TravellingWaveSolutionsofWu–ZhangSystemvia...

$: TravellingWaveSolutionsofWu–ZhangSystemvia …downloads.hindawi.com/journals/ddns/2020/2845841.pdfthe fractional-order Wu–Zhang system by the numerical method and the modiﬁed$
Review ArticleTravelling Wave Solutions of WundashZhang System viaDynamic Analysis

Hang Zheng12 Yonghui Xia 1 Yuzhen Bai 3 and Guo Lei2

1Department of Mathematics Zhejiang Normal University Jinhua 321004 China2Department of Mathematics and Computer Wuyi University Wuyishan 354300 China3School of Mathematical Sciences Qufu Normal University Qufu 273165 China

Correspondence should be addressed to Yonghui Xia xiadoc163com

Received 14 February 2020 Revised 9 April 2020 Accepted 3 June 2020 Published 27 June 2020

Academic Editor Kousuke Kuto

Copyright copy 2020 Hang Zheng et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper based on the dynamical system method we obtain the exact parametric expressions of the travelling wave solutionsof the WundashZhang system Our approach is much different from the existing literature studies on the WundashZhang systemMoreover we also study the fractional derivative of the WundashZhang system Finally by comparison between the integer-orderWundashZhang system and the fractional-order WundashZhang system we see that the phase portrait nonzero equilibrium points andthe corresponding exact travelling wave solutions all depend on the derivative order α Phase portraits and simulations are given toshow the validity of the obtained solutions

1 Introduction

Recently many authors made some efforts on nonlinearpartial differential equations (NPDEs) (see [1ndash14]) Wu andZhang [15] proposed the equation of the form

ϵt + ϵϵx + uϵy + vx 0

ut + ϵux + uuy + vy 0

vt +(ϵv)x +(uv)y +13ϵxxx + ϵxyy + uxxy + uyyy1113872 1113873 0

(1)

where ϵ and u denote the surface velocity of water along thex-direction and the y-direction respectively and v meansthe elevation of water -rough some transformationsequation (1) reduces into the (1 + 1)-dimensional dispersivelong wave equation (WundashZhang system) as follows

ut minus uux minus vx

vt minus vux minus uvx minus13uxxx

(2)

In fact the WundashZhang system can describe thenonlinear water wave availability and many engineersapply it in harbor and coastal design For mathematicalphysics one of the most important topics is to find theexact solutions Many authors proposed various methodsto solve the WundashZhang system numerically We sum-marize as follows the first integral method [16] extendedtanh-function method [17] characteristic functionmethod [18] modified Contersquos invariant Painleve ex-pansion method and truncation of the WTCrsquos approach[19ndash21] elliptic function rational expansion method [22]generalized extended tanh-function method [23] gener-alized extended rational expansion method [24] and soon

However different from the aforementioned methods[16ndash24] in this paper we apply the dynamical systemmethod to study the bifurcation and exact solutions ofNPDEs Dynamical system method is quite different fromthe mentioned methods and it has many successful appli-cations [25ndash39] Another purpose is to study exact solutionsof the fractional-order WundashZhang system [16] It takes theform

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 2845841 9 pageshttpsdoiorg10115520202845841

Dαt u minus uux minus vx

Dαt v minus vux minus uvx minus

13uxxx

(3)

and tgt 0 and 0lt αle 1 So far Khater et al [40 41] studiedthe fractional-order WundashZhang system by the numericalmethod and the modified auxiliary equation method In-spired by [38] we study the fractional-order WundashZhangsystem via the extended three-step method

-e structure of this paper is as follows In Section 2 wefirst consider the bifurcation of phase portraits for the in-teger-order WundashZhang system Correspondingly we cal-culate all possible exact parametric expressions of solutionsto the integer-order WundashZhang system In Section 3 westudy the bifurcation of phase portraits and exact solutionsof the fractional-order WundashZhang system In Section 4comparison of phase portraits and exact solutions betweenthe integer-order and fractional-order WundashZhang system ispresented Finally we end this paper with a conclusion

2 Exact Solutions of Integer-OrderWundashZhang System (2)

In this part we first consider the exact solutions of integer-order WundashZhang system (2)

By the following transformations

u(x t) u(ξ)

v(x t) v(ξ)

ξ x minus lt

(4)

we havezu(x t)

zt minus luξ

zv(x t)

zt minus lvξ

z3u(x t)

zx3 uξξξ

(5)

where uξ and vξ denote the first-order derivative with respectto ξ respectively and uξξξ means the third-order derivativewith respect to ξ We substitute (4) and (5) into system (2)then (2) turns into

luξ uuξ + vξ

lvξ vuξ + uvξ +13uξξξ

(6)

-en by integration over both sides of the first equation(6) and setting constant of integration to zero hence wehave v(ξ) lu(ξ) minus (12)u2(ξ) Substituting it into thesecond equation of (6) we have the following ordinarydifferential equation (ODE)

uξξξ minus92u2uξ + 9luuξ minus 3l

2uξ 0 (7)

Integrating both sides on (7) it follows that

uξξ minus32u3

+92

lu2

minus 3l2u c (8)

Obviously equation (8) reduces to the planar dynamicalsystem

du

dξ y

dy

dξ32u3

minus92

lu2

+ 3l2u + c

(9)

Meanwhile the first integral is written as

H(u y) 12y2

minus38u4

+32

lu3

minus32l2u2

minus cu h (10)

We let g(u) (32)u3 minus (92)lu2 + 3l2u + c thengprime(u) (92)u2 minus 9lu + 3l2 Obviously the roots of g(u)

0 depend on the parameter group (l c) taking differentvalues (i) If cgt (

3

radic3)l3(lgt 0) or clt minus (

3

radic3)l3(lgt 0) or

cgt minus (3

radic3)l3(llt 0) or clt (

3

radic3)l3(llt 0) then g(u) has

only one real root u1 (ii) if c plusmn (3

radic3)l3(lgt 0 or llt 0)

then g(u) has two real roots u2 and u3(u3 lt u2) and (iii) ifminus (

3

radic3)l3 lt clt (

3

radic3)l3(lgt 0) or (

3

radic3)l3 lt clt minus (

3

radic

3)l3(llt 0) then g(u) has three real roots u4 u5 andu6(u6 lt u5 lt u4)

Furthermore noting that

J uj yj1113872 1113873 detM uj yj1113872 1113873 minus gprime(u) minus92u2

minus 9lu + 3l2

1113874 1113875

(11)

where M(uj yj) is the coefficient matrix of the linearizedsystem of (9) at an equilibrium point Ej(j 1 2 6)Ej (uj 0) Using the theory of equilibrium points (see[42]) it is easy to compute that J(u1 0)lt 0 (saddle point)(see Figure 1(a)) J(u3 0)lt 0 (saddle point) J(u2 0) 0(cusp point) (see Figure 1(b)) or J(u2 0)lt 0 (saddle point)J(u3 0) 0 (cusp point) (see Figure 1(c)) and J(u4 0)lt 0(saddle point) J(u5 0)gt 0 (center point) J(u6 0)lt 0(saddle point) (see Figure 1(d)) -en we draw the bi-furcation of phase portraits of system (9) by Maple (seeFigure 1)

From (10) we set integral constant h be fixed whichyields

y2

34u4

minus 3lu3

+ 3l2u2

+ 2cu + 2h 34

G(u) (12)

-en we integrate over a branch of the curve from initialvalue u(t0) u0 -at is

ξ 1113946u

u0

4

3G(s)

1113971

ds (13)

Let hj H(uj 0)(j 1 2 6) for the phase por-traits according to the parameters we get h2 lt h3 andh5 lt h4 lt h6 respectively -en we just consider Figure 1(d)as follows

(i) Firstly if h isin (h5 h4) we get the green curve Itcorresponds to a family of periodic orbits of system(9) surrounding E5(u5 0) -en G(u) (λ1minus

2 Discrete Dynamics in Nature and Society

u)(λ2 minus u)(u minus λ3)(u minus λ4) (λ1 λ2 λ3 and λ4 are thepoints of intersection of the green curve with theu-axis in Figure 1(d)) Using the formula (see 25400in [43]) when λ4 lt λ3 le ult λ2 lt λ1 then we have

1113946u

λ3

dsλ1 minus s( 1113857 λ2 minus s( 1113857 s minus λ3( 1113857 s minus λ4( 1113857

1113969 gsnminus 1 sinφ k1( 1113857

3

radic

2ξ

(14)

where g 2(λ1 minus λ3)(λ2 minus λ4)

1113968and

k21 ((λ2 minus λ3)(λ1 minus λ4))((λ1 minus λ3)(λ2 minus λ4)) Conse-

quently on the basis of (14)snminus 1(sinφ k1) (

3

radic2g)ξ by virtue of

φ sinminus 1 ((λ2 minus λ4)(u minus λ3))((λ2 minus λ3)(u minus λ4))

1113968 it

leads us to sinφ sn(ω1ξ k1) ((λ2minus

1113968

λ4)(u minus λ3))((λ2 minus λ3)(u minus λ4)) whereω1

3(λ1 minus λ3)(λ2 minus λ4)

11139684

-erefore the parametric expression for the periodicorbit of system (9) is given by (see Figure 2(a))

u(ξ) λ4 λ2 minus λ3( 1113857sn2 ω1ξ k1( 1113857 minus λ3 λ2 minus λ4( 1113857

λ2 minus λ3( 1113857sn2 ω1ξ k1( 1113857 minus λ2 minus λ4( 1113857 (15)

where sn(ξ k) is the Jacobian elliptic function Cor-respondingly the exact periodic wave solutions ofequation (2) (see Figure 2(b)) can be written as

6

4

2

0

ndash2

ndash4

ndash6

ndash1 1 2 3 4u

y

(a)

u

y

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

ndash2 ndash1 21 3 4 5

(b)

u

y

8

6

4

2

ndash2

ndash4

ndash6

ndash8

ndash1 0 21 3 4 5

(c)

u

y

ndash1

3

2

1

0

ndash1

ndash2

ndash3

2 3 41

(d)

Figure 1 Bifurcation of phase portraits of system (9) (a) One equilibrium point (b c) Two equilibrium points (d) -ree equilibriumpoints Parameters (a) c 4 l

3

radic (b) c 3 l

3

radic (c) c minus 3 l

3

radic and (d) c 02 l

3

radic

Discrete Dynamics in Nature and Society 3

u(x t) λ4 λ2 minus λ3( 1113857sn2 ω1(x minus lt) k1( 1113857 minus λ3 λ2 minus λ4( 1113857

λ2 minus λ3( 1113857sn2 ω1(x minus lt) k1( 1113857 minus λ2 minus λ4( 1113857

(16)

(ii) -en if h h4 we get the red level curve Now itcorresponds to a homoclinic orbit surroundingE5(u5 0) -us G(u) (λ1 minus u)2(u minus λ2)(u minus λ3)(λ1 λ2 and λ3 are the points of intersection of thered curve with the u-axis in Figure 1(d)) Whenλ3 lt λ2 lt ult λ1 then we have

1113946u

λ2

ds

λ1 minus s( 11138572

s minus λ2( 1113857 s minus λ3( 1113857

1113969

1113946u

λ2

ds

λ1 minus s( 1113857

s minus λ2( 1113857 s minus λ3( 1113857

1113969

3

radic

2ξ

(17)

Let v λ1 minus u equation (17) turns into1113938λ1minus u

λ1minus λ2minus dv(v

(λ1 minus λ2 minus v)(λ1 minus λ3 minus v)

1113968) (

3

radic2)ξ and

by calculating we have

v(ξ) 2 λ2 minus λ1( 1113857 λ3 minus λ1( 1113857

λ2 minus λ3( 1113857cosh ω2ξ( 1113857 minus λ2 + λ3 minus 2λ1( 1113857 (18)

-erefore the exact parametric expression for thehomoclinic orbit of system (9) is obtained by (seeFigure 3(a))

u(ξ) λ1 minus2 λ2 minus λ1( 1113857 λ3 minus λ1( 1113857


where ω2 (34)(λ2 minus λ1)(λ3 minus λ1)

1113968 Hence correspond-

ingly we obtain solitary wave solutions of equation (2) asfollows (see Figure 3(b))

u(x t) λ1 minus2 λ2 minus λ1( 1113857 λ3 minus λ1( 1113857

λ2 minus λ3( 1113857cosh ω2(x minus lt)( 1113857 minus λ2 + λ3 minus 2λ1( 1113857

(20)

-rough the above analysis we get the followingtheorems

Theorem 1 If the parameter group (l c) satisfiesminus (

3

radic3)l3 lt clt (

3

radic3)l3(lgt 0) or

(3


3

radic3)l3(llt 0) and level curves are de-

fined by h isin (h5 h4) then equation (2) has the periodic wavesolutions with the exact parametric expression given by (16)


3

radic3)l3 lt clt (

3


3


3

radic

3)l3(llt 0) and level curves are defined by h h4 thenequation (2) has the solitary wave solutions with the exactparametric expression given by (20)

3 Exact Solutions of Fractional-OrderWundashZhang System (3)

Secondly we consider the fractional-order system Here weuse the conformable fractional derivative proposed by Khalilet al [44] Different from (4) taking ξ x minus ltα we have

Dαt u(x t) t

1minus αzu(x t)

zt t

1minus αdu(ξ)

dt t

1minus αuξdξdt

t1minus α

uξ(minus lα)tαminus 1

minus lαuξ

(21)

Dαt v(x t) t

1minus αzv(x t)

zt t

1minus αdv(ξ)

dt t

1minus αvξdξdt

t1minus α

vξ(minus lα)tαminus 1

minus lαvξ

(22)

ndash10 ndash5 0ξ

5 10

1

11

12

13

14u15

16

17

18

19

(a)

t8 6

4 2 0 ndash2 ndash4ndash6

ndash8

x

2

18

16

14

12

1

151

050

(b)

Figure 2 Periodic wave of (9) and exact periodic wave solutions of (2) when c 02 and l 3

radic (a) Periodic wave (b) Exact periodic wave

solutions


Substituting (21) and (22) into system (3) then (3) turnsinto

lαuξ uuξ + vξ

lαvξ vuξ + uvξ +13uξξξ

(23)

Similar discussion to the integer-order WundashZhangsystem leads us to

uξξ minus32u3

+92

lαu2

minus 3l2α2u c (24)

-en equation (24) reduces to

du

dξ y

dy

dξ32u3

minus92

lαu2

+ 3l2α2u + c

(25)


H(u y) 12y2

minus38u4

+32

lαu3

minus32l2α2u2

minus cu h (26)

It is not difficult to see that the first integral in thefractional-order case depends on the fractional α Conse-quently the equilibrium points 1113957Ei(1113957ui 0)(i 1 2 6) ofsystem (25) change with the fractional α Similarly we let1113957g(u) (32)u3 minus (92)lαu2 + 3l2α2u + c then 1113957gprime(u) (92)u2 minus 9lαu + 3l2α2 -e roots of 1113957g(u) 0 also depend onthe parameter group (l c α) taking different values (i) Ifcgt (

3

radic3)l3α3(lgt 0) or clt minus (

3

radic3)l3α3(lgt 0) or cgt minus

(3

radic3)l3α3(llt 0) or clt (

3

radic3)l3α3(llt 0) then 1113957g(u) has

only one real root 1113957u1 (ii) if c plusmn (3

radic3)l3α3(lgt 0 or llt 0)

then 1113957g(u) has two real roots 1113957u2 and 1113957u3 (1113957u3 lt 1113957u2) and (iii) ifminus (

3

radic3)l3α3 lt clt (

3

radic3)l3α3(lgt 0) or (

3

radic3)l3α3 lt clt minus

(3

radic3)l3α3(llt 0) then 1113957g(u) has three real roots 1113957u4 1113957u5 and

1113957u6 (1113957u6 lt 1113957u5 lt 1113957u4)

-en as the same discussion as before we set 1113957h2 lt 1113957h3 and1113957h5 lt 1113957h4 lt 1113957h6 Obviously it has the similar representation ofperiodic orbits as (15) of the form

u(ξ) 1113957λ4 1113957λ2 minus 1113957λ31113872 1113873sn2 1113957ω1ξ 1113957k11113872 1113873 minus 1113957λ3 1113957λ2 minus 1113957λ41113872 1113873

1113957λ2 minus 1113957λ31113872 1113873sn2 1113957ω1ξ 1113957k11113872 1113873 minus 1113957λ2 minus 1113957λ41113872 1113873 (27)

where 1113957ω1

3(1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)1113969

2 and 1113957k21 ((1113957λ2 minus 1113957λ3)

(1113957λ1 minus 1113957λ4))((1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)) 1113957λ4 lt 1113957λ3 lt 1113957λ2 lt 1113957λ1 Conse-quently we have the exact periodic wave solutions ofsystem (3)

u(x t) 1113957λ4 1113957λ2 minus 1113957λ31113872 1113873sn2 1113957ω1 x minus ltα( ) 1113957k11113872 1113873 minus 1113957λ3 1113957λ2 minus 1113957λ41113872 1113873

1113957λ2 minus 1113957λ31113872 1113873sn2 1113957ω1 x minus ltα( ) 1113957k11113872 1113873 minus 1113957λ2 minus 1113957λ41113872 1113873

(28)We also get similar parametric representation for

a homoclinic orbit of (25) when level curves are defined by1113957h 1113957h4 as follows

u(ξ) 1113957λ1 minus2 1113957λ2 minus 1113957λ11113872 1113873 1113957λ3 minus 1113957λ11113872 1113873

1113957λ2 minus 1113957λ31113872 1113873cosh 1113957ω2ξ( 1113857 minus 1113957λ2 + 1113957λ3 minus 21113957λ11113872 1113873 (29)

where 1113957ω2

(34)(1113957λ2 minus 1113957λ1)(1113957λ3 minus 1113957λ1)1113969

-erefore the exactsolitary wave solutions of system (3) can be written as

u(x t) 1113957λ1 minus2 1113957λ2 minus 1113957λ11113872 1113873 1113957λ3 minus 1113957λ11113872 1113873

1113957λ2 minus 1113957λ31113872 1113873cosh 1113957ω2 x minus ltα( )( 1113857 minus 1113957λ2 + 1113957λ3 minus 21113957λ11113872 1113873

(30)

For the fractional-order situation we have the similartheorems

Theorem 3 If the parameter group (l c α)(0lt αlt 1) sat-isfies minus (

3


3

radic3)l3α3(lgt 0) or

(3

radic3)l3α3 lt clt minus (

3

radic3)l3α3(llt 0) and level curves are

defined by 1113957h isin (1113957h51113957h4) then equation (3) has the periodic

wave solutions with the exact parametric expression given by(28)

3

25

2

15

u

ndash10 ndash5 0 5 10ξ

(a)

3

25

2

15

1

05

50

ndash5ndash10

x

005

115t

10

(b)

Figure 3 Solitary wave of (9) and exact solitary wave solutions of (2) when c 02 and l 3

radic (a) Solitary wave (b) Solitary wave solutions


0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)

Figure 4 Phase portraits of system (9) and (25) when c 02 and l 3

radic (a) α 1 E4(u4 0) E5(u5 0) andE6(u6 0) (b)

α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)

Figure 5 Comparison between the integer-order and fractional-order WundashZhang system when c 02 and l 3

radic (a) α 1 solitary wave

(b) α 1 exact solitary wave solutions (c) α 12 solitary wave (d) α 12 exact solitary wave solutions



3


3


3

radic3)l3α3 lt

clt minus (3

radic3)l3α3(llt 0) and level curves are defined by

1113957h 1113957h4 then equation (3) has the solitary wave solutions withthe exact parametric expression given by (30)

4 Comparison between the Integer-OrderWundashZhang System and the Fractional-OrderWundashZhang System

In this part we compare the phase portraits and exact so-lutions of case (ii) for the integer-order and fractional-orderWundashZhang system Under the same parameters (c 02 andl

3

radic) we take different derivative orders α 1 and

α 12 respectively According to different α we obtain thedifferent phase portraits of system (9) and system (25)

Obviously we see that the nonzero equilibrium pointsare related to the derivative order α -e equilibrium pointsamount to E4(34414364477 0) E5(10524405702 0)and E6(minus 002180859881 0) in Figure 4(a) Neverthelessin Figure 4(b) the equilibrium points amount to1113957E4(16236627178 0) 1113957E5(10524405702 0) and 1113957E6(minus 0078027076777 0) We find that the level curves definedby the first integral h (periodic orbits and homoclinic orbits)all rely on α -us the corresponding exact parametricrepresentations of the WundashZhang system also change alongwith α

-e exact solitary wave solution of the integer-orderWundashZhang system is obtained by (see Figure 5(b))



(31)

However for the fractional-order WundashZhang systemthe exact solitary wave solutions can be written as (seeFigure 5(d))



(32)

We compare Figure 5(a) with Figures 5(c) and 5(b) withFigure 5(d) respectively Certainly the height and opening sizeof the solitary wave are different-e height of the solitary waveof the integer-order WundashZhang system is 33 in Figures 5(a)and 5(b) while the height of the solitary wave of the fractional-order WundashZhang system is almost 165 in Figures 5(c) and5(d) Meanwhile opening size of the integer order is less thanthe fractional order -us the height and opening size of thesolitary wave all depend on the derivative order α

5 Conclusion

-is paper has studied the bifurcation and exact solutions ofthe WundashZhang system We employ the dynamical systemmethod to obtain the solitary wave solutions and periodicwave solutions Moreover we studied the integer-order andfractional-orderWundashZhang system in a united way We find

that the bifurcation of phase portraits nonzero equilibriumpoints and exact solutions for the integer-order and frac-tional-order WundashZhang system all depend on the derivativeorder α

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

H Zheng carried out the computations and figures in theproof L Guo helped to replot the figures and participated inthe revision Y Xia conceived the study and designed anddrafted the manuscript Y Bai participated in the discussionof the project All authors read and approved the finalmanuscript

Acknowledgments

-is work was jointly supported by the Natural ScienceFoundation of Zhejiang Province (Grant no LY20A010016)the National Natural Science Foundation of China (Grantno 11931016) Fujian Province Young Middle-AgedTeachers Education Scientific Research Project (noJT180558) the Foundation of Science and TechnologyProject for the Education Department of Fujian Province(no JA15512) and the Scientific Research Foundation forthe Introduced Senior Talents Wuyi University (Grant noYJ201802)

References

[1] R A M Attia D Lu T Ak and M M A Khater ldquoOpticalwave solutions of the higher-order nonlinear Schrodingerequation with the non-Kerr nonlinear term via modifiedKhater methodrdquo Modern Physics Letters B vol 34 no 5Article ID 2050044 2020

[2] S Chen and W Ma ldquoExact solutions to a generalizedBogoyavlensky-Konopelchenko equation via Maple symboliccomputationsrdquo Complexity vol 2019 Article ID 87874606 pages 2019

[3] E Fan and H Zhang ldquoNew exact solutions to a system ofcoupled kdV equationsrdquo Physics Letters A vol 245 no 5pp 389ndash392 1998

[4] M M A Khater R A M Attia A-H Abdel-AtyM A Abdou H Eleuch and D Lu ldquoAnalytical and semi-analytical ample solutions of the higher-order nonlinearSchrodinger equation with the non-Kerr nonlinear termrdquoResults in Physics vol 16 Article ID 103000 2020

[5] M Khater R Attia and D Baleanu ldquoAbundant new solutionsof the transmission of nerve impulses of an excitable systemrdquogte European Physical Journal Plus vol 135 no 2 pp 1ndash122020

[6] MM A Khater C Park A -HAbdel-Aty R AM Attia andD Lu ldquoOn new computational and numerical solutions of themodified Zakharov-Kuznetsov equation arising in electricalengineeringrdquo Alexandria Engineering Journal vol 59 no 3pp 1099ndash1105 2020

[7] M Khater C Park D Lu and R Attia ldquoAnalytical semi-analytical and numerical solutions for the Cahn-Allen


equationrdquo Advances in Difference Equations vol 2020 no 1pp 1ndash12 2020

[8] M M A Khater A R Seadawy and D Lu ldquoOptical solitonand rogue wave solutions of the ultra-short femto-secondpulses in an optical fiber via two different methods and itsapplicationsrdquo Optik vol 158 pp 434ndash450 2018

[9] M Khater A Seadawy and D Lu ldquoOptical soliton andbright-dark solitary wave solutions of nonlinear complexKundu-Eckhaus dynamical equation of the ultra-short fem-tosecond pulses in an optical fiberrdquo Optical and QuantumElectronics vol 50 no 3 p 155 2018

[10] J Li R A M Attia M M A Khater and D Lu ldquo-e newstructure of analytical and semi-analytical solutions of thelongitudinal plasma wave equation in a magneto-electro-elastic circular rodrdquo Modern Physics Letters B vol 34 no 12Article ID 2050123 2020

[11] W-XMa and Y Liu ldquoInvariant subspaces and exact solutionsof a class of dispersive evolution equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 17 no 10pp 3795ndash3801 2012

[12] C Park M M A Khater R A M Attia W Alharbi andS S Alodhaibi ldquoAn explicit plethora of solution for thefractional nonlinear model of the low-pass electrical trans-mission lines via Atangana-Baleanu derivative operatorrdquoAlexandria Engineering Journal vol 59 no 3 pp 1205ndash12142020

[13] A-M Wazwaz ldquo-e tanh method for travelling wave solu-tions to the Zhiber-Shabat equation and other relatedequationsrdquo Communications in Nonlinear Science and Nu-merical Simulation vol 13 no 3 pp 584ndash592 2008

[14] A-M Wazwaz ldquo-e variable separated ODE and the tanhmethods for solving the combined and the double combinedsinh-cosh-Gordon equationsrdquo Applied Mathematics andComputation vol 177 no 2 pp 745ndash754 2006

[15] T Wu and J Zhang ldquoOn modeling nonlinear long waverdquo inMathematics Is for Solving Problems L P CookV Roytbhurd and M Tulin Eds p 233 SIAM PhiladelphiaPA USA 1996

[16] M Eslami and H Rezazadeh ldquo-e first integral method forWu-Zhang system with conformable time-fractional de-rivativerdquo Calcolo vol 53 no 3 pp 1ndash11 2015

[17] E Fan ldquoExtended tanh-function method and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5pp 212ndash218 2000

[18] M M Helal M L Mekky and E A Mohamed ldquo-echaracteristic function method and its application to (1 + 1)-dimensional dispersive long wave equationrdquo Applied Math-ematics vol 3 no 1 pp 12ndash18 2012

[19] C Chen X Tang and S Lou ldquoSolutions of a (2 + 1)-di-mensional dispersive long wave equationrdquo Physical Review Evol 66 no 3 Article ID 036605 2002

[20] A Pickering ldquoA new truncation in Painleve analysisrdquo Journalof Physics A Mathematical and General vol 26 no 17pp 4395ndash4405 1993

[21] J Weiss M Tabor and G Carnevale ldquo-e Painleve propertyfor partial differential equationsrdquo Journal of MathematicalPhysics vol 24 no 3 pp 522ndash526 1983

[22] Q Wang Y Chen and Z Hongqing ldquoA new Jacobi ellipticfunction rational expansion method and its application to(1 + 1)-dimensional dispersive long wave equationrdquo ChaosSolitons amp Fractals vol 23 no 2 pp 477ndash483 2005

[23] X Zheng Y Chen and H Zhang ldquoGeneralized extendedtanh-function method and its application to (1 + 1)-

dimensional dispersive long wave equationrdquo Physics Letters Avol 311 no 2-3 pp 145ndash157 2003

[24] X Zeng and D-S Wang ldquoA generalized extended rationalexpansion method and its application to (1 + 1)-dimensionaldispersive long wave equationrdquo Applied Mathematics andComputation vol 212 no 2 pp 296ndash304 2009

[25] Z Du J Li and X Li ldquo-e existence of solitary wave solutionsof delayed Camassa-Holm equation via a geometric ap-proachrdquo Journal of Functional Analysis vol 275 no 4pp 988ndash1007 2018

[26] Z Du Z Feng and Z Feng ldquoExistence and asymptotic be-haviors of traveling waves of a modified vector-diseasemodelrdquo Communications on Pure amp Applied Analysis vol 17no 5 pp 1899ndash1920 2018

[27] D Feng J Li and J Jiao ldquoDynamical behavior of singulartraveling waves of (n+ 1)-dimensional nonlinear Klein-Gordon equationrdquo Qualitative gteory of Dynamical Systemsvol 18 no 1 pp 265ndash287 2019

[28] B He and Q Meng ldquoBifurcations and new exact travellingwave solutions for the Gerdjikov-Ivanov equationrdquo Com-munications in Nonlinear Science and Numerical Simulationvol 15 no 7 pp 1783ndash1790 2010

[29] H Liu and J Li ldquoSymmetry reductions dynamical behaviorand exact explicit solutions to the Gordon types of equationsrdquoJournal of Computational and Applied Mathematics vol 257pp 144ndash156 2014

[30] H Liu and J Li ldquoLie symmetry analysis and exact solutions forthe extended mkdV equationrdquo Acta Applicandae Mathema-ticae vol 109 no 3 pp 1107ndash1119 2010

[31] Y Song and X Tang ldquoStability steady-state bifurcations andturing patterns in a predator-prey model with Herd behaviorand prey-taxisrdquo Studies in Applied Mathematics vol 139no 3 pp 371ndash404 2017

[32] Y Song H Jiang Q-X Liu and Y Yuan ldquoSpatiotemporaldynamics of the diffusive mussel-algae model near turing-hopf bifurcationrdquo SIAM Journal on Applied Dynamical Sys-tems vol 16 no 4 pp 2030ndash2062 2017

[33] F Yi J Wei and J Shi ldquoDiffusion-driven instability andbifurcation in the Lengyel-Epstein systemrdquo NonlinearAnalysis Real World Applications vol 9 no 3 pp 1038ndash10512008

[34] T Zhang and J Li ldquoExact torus knot periodic orbits andhomoclinic orbits in a class of three-dimensional flowsgenerated by a planar cubic systemrdquo International Journal ofBifurcation and Chaos vol 27 no 13 pp 1ndash12 2017

[35] T Zhang and J Li ldquoExact solitons periodic peakons andcompactons in an optical soliton modelrdquo Nonlinear Dy-namics vol 91 no 2 pp 1371ndash1381 2018

[36] B Zhang Y Xia W Zhu and Y Bai ldquoExplicit exact travelingwave solutions and bifurcations of the generalized combineddouble sinh-cosh-Gordon equationrdquo Applied Mathematicsand Computation vol 363 p 124576 2019

[37] B Zhang W Zhu Y Xia and Y Bai ldquoA unified analysis ofexact traveling wave solutions for the fractional-order andinteger-order Biswas-Milovic equation via bifurcation theoryof dynamical systemrdquo Qualitative gteory of Dynamical Sys-tems vol 19 no 1 p 11 2020

[38] W Zhu Y Xia B Zhang and Y Bai ldquoExact traveling wavesolutions and bifurcations of the time fractional differentialequations with applicationsrdquo International Journal of Bi-furcation and Chaos vol 29 no 3 Article ID 1950041 2019

[39] W Zhu Y Xia and Y Bai ldquoTraveling wave solutions of thecomplex Ginzburg-Landau equation with Kerr law


nonlinearityrdquo Applied Mathematics and Computationvol 382 Article ID 125342 2020

[40] M M A Khater R A M Attia and D Lu ldquoNumericalsolutions of nonlinear fractional Wu-Zhang system for watersurface versus three approximate schemesrdquo Journal of OceanEngineering and Science vol 4 no 2 pp 144ndash148 2019

[41] M Khater D Lu and R Attia ldquoDispersive long wave ofnonlinear fractional Wu-Zhang system via a modified aux-iliary equation methodrdquo AIP Advances vol 9 no 2 ArticleID 025003 2019

[42] J Li Singular Nonlinear Travelling Wave Equations Bi-furcations and Exact Solutions Science Beijing China 2013

[43] P Byrd and M Friedman Handbook of Elliptic Integrals forEngineers and Scientists Springer-Verlag New York NYUSA 1971

[44] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014


Dαt u minus uux minus vx

Dαt v minus vux minus uvx minus

13uxxx

(3)

and tgt 0 and 0lt αle 1 So far Khater et al [40 41] studiedthe fractional-order WundashZhang system by the numericalmethod and the modified auxiliary equation method In-spired by [38] we study the fractional-order WundashZhangsystem via the extended three-step method

-e structure of this paper is as follows In Section 2 wefirst consider the bifurcation of phase portraits for the in-teger-order WundashZhang system Correspondingly we cal-culate all possible exact parametric expressions of solutionsto the integer-order WundashZhang system In Section 3 westudy the bifurcation of phase portraits and exact solutionsof the fractional-order WundashZhang system In Section 4comparison of phase portraits and exact solutions betweenthe integer-order and fractional-order WundashZhang system ispresented Finally we end this paper with a conclusion

2 Exact Solutions of Integer-OrderWundashZhang System (2)

In this part we first consider the exact solutions of integer-order WundashZhang system (2)

By the following transformations

u(x t) u(ξ)

v(x t) v(ξ)

ξ x minus lt

(4)

we havezu(x t)

zt minus luξ

zv(x t)

zt minus lvξ

z3u(x t)

zx3 uξξξ

(5)

where uξ and vξ denote the first-order derivative with respectto ξ respectively and uξξξ means the third-order derivativewith respect to ξ We substitute (4) and (5) into system (2)then (2) turns into

luξ uuξ + vξ

lvξ vuξ + uvξ +13uξξξ

(6)

-en by integration over both sides of the first equation(6) and setting constant of integration to zero hence wehave v(ξ) lu(ξ) minus (12)u2(ξ) Substituting it into thesecond equation of (6) we have the following ordinarydifferential equation (ODE)

uξξξ minus92u2uξ + 9luuξ minus 3l

2uξ 0 (7)

Integrating both sides on (7) it follows that

uξξ minus32u3

+92

lu2

minus 3l2u c (8)

Obviously equation (8) reduces to the planar dynamicalsystem

du

dξ y

dy

dξ32u3

minus92

lu2

+ 3l2u + c

(9)


H(u y) 12y2

minus38u4

+32

lu3

minus32l2u2

minus cu h (10)

We let g(u) (32)u3 minus (92)lu2 + 3l2u + c thengprime(u) (92)u2 minus 9lu + 3l2 Obviously the roots of g(u)

0 depend on the parameter group (l c) taking differentvalues (i) If cgt (

3

radic3)l3(lgt 0) or clt minus (

3

radic3)l3(lgt 0) or

cgt minus (3

radic3)l3(llt 0) or clt (

3

radic3)l3(llt 0) then g(u) has

only one real root u1 (ii) if c plusmn (3

radic3)l3(lgt 0 or llt 0)

then g(u) has two real roots u2 and u3(u3 lt u2) and (iii) ifminus (

3

radic3)l3 lt clt (

3


3


3

radic

3)l3(llt 0) then g(u) has three real roots u4 u5 andu6(u6 lt u5 lt u4)

Furthermore noting that

J uj yj1113872 1113873 detM uj yj1113872 1113873 minus gprime(u) minus92u2

minus 9lu + 3l2

1113874 1113875

(11)

where M(uj yj) is the coefficient matrix of the linearizedsystem of (9) at an equilibrium point Ej(j 1 2 6)Ej (uj 0) Using the theory of equilibrium points (see[42]) it is easy to compute that J(u1 0)lt 0 (saddle point)(see Figure 1(a)) J(u3 0)lt 0 (saddle point) J(u2 0) 0(cusp point) (see Figure 1(b)) or J(u2 0)lt 0 (saddle point)J(u3 0) 0 (cusp point) (see Figure 1(c)) and J(u4 0)lt 0(saddle point) J(u5 0)gt 0 (center point) J(u6 0)lt 0(saddle point) (see Figure 1(d)) -en we draw the bi-furcation of phase portraits of system (9) by Maple (seeFigure 1)

From (10) we set integral constant h be fixed whichyields

y2

34u4

minus 3lu3

+ 3l2u2

+ 2cu + 2h 34

G(u) (12)

-en we integrate over a branch of the curve from initialvalue u(t0) u0 -at is

ξ 1113946u

u0

4

3G(s)

1113971

ds (13)

Let hj H(uj 0)(j 1 2 6) for the phase por-traits according to the parameters we get h2 lt h3 andh5 lt h4 lt h6 respectively -en we just consider Figure 1(d)as follows

(i) Firstly if h isin (h5 h4) we get the green curve Itcorresponds to a family of periodic orbits of system(9) surrounding E5(u5 0) -en G(u) (λ1minus



1113946u

λ3


1113969 gsnminus 1 sinφ k1( 1113857

3

radic

2ξ

(14)


1113968and



3



1113968 it


1113968



11139684





6

4

2

0

ndash2

ndash4

ndash6

ndash1 1 2 3 4u

y

(a)

u

y

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8


(b)

u

y

8

6

4

2

ndash2

ndash4

ndash6

ndash8

ndash1 0 21 3 4 5

(c)

u

y

ndash1

3

2

1

0

ndash1

ndash2

ndash3

2 3 41

(d)


3

radic (b) c 3 l

3


3


3

radic




(16)


1113946u

λ2

ds

λ1 minus s( 11138572

s minus λ2( 1113857 s minus λ3( 1113857

1113969

1113946u

λ2

ds

λ1 minus s( 1113857

s minus λ2( 1113857 s minus λ3( 1113857

1113969

3

radic

2ξ

(17)




1113968) (

3

radic2)ξ and












(20)



3

radic3)l3 lt clt (

3

radic3)l3(lgt 0) or

(3


3




3

radic3)l3 lt clt (

3


3


3

radic




Dαt u(x t) t

1minus αzu(x t)

zt t

1minus αdu(ξ)

dt t

1minus αuξdξdt

t1minus α


minus lαuξ

(21)

Dαt v(x t) t

1minus αzv(x t)

zt t

1minus αdv(ξ)

dt t

1minus αvξdξdt

t1minus α


minus lαvξ

(22)

ndash10 ndash5 0ξ

5 10

1

11

12

13

14u15

16

17

18

19

(a)

t8 6


ndash8

x

2

18

16

14

12

1

151

050

(b)



solutions



lαuξ uuξ + vξ


(23)


uξξ minus32u3

+92

lαu2



du

dξ y

dy

dξ32u3

minus92

lαu2

+ 3l2α2u + c

(25)


H(u y) 12y2

minus38u4

+32

lαu3

minus32l2α2u2

minus cu h (26)


3


3


(3


3





3


3


3


(3


1113957u6 (1113957u6 lt 1113957u5 lt 1113957u4)




where 1113957ω1

3(1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)1113969

2 and 1113957k21 ((1113957λ2 minus 1113957λ3)








where 1113957ω2

(34)(1113957λ2 minus 1113957λ1)(1113957λ3 minus 1113957λ1)1113969




(30)



3


3


(3


3




3

25

2

15

u


(a)

3

25

2

15

1

05

50

ndash5ndash10

x

005

115t

10

(b)




0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)



α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)






3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References




















































1113946u

λ3


1113969 gsnminus 1 sinφ k1( 1113857

3

radic

2ξ

(14)


1113968and



3



1113968 it


1113968



11139684





6

4

2

0

ndash2

ndash4

ndash6

ndash1 1 2 3 4u

y

(a)

u

y

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8


(b)

u

y

8

6

4

2

ndash2

ndash4

ndash6

ndash8

ndash1 0 21 3 4 5

(c)

u

y

ndash1

3

2

1

0

ndash1

ndash2

ndash3

2 3 41

(d)


3

radic (b) c 3 l

3


3


3

radic




(16)


1113946u

λ2

ds

λ1 minus s( 11138572

s minus λ2( 1113857 s minus λ3( 1113857

1113969

1113946u

λ2

ds

λ1 minus s( 1113857

s minus λ2( 1113857 s minus λ3( 1113857

1113969

3

radic

2ξ

(17)




1113968) (

3

radic2)ξ and












(20)



3

radic3)l3 lt clt (

3

radic3)l3(lgt 0) or

(3


3




3

radic3)l3 lt clt (

3


3


3

radic




Dαt u(x t) t

1minus αzu(x t)

zt t

1minus αdu(ξ)

dt t

1minus αuξdξdt

t1minus α


minus lαuξ

(21)

Dαt v(x t) t

1minus αzv(x t)

zt t

1minus αdv(ξ)

dt t

1minus αvξdξdt

t1minus α


minus lαvξ

(22)

ndash10 ndash5 0ξ

5 10

1

11

12

13

14u15

16

17

18

19

(a)

t8 6


ndash8

x

2

18

16

14

12

1

151

050

(b)



solutions



lαuξ uuξ + vξ


(23)


uξξ minus32u3

+92

lαu2



du

dξ y

dy

dξ32u3

minus92

lαu2

+ 3l2α2u + c

(25)


H(u y) 12y2

minus38u4

+32

lαu3

minus32l2α2u2

minus cu h (26)


3


3


(3


3





3


3


3


(3


1113957u6 (1113957u6 lt 1113957u5 lt 1113957u4)




where 1113957ω1

3(1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)1113969

2 and 1113957k21 ((1113957λ2 minus 1113957λ3)








where 1113957ω2

(34)(1113957λ2 minus 1113957λ1)(1113957λ3 minus 1113957λ1)1113969




(30)



3


3


(3


3




3

25

2

15

u


(a)

3

25

2

15

1

05

50

ndash5ndash10

x

005

115t

10

(b)




0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)



α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)






3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References





















































(16)


1113946u

λ2

ds

λ1 minus s( 11138572

s minus λ2( 1113857 s minus λ3( 1113857

1113969

1113946u

λ2

ds

λ1 minus s( 1113857

s minus λ2( 1113857 s minus λ3( 1113857

1113969

3

radic

2ξ

(17)




1113968) (

3

radic2)ξ and












(20)



3

radic3)l3 lt clt (

3

radic3)l3(lgt 0) or

(3


3




3

radic3)l3 lt clt (

3


3


3

radic




Dαt u(x t) t

1minus αzu(x t)

zt t

1minus αdu(ξ)

dt t

1minus αuξdξdt

t1minus α


minus lαuξ

(21)

Dαt v(x t) t

1minus αzv(x t)

zt t

1minus αdv(ξ)

dt t

1minus αvξdξdt

t1minus α


minus lαvξ

(22)

ndash10 ndash5 0ξ

5 10

1

11

12

13

14u15

16

17

18

19

(a)

t8 6


ndash8

x

2

18

16

14

12

1

151

050

(b)



solutions



lαuξ uuξ + vξ


(23)


uξξ minus32u3

+92

lαu2



du

dξ y

dy

dξ32u3

minus92

lαu2

+ 3l2α2u + c

(25)


H(u y) 12y2

minus38u4

+32

lαu3

minus32l2α2u2

minus cu h (26)


3


3


(3


3





3


3


3


(3


1113957u6 (1113957u6 lt 1113957u5 lt 1113957u4)




where 1113957ω1

3(1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)1113969

2 and 1113957k21 ((1113957λ2 minus 1113957λ3)








where 1113957ω2

(34)(1113957λ2 minus 1113957λ1)(1113957λ3 minus 1113957λ1)1113969




(30)



3


3


(3


3




3

25

2

15

u


(a)

3

25

2

15

1

05

50

ndash5ndash10

x

005

115t

10

(b)




0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)



α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)






3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References




















































lαuξ uuξ + vξ


(23)


uξξ minus32u3

+92

lαu2



du

dξ y

dy

dξ32u3

minus92

lαu2

+ 3l2α2u + c

(25)


H(u y) 12y2

minus38u4

+32

lαu3

minus32l2α2u2

minus cu h (26)


3


3


(3


3





3


3


3


(3


1113957u6 (1113957u6 lt 1113957u5 lt 1113957u4)




where 1113957ω1

3(1113957λ1 minus 1113957λ3)(1113957λ2 minus 1113957λ4)1113969

2 and 1113957k21 ((1113957λ2 minus 1113957λ3)








where 1113957ω2

(34)(1113957λ2 minus 1113957λ1)(1113957λ3 minus 1113957λ1)1113969




(30)



3


3


(3


3




3

25

2

15

u


(a)

3

25

2

15

1

05

50

ndash5ndash10

x

005

115t

10

(b)




0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)



α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)






3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References



















































0 1 2 3 4ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(a)

0 1 2ndash1u

ndash3

ndash2

ndash1

y

1

2

3

(b)



α 12 1113957E4(1113957u4 0) 1113957E5(1113957u5 0) and 1113957E6(1113957u6 0)

3

25

2

15

u


(a)

3

25

2

15

1

050

051

1510

50

ndash5 ndash10

t x

(b)

16

15

14

13

12

11

10

09

08


(c)

005

115t

105

0ndash5

ndash10

x

161514131211100908

(d)






3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References




















































3


3


3

radic3)l3α3 lt

clt minus (3





3







(31)




(32)


5 Conclusion







Acknowledgments


References



















































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