Ahmet Bekir

New solitons and periodic wave solutions for some nonlinear physical models by using the sine–cosine method

In this paper, we establish exact solutions for nonlinear equations. The sine–cosine method is used to construct periodic and soliton solutions of nonlinear physical models. Many new families of exact travelling wave solutions of the symmetric regularized long-wave (SRLW) and the Klein–Gordon–Zakharov (KGZ) equations are successfully obtained. These solutions may be important 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.

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.

The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations

In this paper, we establish exact solutions for some nonlinear fractional differential equations (FDEs). The first integral method with help of the fractional complex transform (FCT) is used to obtain exact solutions for the time fractional modified Korteweg–de Vries (fmKdV) equation and the space–time fractional modified Benjamin–Bona–Mahony (fmBBM) equation. This method is efficient and powerful in solving kind of other nonlinear FDEs.

Travelling wave solutions of the Cahn–Allen equation by using first integral method

Abstract In this paper, we established travelling wave solutions of the nonlinear equation. The first integral method was used to construct travelling wave solutions of the Cahn–Allen equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed.

Application of Exp-function method for (3+1)-dimensional nonlinear evolution equations

In this paper, the Exp-function method is used to construct solitary and periodic solutions of nonlinear partial differential equations. We choose an example, which includes Kadomstev-Petviashvili equation and potantial-YTSF equation, to illustrate the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave equations.

Application of the Exp-function method for nonlinear differential-difference equations

In this paper, we established abundant travelling wave solutions for some nonlinear differential-difference equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics.

Exponential rational function method for space–time fractional differential equations

In this paper, exponential rational function method is applied to obtain analytical solutions of the space–time fractional Fokas equation, the space–time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space–time fractional coupled Burgers’ equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.

Exact solutions of coupled nonlinear Klein-Gordon equations

In this paper, we employ the tanh method for traveling wave solutions of coupled nonlinear Klein-Gordon equations. Based on the idea of the tanh method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The solutions obtained include solitons and periodic solutions.

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.

A travelling wave solution to the Ostrovsky equation

In this paper, we consider a nonlinear evolution equation like Ostrovsky equation. By using the hyperbolic tangent method and an exponential function approach, a travelling wave solution for the Ostrovsky equation is presented. It is observed that both methods lead to the same type of solution.