Generalized solitary and periodic wave solutions to a (2 + 1)-dimensional Zakharov-Kuznetsov equation

In this paper, the Exp-function method is employed to the Zakharov-Kuznetsov equation as a (2 + 1)-dimensional model for nonlinear Rossby waves. The observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient. The obtained results may be useful for better understanding the properties of two-dimensional coherent structures such as atmospheric blocking events.     

Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

In this paper, new exact solutions with two arbitrary functions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by means of the Riccati equation and its generalized solitary wave solutions constructed by the Exp-function method. It is shown that the Exp-function method provides us with a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.     

A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper the (2 + 1)-dimensional Boiti-Leon-Pempinelli (BLP) equation will be studied. The tanh-coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures.     

Analytic investigation of the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation for traveling wave solutions

By means of the two distinct methods, the Exp-function method and the extended (G′/G)-expansion method, we successfully performed an analytic study on the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.     

Explicit periodic solitary wave solutions for the (2 + 1)-dimensional Boussinesq equation

In this paper, (2 + 1)-dimensional Boussinesq equation is investigated. By using homoclinic test method with the aid of Maple, new explicit periodic solitary wave solutions are obtained. Moreover, mechanical feature of wave is exhibited.     

Finding general and explicit solutions (2 + 1) dimensional Broer-Kaup-Kupershmidt system nonlinear equation by exp-function method

In this work, we implement a relatively new analytical technique, the exp-function method, for solving nonlinear special form of generalized nonlinear (2 + 1) dimensional Broer-Kaup-Kupershmidt equation, which may contain high nonlinear terms. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and reliability of exp method. It is predicted that exp-function method can be found widely applicable in engineering.     

Exact solutions and conservation laws of (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper, we obtain the symmetry group theorem by using the modified CK’s direct method, and some new exact solutions of (2 + 1)-dimensional BLP equation. Also we derive the corresponding Lie algebra and the conservation laws of BLP equation.     

Exact solutions of the Drinfel’d-Sokolov-Wilson equation using the Exp-function method

The generalized solitary solutions of the classical Drinfel’d-Sokolov-Wilson equation (DSWE) are obtained using the Exp-function method. Then, some of these solutions are easily converted into kink-shaped solutions and blow-up solutions by a simple transformation.     

A (3 + 1)-dimensional nonlinear evolution equation with multiple soliton solutions and multiple singular soliton solutions

In this work, a (3 + 1)-dimensional nonlinear evolution equation is investigated. The Hirota’s bilinear method is applied to determine the necessary conditions for the complete integrability of this equation. Multiple soliton solutions are established to confirm the compatibility structure. Multiple singular soliton solutions are also derived. The resonance phenomenon does not exist for this model.     

Explicit solutions of the (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients

Using an improved direct reduction method, we find the equivalence transformations of (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients, and obtain the corresponding relationship between explicit solutions of AKNS equation and those of the corresponding reduced equation. In addition, we get some new explicit solutions of AKNS equation by applying Lie symmetry method.     

Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method

In this paper, we establish exact solutions for (2 + 1)-dimensional nonlinear evolution equations. The sine-cosine method is used to construct exact periodic and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Boussinesq, breaking soliton and BKP equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.     

Exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation using Lie symmetry analysis

In this paper, Lie group analysis is employed to derive some exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation which describes the dynamics of solitons and nonlinear waves in plasmas and superfluids.     

F展开法结合指数函数法求解Zhiber-Shabat 方程的精确解

利用F展开法与指数函数法相结合的方法,在相关文献的基础上,重新研究了Zhiber-Shabat方程,获得了许多与现有文献中解的表达式不相同的各种精确解.这些解同样具有孤立波解,纽子波解和周期波解的各种动力学特征.从而丰富了相关文献中关于Zhiber-Shabat波方程的孤立子解和周期解的种类.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

Exact solitary wave solutions of the generalized (2 + 1) dimensional Boussinesq equation

Bifurcation method of dynamical systems is employed to investigate bifurcation of solitary waves in the generalized (2 + 1) dimensional Boussinesq equation. Numbers of solitary waves are given for each parameter condition. Under some parameter conditions, exact solitary wave solutions are obtained.     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

New exact solutions to the (2 + 1)-dimensional Ito equation: Extended homoclinic test technique

Exact soliton solutions to the (2 + 1)-dimensional Ito equation are studied based on the idea of extended homoclinic test and bilinear method. Some explicit solutions, such as triangle function solutions, soliton solutions, doubly-periodic wave solutions and periodic solitary wave solutions, are obtained. It shows that the (2 + 1)-dimensional Ito equation has richer solutions. Besides, the elastic interactions of the solutions and their corresponding physical meaning are discussed.     

Four (2 + 1)-dimensional integrable extensions of the Kadomtsev-Petviashvili equation

In this work, four (2 + 1)-dimensional nonlinear extensions of the Kadomtsev-Petviashvili (KP) equation are developed. The complete integrability of these models are investigated. Multiple-soliton solutions and multiple singular soliton solutions are determined to demonstrate the compatibility of these models. The resonance phenomenon does not exist for any of the derived models.     

New exact traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation

In this paper, some novel solitary wave solutions, including solitary-like wave solution, x-periodic soliton solution, y-periodic soliton solution, doubly periodic solution, rational solution, and new non-traveling wave solution, are obtained for (2 + 1)-dimensional Burgers equation by means of the generalized direct ansätz method and different test functions.