Min-Yuan Liu(劉敏遠(yuǎn)), Hui Xu(許慧), and Zeng-Gui Wang(王增桂)

School of Mathematical Sciences,Liaocheng University,Liaocheng 252059,China

Keywords: the time-fractional modified KdV equation,bifurcation analysis,exact solutions

1.Introduction

Fractional-order partial differential equations are prevalently employed to model numerous complex phenomena involving viscoelasticity, electromagnetism, materials science,electrical networks, and fluid dynamics, etc., and interpreted best in inhomogeneous media particularly.The exact solutions of nonlinear evolution equations, which aid us in perusing the hidden physical properties, have been continuously constructed.So far, a multitude of effective methods of solving for exact solutions, including Hirota bilinear transformation,[1,2]homotopy perturbation method,[3]bifurcation method,[4-6]Darboux transformation,[7-9]generalized Kudryashov method,[10-12]inverse scattering method,[13,14]Lie symmetry analysis,[15]and so on, have been popularly adopted.

Recently, Ma[16-19]have provided some new wave equations, including integrable nonlinear Schr¨odinger type equations, AKNS type integrable equation and modified Korteweg-de Vries (KdV) type integrable equations.In this paper, we investigate a time-fractional order modified KdV equation[20]

whereaandbare arbitrary constants.Equation (1) is well known in the explanation of physical science phenomena such as particle vibrations in lattices, thermodynamics and current flow.

Until now, (G′/G)-expansion method and improved(G′/G)-expansion method were adopted by Sahoo and Ray[20]to find soliton solutions, and the expafunction method was carried out to Eq.(1) by Zafar.[21]Based on shifted Jacobi polynomials, Bhrawyet al.[22]calculated high accurate approximate solutions via a numerical method.Numerical solutions were generated via using the modified homotopy analysis Laplace transform method by Liet al.,[23]and semi-analytical solutions were yielded with a new variational method.[24]Alternatively,Akbulut and Kaplan[25]utilized auxiliary equation method to seek exact solutions successfully.Wang and Xu[26]provided Lie group analysis of Eq.(1)and constructed ecplicit solutions through symmetry reduction.

Bifurcation analysis[27-30]is a geometrically intuitive method to present exact traveling wave solutions.Under bifurcation theory,the types of orbits are highly tied to the sorts of solutions, which benefits us to classify solutions theoretically.Generally,the existence of periodic and homoclinic orbits signifies that Eq.(1)exists periodic and soliton solutions,correspondingly.Further, kink (or anti-kink), breaking wave solutions are separately attached to open and heteroclinic orbits.Moreover,supposing thatu(?)(?=?(x,t))is a continuous solution of Eq.(1) and satisfyingwe can say thatuis a soliton solution forn+=n-.Otherwise,uis a kink(or anti-kink)solution.As far as we know,Liang and Tang[31]only performed bifurcation analysis when integration constant is zero.Here, we consider a more general case, i.e.,we provide related discussion when the integration constant is an arbitrary constant.

LetG: [0,∞]→R, the conformable fractional derivative of orderβ[32]is defined as

The continuation plots are designed as follows: In Section 2, we analyze the phase portraits of the time-fractional modified KdV equation.Based on different orbits of the phase portraits, three types of new exact solutions are obtained in Section 3.In Section 4, we graphically illustrate some solutions via presenting solution profiles, 3D and density plots.Section 5 gives a summary.

2.Phase portraits and bifurcation

Equation(1)is executed to perform traveling wave transformation

Via qualitative theory[4]of differential equations, we know that (?i,0) is a center point when?′(?i)< 0; when?′(?i)>0,(?i,0)is defined as a saddle point;(?i,0)becomes a degraded saddle point when?′(?)=0.

2.1.Analysis for M>0 and N>0

The phase portraits of system (5) are graphically summarized forM>0,N>0.Depending on the values ofD,we present seven situations: (a)D<-D0, (b)D=-D0, (c)-D0D0in Fig.1.

Here,we make the situations of Figs.1(d),1(e),and 1(f)as examples and discuss them in detail.The classification of equilibrium points is established through qualitative theory,as listed in Table 1.

Table 1.Classification of equilibrium points with different values of D.

Based on Fig.1 and Table 1, the conclusions about the relationship betweenρa(bǔ)nd the types of orbits are exhibited.

Conclusion 1 WhenD=0,see Fig.1(d).

(1)Forρ ≤ρ2, system(5)remains bounded open orbits with hyperbolic shape in black and pink, which states that Eq.(1)has breaking wave solutions.

(2)Forρ2<ρ<ρ1,a periodic orbit enveloping the center pointAand matching a periodic solution is outlined visually.There exist two bounded open orbits in blue which are relevant to breaking wave solutions.

(3)Forρ=ρ1, there are two heteroclinic orbits running through saddle pointsB,Cand two special orbits in red,which refers that kink (or anti-kink), singular solutions are yielded,individually.

(4)Forρ>ρ1,no closed orbit is produced.

Conclusion 2 When 0

(1) Forρ ≤ρ2, bounded open orbits in gold and black associated with breaking wave solutions are pointed out.

(2)Forρ2<ρ<ρ3, Eq.(1)contains periodic, breaking wave solutions which correspond to periodic, open orbits in pink,respectively.

(3)Forρ=ρ3,homoclinic orbit in red going through saddle pointBis described,which indicates there lies soliton solution.In addition,open and special orbits are revealed.

(4)Forρ>ρ3,there is no closed orbit.

Conclusion 3 WhenD=D0,see Fig.1(f).

(1)Forρ=ρ2,two special orbits in red are shown,which demonstrates that Eq.(1)produces singular solutions.

(2)Forρ<ρ2,system(5)yields two open orbits in green matching with breaking wave solutions.

(3)Forρ2<ρ<ρ1,there exist two open orbits in yellow.(4)Forρ ≥ρ1,no closed orbit is obtained.

2.2.Analysis for M<0 and N<0

ForM<0,N<0,we draw the following phase portraits in Fig.2.Analogously,we classify the equilibrium points that are arranged in Table 2.According to Fig.2 and Table 2, we illustrate the following conclusions.

Conclusion 4 WhenD=0,see Fig.2(d).

(1)Forρ=ρ2,system(5)contains two homoclinic orbits in black passing through saddle pointA,referring that soliton solutions are owned.

(2)Forρ1<ρ<ρ2,two periodic orbits in pink encircling center pointsB,Care drawn,so Eq.(1)has periodic solutions.

(3)Forρ>ρ2,there exists a periodic orbit in blue laying outside the homoclinic orbits.

(4)Forρ ≤ρ1,no closed orbit is pointed out.

Conclusion 5 When 0

(1)Forρ=ρ2,two homoclinic orbits in red meet at saddle pointA, from which we perceive that Eq.(1) has soliton solutions.

(2)Forρ1<ρ<ρ2,system(5)reveals periodic orbits in pink matching periodic solutions and surrounding the center pointsB,C,separately.

(3)Forρ3<ρ ≤ρ1,periodic orbits in green and in yellow are shown.

(4) Forρ>ρ2, a periodic orbit in blue lies outside the homoclinic orbits.

(5)Forρ ≤ρ3,there is no closed orbit.

Conclusion 6 WhenD=D0,see Fig.2(f).

(1)Forρ=ρ1,a homoclinic orbit in red is plotted,which implies that Eq.(1)has soliton solution.

(2) Forρ3<ρ<ρ1, a periodic orbit corresponding to periodic solution in blue is yielded.

(3)Forρ>ρ1,a periodic orbit in green is displayed outside.

(4)Forρ ≤ρ3,no closed orbit is obtained.

Fig.1.Phase portraits of system (5) with b=-1, k=1, a=3, α = , v=4: (a) c=1.5, (b) c= (c) c=0.3, (d) c=0,(e)c=-0.3,(f)c=-(g)c=-1.5.

Table 2.Classification of equilibrium points with different values of D.

2.3.Analysis for MN<0

ForD=0 andM>0,N<0 orM<0,N>0,the phase portraits are plotted in Fig.3.

Fig.3.Phase portraits of system (5) with b = -1, k = 1, a = 3, α =1/2, v=-4.

3.1.Solutions for M>0 and N>0

Family I Periodic solutions.

Theorem 1 See orbitL10from Fig.1(e)corresponding to individualρ2<ρ<ρ3.Periodic solution of Eq.(1) is enumerated as

The relevant discussion has been carried out in Ref.[31]and is not be described here.

3.Exact solutions

For a particularρ,it is well known thatH=ρmaps to a level curve of system(5)that determines a solution of Eq.(1).According to the phase portraits, we aim to build parametric representations of exact solutions of Eq.(1), which are not mentioned in Ref.[31].

Firstly, the first equation of system (5) is inserted into Eq.(7),which yields

Proof This refers to the orbitL8in Fig.1(e).The?axis is intersected in two points and the pointBis double.Therefore,by considering{(?,?)|?2=ψ(?),?5

(ii) At this point, the periodic solution corresponding toL10has been established before,and we now focus onL11,L12.Via utilizing{(?,?)|?2=ψ(?),?1

Solving them,we obtain solutions(26)and(27).

Remark The parametric expressions of the kink(or antikink) and periodic solutions in Fig.1(e) have been listed in Ref.[31],we do not repeat the discussion here.

3.2.Solutions for M<0 and N<0

Here, we consider the homoclinic orbits as examples to construct all relevant exact parametric expressions of soliton solutions.

Theorem 4(1)See Fig.2(d).Whenρ=ρ2,two soliton solutions are yielded,

(2) See Fig.2(e).Whenρ=ρ2, Eq.(1) produces two soliton solutions

Inserting Eq.(49) into Eq.(12) and integrating alongT5, we verify soliton solution(44).

4.Graphic representation

This section pursuits to provide relevant graphs of some solutions in Figs.4-7.

Fig.4.Dark soliton solution(15)with η2=-1.

Fig.5.Singular solution(18)with η3=0.

Fig.6.Singular solution(26)with η11=-2.

Fig.7.Bright soliton solution(44).

For Figs.4-6,choosing appropriate integration constantsηiandα=0.5,v=4,a=3,b=-1,k=1, which impliesM=1,N=2, we exhibit graphically dark soliton solutionu2,periodic-singular solutionu3and singular solutionu12with the corresponding 3D,density plots and solution profiles,separately.Figure 7 records the behavior of dark solutionu19by selecting parametersα=0.5,η17=-2,v=4,a=3,b=1,k=1,which indicatesM=-1 andN=-2.

5.Conclusion

This work is dedicated to research of exact solutions of the time-fractional modified KdV equation via bifurcation analysis.According to the different orbits of the phase portraits drawn with suitable parameters, the relevant qualitative analysis is concluded.Then,exact solutions of Eq.(1)matching to different types of orbits are constructed successively.Finally,we perform some solutions via presenting solution profiles,3D and density plots with suitable parametric values.

Acknowledgements

Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084), the Natural Science Foundation of Liaocheng University (Grant No.318012025), and Discipline with Strong Characteristics of Liaocheng University-Intelligent Science and Technology(Grant No.319462208).