用改进的代数方法构造(2+1)维ZK-MEW方程的精确行波解

利用一种改进的统一代数方法将构造(2+1)维ZK MEW((2+1)-dimensionalZakharov-Kuznetsovmodifiedequalwidth)方程精确行波解的问题转化为求解一组非线性的代数方程组．再借助于符号计算系统Mathematica求解所得到的非线性代数方程组,最终获得了方程的多种形式的精确行波解．其中包括有理解,三角函数解,双曲函数解,双周期Jacobi椭圆函数解,双周期Weierstrass椭圆形式解等．并给出了部分解的图形.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

(2+1)维色散长波方程新的类孤子解

通过一个简单的变换,将(2+1)维色散长波方程简化为人们熟知的带强迫项Burgers方程,借助Mathematica软件,利用齐次平衡原则和变系数投影Riccati方程法,求出了(2+1)维色散长波方程新的精确解.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

Jacobi Elliptic Function Solutions and Traveling Wave Solutions of the (2 + 1)-Dimensional Gardner-KP Equation

The fully integrable KP equation is one of the models that describes the evolution of nonlinear waves, the expansion of the well-known KdV equation, where the impacts of surface tension and viscosity are negligible. This paper uses the Modified Extended Direct Algebraic (MEDA) method to build fresh exact, periodic, trigonometric, hyperbolic, rational, triangular and soliton alternatives for the (2 + 1)-dimensional Gardner KP equation. These solutions that we discover in this article will help us understand the phenomena of the (2 + 1)-dimensional Gardner KP equation. Comparing the study in this paper and existing work, we find more exact solutions with soliton and periodic structures and the rational function solution in this paper is more general than the rational solution in existing literature. Most of the Jacobi elliptic function solutions and the mixed Jacobi elliptic function solutions to the (2 + 1)-dimensional Gardner KP equation discovered in this paper, to the best of our highest understanding are not seen in any existing paper until now.     

分离变量法在变系数(2+1)维方程求解中的应用

分离变量法是求解具有局域相干结构解的有效解析方法.考虑到传播介质的非均匀性和边界的不一致性,变系数(2+1)色散长波方程可以实际地描述宽广的河道或有限深的远海中非线性波的传播.解析研究了变系数(2+1)维色散长波方程.通过分离变量法,得到了该方程组的具有丰富结构的分离变量解.     

Bifurcations of travelling wave solutions for the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation

By using the bifurcation theory of dynamical systems, we study the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, the existence of solitary wave solutions, compacton solutions, periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

关于(2+1)-维色散的长波方程求解的一个注记

在文献[3]的基础上，根据一些简单方程的特征，导出了(2十1)—维色散的长波方程的新的精确解，其中包含了已有文献中的孤子解，多孤子解等．     

(2+1)维Broer-Kaup-Kupershmidt方程组的行波解

借助于计算机代数系统Mathematica,利用推广的简单方程方法构造了(2+1)维Broer-Kaup-Kupershmidt方程组的新的精确行波解,分别以含有双参数的用双曲函数,三角函数和有理函数表示,其中双曲函数表.示的行波解中参数取特殊值时可得到文献已有的孤波解.方法也适用于其它非线性发展方程(组).     

(2+1)维Burgers方程新的复合解

In this paper, a new generalized compound Riccati equations rational expansion method （GCRERE） is proposed. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method is not only recover some known solutions, but also find some new and general complexiton solutions. Being concise and straightforward, it is applied to the （2＋1）-dimensional Burgers equation. As a result, eight families of new exact analytical solutions for this equation are found. The method can also be applied to other nonlinear partial differential equations.     

On some new analytical solutionsforthe (2+1)-dimensional burgers equation and the special type of dodd_bullough_Mikhailov equation

Some new travelling wave transform methods are very importantfor obtaining analytical solutions of special type of nonlinear partial differentialequations (NLPDEs). Some of these solutions of NLPDEs may be inthe different forms such as rational function solutions, trigonometric functionsolutions, hyperbolic function solutions, exponential function solutions andJacobi elliptic function solutions. These forms tell us the various propertiesof the NLPDEs from scientifical applications to engineering.In this research, we have studied to obtain the analytical solution ofthe nonlinear (2+1)-dimensional Burgers equation which is named from JohannesMartinus Burgers and the nonlinear special type of the Dodd-Bullough-Mikhailov equation introduced to the literature by Roger Dodd, Robin Bullough,and Alexander Mikhailov.     

The breather wave solutions, M-lump solutions and semi-rational solutions to a (2+1)-dimensional generalized Korteweg-de Vries equation

Under investigation in this work is a (2+1)-dimensional generalized Korteweg-de Vries equation, which can be used to describe many nonlinear phenomena in plasma physics. By using the properties of Bell"s polynomial, we obtain the bilinear formalism of this equation. The expression of $N$-soliton solution is established in terms of the Hirota"s bilinear method. Based on the resulting $N$-soliton solutions, we succinctly show its breather wave solutions. Furthermore, with the aid of the corresponding soliton solutions, the $M$-lump solutions are well presented by taking a long wave limit. Two types of hybrid solutions are also represented in detail. Finally, some graphic analysis are provided in order to better understand the propagation characteristics of the obtained solutions.     

Lump Solutions, Lump-soliton Interaction Phenomena and Breather-soliton Solutions to a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli-like Equation

The (3+1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (BLMP-like equation) is introduced by the generalized bilinear operators $D_{p}$ associated with $p=3$. The lump solutions, lump-soliton interaction phenomena and breather-soliton solutions are discussed to the (3+1)-dimensional BLMP-like equation based on the generalized bilinear method with symbolic computation system Mathematica. In order to observe the behavior of those solutions, we fix the value of $z$, then give the 3D-graphs of some solutions at different times. We find a lump solution moved in oblique direction; a lump-soliton interaction phenomenon is appeared and disappeared along with the time. We also see a kink-breather soliton moved in oblique direction.     

Rogue wave and interaction phenomenon to (1+1)-dimensional Ito equation

A novel type of exact rogue wave is found for the (1+1)-dimensional Ito equation, which is generated by the interaction solution between an algebraic localized soliton (named “lump”) and an exponentially localized twin soliton. In addition, the interaction solution among triangular periodic wave and twin soliton is also proposed. Three special interaction phenomenons are displayed by some visual figures, respectively.     

Wick型随机(2+1)维Broer-Kaup方程的B(a)cklund变换和白噪声泛函解

本文对(2+1)维变系数Broer-Kaup方程和wick型随机(2+1)维Broer-Kaup方程进行了研究,利用Hermite变换、齐次平衡法以及tanh函数法给出了wick型随机(2+1)维Broer-Kaup方程的B(a)cklund变换和白噪声泛函解.     

一个2+1维变形Boussinesq方程的N孤子解

研究了一个2＋1维变形Boussinesq非线性发展方程：utt-uxx-uyy-3（u^2）xx-uxxxx=0,运用Hirota双线性方法得到它的N孤子解.     

Painlevé analysis,lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painlevé–Bäcklund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.     

Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.