Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

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 α. Phase portraits and simulations are given to show the validity of the obtained solutions.


Introduction
Recently, many authors made some efforts on nonlinear partial differential equations (NPDEs) (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Wu and Zhang [15] proposed the equation of the form where ϵ and u denote the surface velocity of water along the x-direction and the y-direction, respectively, and v means the elevation of water. rough some transformations, equation (1) reduces into the (1 + 1)-dimensional dispersive long wave equation (Wu-Zhang system) as follows: 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 [16], extended tanh-function method [17], characteristic function method [18], modified Conte's invariant Painlevé expansion method and truncation of the WTC's approach [19][20][21], elliptic function rational expansion method [22], generalized extended tanh-function method [23], generalized extended rational expansion method [24], and so on.
However, different from the aforementioned methods [16][17][18][19][20][21][22][23][24], in this paper, we apply the dynamical system method to study the bifurcation and exact solutions of NPDEs. Dynamical system method is quite different from the mentioned methods, and it has many successful applications [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Another purpose is to study exact solutions of the fractional-order Wu-Zhang system [16]. It takes the form and t > 0 and 0 < α ≤ 1. So far, Khater et al. [40,41] studied the fractional-order Wu-Zhang system by the numerical method and the modified auxiliary equation method. Inspired by [38], we study the fractional-order Wu-Zhang system via the extended three-step method. e structure of this paper is as follows. In Section 2, we first consider the bifurcation of phase portraits for the integer-order Wu-Zhang system. Correspondingly, we calculate all possible exact parametric expressions of solutions to the integer-order Wu-Zhang system. In Section 3, we study the bifurcation of phase portraits and exact solutions of the fractional-order Wu-Zhang system. In Section 4, comparison of phase portraits and exact solutions between the integer-order and fractional-order Wu-Zhang system is presented. Finally, we end this paper with a conclusion.

Exact Solutions of Integer-Order Wu-Zhang System (2)
In this part, we first consider the exact solutions of integerorder Wu-Zhang system (2). By the following transformations, we have where u ξ and v ξ denote the first-order derivative with respect to ξ, respectively, and u ξξξ means the third-order derivative with respect to ξ. We substitute (4) and (5) into system (2); then, (2) turns into en, by integration over both sides of the first equation (6) and setting constant of integration to zero, hence, we have v(ξ) � lu(ξ) − (1/2)u 2 (ξ). Substituting it into the second equation of (6), we have the following ordinary differential equation (ODE): Integrating both sides on (7), it follows that Obviously, equation (8) reduces to the planar dynamical system: Meanwhile, the first integral is written as We let g(u) � (3/2)u 3 − (9/2)lu 2 + 3l 2 u + c; then, g ′ (u) � (9/2)u 2 − 9lu + 3l 2 . Obviously, the roots of g(u) � 0 depend on the parameter group (l, c) taking different values.
From (10), we set integral constant h be fixed, which yields en, we integrate over a branch of the curve from initial value u(t 0 ) � u 0 . at is, Let h j � H(u j , 0)(j � 1, 2, . . . , 6); for the phase portraits, according to the parameters, we get h 2 < h 3 and h 5 < h 4 < h 6 , respectively. en, we just consider Figure 1(d) as follows: h 4 ), we get the green curve. It corresponds to a family of periodic orbits of system , and λ 4 are the points of intersection of the green curve with the u-axis in Figure 1(d)). Using the formula (see 254.00 in [43]), when where where . erefore, the parametric expression for the periodic orbit of system (9) is given by (see Figure 2(a)) where sn(ξ, k) is the Jacobian elliptic function. Correspondingly, the exact periodic wave solutions of equation (2) (see Figure 2(b)) can be written as Discrete Dynamics in Nature and Society 3 (ii) en, if h � h 4 , we get the red level curve. Now, it corresponds to a homoclinic orbit surrounding E 5 (u 5 , 0). us, G(u) � (λ 1 − u) 2 (u − λ 2 )(u − λ 3 ) (λ 1 , λ 2 , and λ 3 are the points of intersection of the red curve with the u-axis in Figure 1(d)). When λ 3 < λ 2 < u < λ 1 , then we have Let erefore, the exact parametric expression for the homoclinic orbit of system (9) is obtained by (see Figure 3(a)) where ω 2 � ������������������� (3/4)(λ 2 − λ 1 )(λ 3 − λ 1 ). Hence, correspondingly, we obtain solitary wave solutions of equation (2) as follows (see Figure 3(b)): rough the above analysis, we get the following theorems: /3)l 3 (l < 0) and level curves are defined by h ∈ (h 5 , h 4 ), then equation (2) has the periodic wave solutions with the exact parametric expression given by (16). 3 (l < 0) and level curves are defined by h � h 4 , then equation (2) has the solitary wave solutions with the exact parametric expression given by (20).

Exact Solutions of Fractional-Order Wu-Zhang System (3)
Secondly, we consider the fractional-order system. Here, we use the conformable fractional derivative proposed by Khalil et al. [44]. Different from (4), taking ξ � x − lt α , we have  Discrete Dynamics in Nature and Society Substituting (21) and (22)
/3)l 3 α 3 (l < 0), then g(u) has only one real root u 1 ; (ii) if c � ± ( � 3 √ /3)l 3 α 3 (l > 0 or l < 0), then g(u) has two real roots: u 2 and u 3 (u 3 < u 2 ); and (iii) if /3)l 3 α 3 (l < 0), then g(u) has three real roots: u 4 , u 5 , and u 6 (u 6 < u 5 < u 4 ). en, as the same discussion as before, we set h 2 < h 3 and h 5 < h 4 < h 6 . Obviously, it has the similar representation of periodic orbits as (15) of the form where Consequently, we have the exact periodic wave solutions of system (3): We also get similar parametric representation for a homoclinic orbit of (25) when level curves are defined by h � h 4 as follows: where . erefore, the exact solitary wave solutions of system (3) can be written as For the fractional-order situation, we have the similar theorems: /3)l 3 α 3 (l < 0) and level curves are defined by h ∈ (h 5 , h 4 ), then equation (3) has the periodic wave solutions with the exact parametric expression given by (28).  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 (c � 0.2 and l � � 3 √ ), we take different derivative orders α � 1 and α � 1/2, respectively. According to different α, we obtain the different phase portraits of system (9) and system (25).
us, the corresponding exact parametric representations of the Wu-Zhang system also change along with α. e exact solitary wave solution of the integer-order Wu-Zhang system is obtained by (see Figure 5 However, for the fractional-order Wu-Zhang system, the exact solitary wave solutions can be written as (see Figure 5(d)) We compare Figure 5

Conclusion
is 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 α.

Conflicts of Interest
e authors declare that they have no conflicts of interest.