction solutions to the(3+1)-dimensional nonlinear evolution equation

In this paper, a new (3+1)-dimensional nonlinear evolution equation is introduced, through the generalized bilinear operators based on prime number p=3. By Maple symbolic calculation, one-, two-lump, and breather-type periodic soliton solutions are obtained, where the condition of positiveness and analyticity of the lump solution are considered. The interaction solutions between the lump and multi-kink soliton, and the interaction between the lump and breather-type periodic soliton are derived, by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one. In addition, new interaction solutions between a lump, periodic-solitary waves, and one-, two- or even three-kink solitons are constructed by using the ansatz technique. Finally, the characteristics of these various solutions are exhibited and illustrated graphically.     

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

In this paper, based on the Hirota's bilinear method, the Wronskian and Grammian technique, as well as several properties of determinant, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented, which guarantees that the Wronskian determinant and the Grammian determinant solve the (3+1)-dimensional Jimbo-Miwa equation in the bilinear form, and 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.     

Explicit and exact travelling plane wave solutions of the （2＋1）—dimensional Boussinesq equation

The deformation mapping method is applied to solve a system of (2+1)-dimensional Boussinesq equations. Many types of explicit and exact travelling plane wave solutions, which contain solitary wave solutions,periodic wave solutions,Jacobian elliptic function solutions and others exact solutions, are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and the cubic nonlinear Klein－Gordon equation.     

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.     

(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     

Symmetry reduction and exact solutions of the (3+1)-dimensional Zakharovben Kuznetsov equation

By means of the classical method, we investigate the (3+1)-dimensional Zakharov-Kuznetsov equation. The symmetry group of the (3+1)-dimensional Zakharov-Kuznetsov equation is studied first and the theorem of group invariant solutions is constructed. Then using the associated vector fields of the obtained symmetry, we give the one-, two-, and three-parameter optimal systems of group-invariant solutions. Based on the optimal system, we derive the reductions and some new solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation.     

一个(3+1)维孤子方程的周期解

利用Hirota方法及Riemann theta函数得到了一个(3+1)维孤子方程的周期解.在极限情况下,该周期解退化为孤子解.另外,利用计算机技术和Mathematica绘制了解的三维曲面图.     

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.     

The Soliton Solution of （3＋1）—Dimensional Kadomtsev—Petviashvili Equations

A simple algebraic transformation relation of a special type of solution between the (3 1)-dimensional Kadomtsev-petviashvili(KP) equation and the cubic nonlinear Klein-Gordon equation (NKG) is established.Using known solutions of the NKG equation,we can obtain many soliton solutions and periodic solution of the (3 1)-dimensional KP equation.     

Solving (2+1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ordinary differential equation method is presented for solving the (2+1)-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the (2+1)-dimensional sine-Poisson equation are derived in a simple manner by this technique.     

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.     

Solutions of novel soliton molecules and their interactions of (2 + 1)-dimensional potential Boiti-Leon-Manna-Pempinelli equation

The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations. In this paper, we use a new form of variable separation to study novel soliton molecules and their interactions in (2+1)-dimensional potential Boiti-Leon-Manna-Pempinelli equation. Dromion molecules, ring molecules, lump molecules, multi-instantaneous molecules, and their interactions are obtained. Then we draw corresponding images with maple software to study their dynamic behavior.     

Rogue waves of a (3+1)-dimensional BKP equation

We investigate certain rogue waves of a (3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method. We obtain semi-rational solutions in the determinant form, which contain two special interactions: (i) one lump develops from a kink soliton and then fuses into the other kink one; (ii) a line rogue wave arises from the segment between two kink solitons and then disappears quickly. We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time, which performs like a rogue wave. Furthermore, the higher-order semi-rational solutions describing the interaction between two lumps (one line rogue wave) and three kink solitons are presented.     

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.     

Forced(2+1)-dimensional discrete three-wave equation

We generalize the ■-dressing method to investigate a(2+1)-dimensional lattice,which can be regarded as a forced(2+1)-dimensional discrete three-wave equation.The soliton solutions to the(2+1)-dimensional lattice are given through constructing different symmetry conditions.The asymptotic analysis of one-soliton solution is discussed.For the soliton solution,the forces are zero.     

(3+1)维非线性Burgers系统的新的分离变量解及其局域激发结构与分形结构

将扩展的Riccati方程映射法推广到了(3+1)维非线性Burgers系统，得到了系统的分离变量解；由于在解中含有一个关于自变量(x,y,z,t)的任意函数，通过对这个任意函数的适当选取，并借助于数学软件Mathematica进行数值模拟，得到了系统的新而丰富的局域激发结构和分形结构.结果表明，扩展的Riccati方程映射法在求解高维非线性系统时，仍然是一种行之有效的方法，并且可以得到比(2+1)维非线性系统更为丰富的局域激发结构. 关键词： 扩展的Riccati方程映射法 (3+1)维非线性Burgers方程 局域激发结构 分形结构     

New lump,lump-kink,breather waves and other interaction solutions to the (3+1)-dimensional soliton equation

This study investigates the (3+1)-dimensional soliton equation via the Hirota bilinear approach and symbolic computations. We successfully construct some new lump, lump-kink, breather wave, lump periodic, and some other new interaction solutions. All the reported solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica package. The solution's visual characteristics are graphically represented in order to shed more light on the results obtained. The findings obtained are useful in understanding the basic nonlinear fluid dynamic scenarios as well as the dynamics of computational physics and engineering sciences in the related nonlinear higher dimensional wave fields.     

Applications of Extended Mapping Deformation Method in Two (3+1)-Dimensional Nonlinear Models

Based on the extended mapping deformation method and symbolic computation, many exact travelling wave solutions are found for the (3+1)-dimensional JM equation and the (3+1)-dimensional KP equation. The obtained solutions include solitary solution, periodic wave solution, rational travelling wave solution, and Jacobian and Weierstrass function solution, etc.     

(3+1)维Nizhnik-Novikov-Veselov方程的孤子解和周期解

采用行波法约化方程,建立一种变换关系,把求解(3+1)维NizhnikNovikovVeselov(NNV)方程的解转化为求解一维非线性KleinGordon方程的解,从而得到了(3+1)维NNV方程的孤子解和周期解. 关键词： (3+1)维Nizhnik-Novikov-Veselov方程 非线性Klein-Gordon方程 孤子解 周期解