# Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

Authors: Hang Zheng; Yonghui Xia; Yuzhen Bai; Guo Lei
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

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


---

## Body

## 1. Introduction

Recently, many authors made some efforts on nonlinear partial differential equations (NPDEs) (see [1–14]). Wu and Zhang [15] proposed the equation of the form(1)ϵt+ϵϵx+uϵy+vx=0,ut+ϵux+uuy+vy=0,vt+ϵvx+uvy+13ϵxxx+ϵxyy+uxxy+uyyy=0,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. Through some transformations, equation (1) reduces into the (1 + 1)-dimensional dispersive long wave equation (Wu–Zhang system) as follows:(2)ut=−uux−vx,vt=−vux−uvx−13uxxx.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 [19–21], 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 [16–24], 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 [25–39]. Another purpose is to study exact solutions of the fractional-order Wu–Zhang system [16]. It takes the form(3)Dtαu=−uux−vx,Dtαv=−vux−uvx−13uxxx,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.The structure of this paper is as follows. In Section2, 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.


## 2. Exact Solutions of Integer-Order Wu–Zhang System (2)

In this part, we first consider the exact solutions of integer-order Wu–Zhang system (2).By the following transformations,(4)ux,t=uξ,vx,t=vξ,ξ=x−lt,we have(5)∂ux,t∂t=−luξ,∂vx,t∂t=−lvξ,∂3ux,t∂x3=uξξξ,where uξ and vξ denote the first-order derivative with respect to ξ, respectively, and uξξξ means the third-order derivative with respect to ξ. We substitute (4) and (5) into system (2); then, (2) turns into(6)luξ=uuξ+vξ,lvξ=vuξ+uvξ+13uξξξ.Then, by integration over both sides of the first equation (6) and setting constant of integration to zero, hence, we have vξ=luξ−1/2u2ξ. Substituting it into the second equation of (6), we have the following ordinary differential equation (ODE):(7)uξξξ−92u2uξ+9luuξ−3l2uξ=0.Integrating both sides on (7), it follows that(8)uξξ−32u3+92lu2−3l2u=c.Obviously, equation (8) reduces to the planar dynamical system:(9)dudξ=y,dydξ=32u3−92lu2+3l2u+c.Meanwhile, the first integral is written as(10)Hu,y=12y2−38u4+32lu3−32l2u2−cu=h.We letgu=3/2u3−9/2lu2+3l2u+c; then, g′u=9/2u2−9lu+3l2. Obviously, the roots of gu=0 depend on the parameter group l,c taking different values. (i) If c>3/3l3l>0 or c<−3/3l3l>0 or c>−3/3l3l<0 or c<3/3l3l<0, then gu has only one real root u1; (ii) if c=±3/3l3l>0orl<0, then gu has two real roots: u2 and u3u3<u2; and (iii) if −3/3l3<c<3/3l3l>0 or 3/3l3<c<−3/3l3l<0, then gu has three real roots: u4, u5, and u6u6<u5<u4.Furthermore, noting that(11)Juj,yj=detMuj,yj=−g′u=−92u2−9lu+3l2,where Muj,yj is the coefficient matrix of the linearized system of (9) at an equilibrium point Ejj=1,2,…,6, Ej=uj,0. Using the theory of equilibrium points (see [42]), it is easy to compute that Ju1,0<0 (saddle point) (see Figure 1(a)), Ju3,0<0 (saddle point), Ju2,0=0 (cusp point) (see Figure 1(b)) or Ju2,0<0 (saddle point), Ju3,0=0 (cusp point) (see Figure 1(c)), and Ju4,0<0 (saddle point), Ju5,0>0 (center point), Ju6,0<0 (saddle point) (see Figure 1(d)). Then, we draw the bifurcation of phase portraits of system (9) by Maple (see Figure 1).Figure 1
Bifurcation of phase portraits of system (9). (a) One equilibrium point. (b, c) Two equilibrium points. (d) Three equilibrium points. Parameters: (a) c=4, l=3, (b) c=3, l=3, (c) c=−3, l=3, and (d) c=0.2, l=3.
(a)(b)(c)(d)Figure 2
Periodic wave of (9) and exact periodic wave solutions of (2) when c=0.2 and l=3. (a) Periodic wave. (b) Exact periodic wave solutions.
(a)(b)From (10), we set integral constant h be fixed, which yields(12)y2=34u4−3lu3+3l2u2+2cu+2h=34Gu.Then, we integrate over a branch of the curve from initial valueut0=u0. That is,(13)ξ=∫u0u43Gsds.Lethj=Huj,0j=1,2,…,6; for the phase portraits, according to the parameters, we get h2<h3 and h5<h4<h6, respectively. Then, we just consider Figure 1(d) as follows:(i)
Firstly, ifh∈h5,h4, we get the green curve. It corresponds to a family of periodic orbits of system (9) surrounding E5u5,0. Then, Gu=λ1−uλ2−uu−λ3u−λ4 (λ1,λ2,λ3,andλ4 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 λ4<λ3≤u<λ2<λ1, then we have(14)∫λ3udsλ1−sλ2−ss−λ3s−λ4=gsn−1sinφ,k1=32ξ,whereg=2/λ1−λ3λ2−λ4 and k12=λ2−λ3λ1−λ4/λ1−λ3λ2−λ4. Consequently, on the basis of (14), sn−1sinφ,k1=3/2gξ; by virtue of φ=sin−1λ2−λ4u−λ3/λ2−λ3u−λ4, it leads us to sinφ=snω1ξ,k1=λ2−λ4u−λ3/λ2−λ3u−λ4, where ω1=3λ1−λ3λ2−λ4/4.Therefore, the parametric expression for the periodic orbit of system (9) is given by (see Figure 2(a))(15)uξ=λ4λ2−λ3sn2ω1ξ,k1−λ3λ2−λ4λ2−λ3sn2ω1ξ,k1−λ2−λ4,wheresnξ,k is the Jacobian elliptic function. Correspondingly, the exact periodic wave solutions of equation (2) (see Figure 2(b)) can be written as(16)ux,t=λ4λ2−λ3sn2ω1x−lt,k1−λ3λ2−λ4λ2−λ3sn2ω1x−lt,k1−λ2−λ4.(ii)
Then, ifh=h4, we get the red level curve. Now, it corresponds to a homoclinic orbit surrounding E5u5,0. Thus, Gu=λ1−u2u−λ2u−λ3 (λ1,λ2,andλ3 are the points of intersection of the red curve with the u-axis in Figure 1(d)). When λ3<λ2<u<λ1, then we have(17)∫λ2udsλ1−s2s−λ2s−λ3=∫λ2udsλ1−ss−λ2s−λ3=32ξ.Letv=λ1−u; equation (17) turns into ∫λ1−λ2λ1−u−dv/vλ1−λ2−vλ1−λ3−v=3/2ξ, and by calculating, we have(18)vξ=2λ2−λ1λ3−λ1λ2−λ3coshω2ξ−λ2+λ3−2λ1.Therefore, the exact parametric expression for the homoclinic orbit of system (9) is obtained by (see Figure 3(a))(19)uξ=λ1−2λ2−λ1λ3−λ1λ2−λ3coshω2ξ−λ2+λ3−2λ1,where ω2=3/4λ2−λ1λ3−λ1. Hence, correspondingly, we obtain solitary wave solutions of equation (2) as follows (see Figure 3(b)):(20)ux,t=λ1−2λ2−λ1λ3−λ1λ2−λ3coshω2x−lt−λ2+λ3−2λ1.Figure 3
Solitary wave of (9) and exact solitary wave solutions of (2) when c=0.2 and l=3. (a) Solitary wave. (b) Solitary wave solutions.
(a)(b)Through the above analysis, we get the following theorems:Theorem 1.
If the parameter groupl,c satisfies −3/3l3<c<3/3l3l>0 or 3/3l3<c<−3/3l3l<0 and level curves are defined by h∈h5,h4, then equation (2) has the periodic wave solutions with the exact parametric expression given by (16).Theorem 2.
If the parameter groupl,c satisfies −3/3l3<c<3/3l3l>0 or 3/3l3<c<−3/3l3l<0 and level curves are defined by h=h4, then equation (2) has the solitary wave solutions with the exact parametric expression given by (20).


## 3. 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−ltα, we have(21)Dtαux,t=t1−α∂ux,t∂t=t1−αduξdt=t1−αuξdξdt=t1−αuξ−lαtα−1=−lαuξ,(22)Dtαvx,t=t1−α∂vx,t∂t=t1−αdvξdt=t1−αvξdξdt=t1−αvξ−lαtα−1=−lαvξ.Substituting (21) and (22) into system (3), then (3) turns into(23)lαuξ=uuξ+vξ,lαvξ=vuξ+uvξ+13uξξξ.Similar discussion to the integer-order Wu–Zhang system leads us to(24)uξξ−32u3+92lαu2−3l2α2u=c.Then, equation (24) reduces to(25)dudξ=y,dydξ=32u3−92lαu2+3l2α2u+c.Meanwhile, the first integral is written as(26)Hu,y=12y2−38u4+32lαu3−32l2α2u2−cu=h.It is not difficult to see that the first integral in the fractional-order case depends on the fractionalα. Consequently, the equilibrium points E˜iu˜i,0i=1,2,…,6 of system (25) change with the fractional α. Similarly, we let g˜u=3/2u3−9/2lαu2+3l2α2u+c; then, g˜′u=9/2u2−9lαu+3l2α2. The roots of g˜u=0 also depend on the parameter group l,c,α taking different values. (i) If c>3/3l3α3l>0 or c<−3/3l3α3l>0 or c>−3/3l3α3l<0 or c<3/3l3α3l<0, then g˜u has only one real root u˜1; (ii) if c=±3/3l3α3l>0 or l<0, then g˜u has two real roots: u˜2 and u˜3u˜3<u˜2; and (iii) if −3/3l3α3<c<3/3l3α3l>0 or 3/3l3α3<c<−3/3l3α3l<0, then g˜u has three real roots: u˜4, u˜5, and u˜6u˜6<u˜5<u˜4.Then, as the same discussion as before, we seth˜2<h˜3 and h˜5<h˜4<h˜6. Obviously, it has the similar representation of periodic orbits as (15) of the form(27)uξ=λ˜4λ˜2−λ˜3sn2ω˜1ξ,k˜1−λ˜3λ˜2−λ˜4λ˜2−λ˜3sn2ω˜1ξ,k˜1−λ˜2−λ˜4,where ω˜1=3λ˜1−λ˜3λ˜2−λ˜4/2andk˜12=λ˜2−λ˜3λ˜1−λ˜4/λ˜1−λ˜3λ˜2−λ˜4, λ˜4<λ˜3<λ˜2<λ˜1. Consequently, we have the exact periodic wave solutions of system (3):(28)ux,t=λ˜4λ˜2−λ˜3sn2ω˜1x−ltα,k˜1−λ˜3λ˜2−λ˜4λ˜2−λ˜3sn2ω˜1x−ltα,k˜1−λ˜2−λ˜4.We also get similar parametric representation for a homoclinic orbit of (25) when level curves are defined by h˜=h˜4 as follows:(29)uξ=λ˜1−2λ˜2−λ˜1λ˜3−λ˜1λ˜2−λ˜3coshω˜2ξ−λ˜2+λ˜3−2λ˜1,where ω˜2=3/4λ˜2−λ˜1λ˜3−λ˜1. Therefore, the exact solitary wave solutions of system (3) can be written as(30)ux,t=λ˜1−2λ˜2−λ˜1λ˜3−λ˜1λ˜2−λ˜3coshω˜2x−ltα−λ˜2+λ˜3−2λ˜1.For the fractional-order situation, we have the similar theorems:Theorem 3.
If the parameter groupl,c,α0<α<1 satisfies −3/3l3α3<c<3/3l3α3l>0 or 3/3l3α3<c<−3/3l3α3l<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).Theorem 4.
If the parameter groupl,c,α0<α<1 satisfies −3/3l3α3<c<3/3l3α3l>0 or 3/3l3α3<c<−3/3l3α3l<0 and level curves are defined by h˜=h˜4, then equation (3) has the solitary wave solutions with the exact parametric expression given by (30).


## 4. Comparison between the Integer-Order Wu–Zhang System and the Fractional-Order Wu–Zhang System

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).Obviously, we see that the nonzero equilibrium points are related to the derivative orderα. The equilibrium points amount to E43.4414364477,0, E51.0524405702,0, and E6−0.02180859881,0 in Figure 4(a). Nevertheless, in Figure 4(b), the equilibrium points amount to E˜41.6236627178,0, E˜51.0524405702,0, and E˜6−0.078027076777,0. We find that the level curves defined by the first integral h (periodic orbits and homoclinic orbits) all rely on α. Thus, the corresponding exact parametric representations of the Wu–Zhang system also change along with α.Figure 4
Phase portraits of system (9) and (25) when c=0.2 and l=3. (a) α=1,E4u4,0,E5u5,0,andE6u6,0. (b) α=1/2,E˜4u˜4,0,E˜5u˜5,0,andE˜6u˜6,0.
(a)(b)The exact solitary wave solution of the integer-order Wu–Zhang system is obtained by (see Figure5(b))(31)ux,t=λ1−2λ2−λ1λ3−λ1λ2−λ3coshω2x−lt−λ2+λ3−2λ1.Figure 5
Comparison between the integer-order and fractional-order Wu–Zhang system whenc=0.2 and l=3. (a) α=1 solitary wave. (b) α=1 exact solitary wave solutions. (c) α=1/2 solitary wave. (d) α=1/2 exact solitary wave solutions.
(a)(b)(c)(d)However, for the fractional-order Wu–Zhang system, the exact solitary wave solutions can be written as (see Figure5(d))(32)ux,t=λ˜1−2λ˜2−λ˜1λ˜3−λ˜1λ˜2−λ˜3coshω˜2x−ltα−λ˜2+λ˜3−2λ˜1.We compare Figure5(a) with Figures 5(c) and 5(b) with Figure 5(d), respectively. Certainly, the height and opening size of the solitary wave are different. The height of the solitary wave of the integer-order Wu–Zhang system is 3.3 in Figures 5(a) and 5(b), while the height of the solitary wave of the fractional-order Wu–Zhang system is almost 1.65 in Figures 5(c) and 5(d). Meanwhile, opening size of the integer order is less than the fractional order. Thus, the height and opening size of the solitary wave all depend on the derivative order α.


## 5. Conclusion

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 orderα.



---

*Source: 2845841-2020-06-27.xml*