We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

In the recent years, investigations of exact solutions to nonlinear partial differential equations (NPDEs) play an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geochemistry. To obtain traveling wave solutions, many powerful methods have been presented, such as the inverse scattering method [

In this paper, we will use the generalized projective Riccati equations method to construct exact solutions for the following three nonlinear evolution equations with higher-order nonlinear terms:

(i) the nonlinear Pochhammer-Chree equation [

(ii) the nonlinear Burgers equation [

(iii) the nonlinear generalized Zakharov-Kuznetsov equation [

Zuo [

Consider we have the following NPDE:

We use the wave transformation

We assume that (

If

We determine the positive integer

(a) When

(b) When

Substitute (

It is well known [

(i) If

(ii) If

(iii) If

Substituting the values of

We close this section with the remark that without loss of generality we take

In this section, we will apply the proposed method described in Section

In this example, we find the exact solutions of (

Substituting (

We have

We have

We have

In this example, we study the Burgers equation with power-law nonlinearity (

Substituting (

We have

We have

In this example, we study the generalized Zakharov-Kuznetsov equation with power-law nonlinearity (

Substituting (

We have

We have

We have

We have

We have

In this section, we have presented some graphs of the obtained solutions constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation. Using mathematical software Maple, three-dimensional plots of some obtained exact solutions have been shown in Figures

The plot of solution (

The plot of solution (

The plot of solution (

The plot of solution (

The plot of solution (

The plot of solution (

The plot of solution (

The plot of solution (

The obtained solutions for this equation are hyperbolic. From these explicit results it is easy to say that the solution (

From the obtained solutions for the nonlinear Burgers equation (

From the obtained solutions for the generalized nonlinear Zakharov-Kuznetsov equation (

The generalized projective Riccati equations method is used in this paper to obtain some new exact solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation, and the generalized Zakharov-Kuznetsov equation. On comparing our results in this paper with the well-known results obtained in [

The authors declare that there is no conflict of interests regarding the publication of this paper.