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In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order

Recently, many authors made some efforts on nonlinear partial differential equations (NPDEs) (see [

In fact, the Wu–Zhang system can describe the nonlinear water wave availability, and many engineers apply it in harbor and coastal design. For mathematical physics, one of the most important topics is to find the exact solutions. Many authors proposed various methods to solve the Wu–Zhang system numerically. We summarize as follows: the first integral method [

However, different from the aforementioned methods [

The structure of this paper is as follows. In Section

In this part, we first consider the exact solutions of integer-order Wu–Zhang system (

By the following transformations,

Then, by integration over both sides of the first equation (

Integrating both sides on (

Obviously, equation (

Meanwhile, the first integral is written as

We let

Furthermore, noting that

Bifurcation of phase portraits of system (

Periodic wave of (

From (

Then, we integrate over a branch of the curve from initial value

Let

Firstly, if

where

Therefore, the parametric expression for the periodic orbit of system (

where

Then, if

Let

Therefore, the exact parametric expression for the homoclinic orbit of system (

Solitary wave of (

Through the above analysis, we get the following theorems:

If the parameter group

If the parameter group

Secondly, we consider the fractional-order system. Here, we use the conformable fractional derivative proposed by Khalil et al. [

Substituting (

Similar discussion to the integer-order Wu–Zhang system leads us to

Then, equation (

Meanwhile, the first integral is written as

It is not difficult to see that the first integral in the fractional-order case depends on the fractional

Then, as the same discussion as before, we set

We also get similar parametric representation for a homoclinic orbit of (

For the fractional-order situation, we have the similar theorems:

If the parameter group

If the parameter group

In this part, we compare the phase portraits and exact solutions of case (ii) for the integer-order and fractional-order Wu–Zhang system. Under the same parameters (

Obviously, we see that the nonzero equilibrium points are related to the derivative order

Phase portraits of system (

The exact solitary wave solution of the integer-order Wu–Zhang system is obtained by (see Figure

Comparison between the integer-order and fractional-order Wu–Zhang system when

However, for the fractional-order Wu–Zhang system, the exact solitary wave solutions can be written as (see Figure

We compare Figure

This paper has studied the bifurcation and exact solutions of the Wu–Zhang system. We employ the dynamical system method to obtain the solitary wave solutions and periodic wave solutions. Moreover, we studied the integer-order and fractional-order Wu–Zhang system in a united way. We find that the bifurcation of phase portraits, nonzero equilibrium points, and exact solutions for the integer-order and fractional-order Wu–Zhang system all depend on the derivative order

The authors declare that they have no conflicts of interest.

H. Zheng carried out the computations and figures in the proof. L. Guo helped to replot the figures and participated in the revision. Y. Xia conceived the study and designed and drafted the manuscript. Y. Bai participated in the discussion of the project. All authors read and approved the final manuscript.

This work was jointly supported by the Natural Science Foundation of Zhejiang Province (Grant no. LY20A010016), the National Natural Science Foundation of China (Grant no. 11931016), Fujian Province Young Middle-Aged Teachers Education Scientific Research Project (no. JT180558), the Foundation of Science and Technology Project for the Education Department of Fujian Province (no. JA15512), and the Scientific Research Foundation for the Introduced Senior Talents, Wuyi University (Grant no. YJ201802).