DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi10.1155/2021/53032955303295Research ArticleBifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equationshttps://orcid.org/0000-0002-2502-5330LiZhao12LiPeng3https://orcid.org/0000-0003-3942-9480HanTianyong12KhanAbdul Qadeer1College of Computer ScienceChengdu UniversityChengdu 610106Chinacdu.edu.cn2Key Laboratory of Pattern Recognition and Intelligent Information ProcessingInstitutions of Higher Education of Sichuan ProvinceChengdu UniversityChengdu 610106Chinacdu.edu.cn3North China Electric Power Test and Research InstituteChina Datang Corporation Science and Technology Research Institute CoBeijing 100040China2021181020212021306202131720216102021181020212021Copyright © 2021 Zhao Li et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

Institutions of Higher Education of Sichuan ProvinceMSSB-2021-13
1. Introduction

In this paper, we consider the fractional generalized Hirota–Satsuma coupled KdV (FGHSCKdV) equations as follows :(1)Dtαu=14Dx3βu+3uDxβu+3Dxβv2+w,Dtαv=12Dx3βv3uDxβv,Dtαw=12Dx3βw3uDxβw,where u=ut,x, v=vt,x, and w=wt,x represent unknown functions about time t and space x, respectively, and Dtαu and Dxβu represent the conformable fractional derivatives of u with respect to time t of order α and space x of order β, respectively. The FGHSCKdV equations are related to most types of long waves, acoustic waves, and planetary waves in geophysical hydrodynamics.

In addition, when α=β=1, equation (1) is reduced to the well-known generalized Hirota–Satsuma coupled KdV (GHSCKdV) equations as follows :(2)ut=14uxxx+3uux+3v2+wx,vt=12vxxx3uvx,wt=12wxxx3uwx,where u, v, and w are the unknown functions of t and x. System (2) is usually used to explain the interactions between two long waves with different dispersive relations in physics, which was first proposed in Ref. . When w=0, system (2) is reduced to the coupled KdV equation, which was first introduced by Satsuma and Hirota .

Equation (2) is a very important mathematical physical equation. In recent years, many scholars and experts have devoted themselves to study equation (2), especially the property of orbits, the bifurcation of phase portrait, and the exact solution of these equations. It has become a very important subject in the field of physics and engineering technology because it can further help physicists to explain the dynamic behavior of GHSCKdV equations. Recently, many important methods have been established to construct the traveling wave solutions of equation (2), such as the subequation method, the Hirota direct method, the Hirota bilinear method, plane dynamic system analysis method, and so on (see  and the references therein).

Fractional partial differential equation (FPDE), usually used to simulate phenomena in natural science, plays a very important role in the field of applied sciences (such as fluid mechanics, nonlinear optics, and so on). The most commonly used definitions of fractional derivatives include Caputo derivative, Riemann–Liouville derivative, conformable derivative, etc. But, for FGHSCKdV equations, the research progress is very slow due to the complexity of fractional derivative. The main reason is that most of the fractional derivatives do not obey the chain rule. For example, in Ref.  and Ref. , the researchers applied Jumarie’s modified Riemann–Liouville derivative to seek the exact solutions of FGHSCKdV. With the development of fractional calculus, in 2014, Khalil et al.  gave a new definition of conformable fractional derivative, which satisfies the chain rule and Leibniz formula. We can easily simplify the FPDE into the nonlinear ordinary differential equation (NLODE). Therefore, the method of constructing the exact solution of PDE is also suitable for finding the exact solution of FPDE. In Ref. , Ali et al. obtained the exact solutions of the time-fractional GHSCKdV in the sense of the conformable derivative by the subequation method and residual power series method. In Ref , although Sirisubtawee and his collaborators studied FGHSCKdV by the G/G,1/G-expansion method, the obtained solutions only include the hyperbolic function solutions, rational function solutions, and trigonometric function solutions. Based on the lack of research on the FGHSCKdV equations, the main purpose of this paper is to study the property of orbits, the bifurcation of phase portrait, the exact solution, and stability of FGHSCKdV equations.

This paper is organized as follows. In Section 2, phase portraits of system (1) are drawn. In Section 3, traveling wave solutions of system (1) are constructed. In Section 4, the stability criteria of system (1) are given. Finally, some conclusions are discussed in Section 5.

2. Phase Portraits of System (<xref ref-type="disp-formula" rid="EEq1">1</xref>)

We first do the following traveling wave transformation:(3)ut,x=Uξ,vt,x=Vξ,wt,x=Wξ,ξ=kxββctαα,where α and β are nonzero arbitrary constants and c stands for the wave speed.

Substituting (3) into (1), we get the NLODE as follows:(4)ckU=14k3U+3kUU+3kV2+W,ckV=12k3V3kUV,ckW=12k3W3kUW.

In order to eliminate the coupling relationship of the above equations, we assume that(5)U=a1V2+b1V+c1,W=a2V+b2,where a1,b1,c1, a2, and b2 are constants.

Substituting (5) into (4) and then integrating once, we find that equation (4) has the same result as follows:(6)k2V=2a1V33b1V2+2c3c1V+d1,where d1 is the integral constant.

Multiplying equation (6) by V and then integrating equation (6), we obtain(7)k2V2=a1V42b1V3+2c3c1V2+d1V+d2,where d2 is the integral constant.

On the one hand, we obtain the following from (4), (6), and (7):(8)k2U=2a1V2+2a1V+b1V=6a12V412a1b1V3+8a1c24a1c13b12+6a1d1+2b1c6b1c1+b1d1+2a1d2.

On the other hand, integrating (4) once, we obtain(9)14k2U+32U2+cU+3V2+W+d3=0,where d3 is the integral constant.

Therefore, by substituting (5) and (8) into (9), we can get(10)3a1c3a1c1+34b13=0,12a1d1+b1c+b1c1+a2=0,142a1d2+b1d1+32c12+c1c+3b2+d3=0.

We find from (10) that(11)a1=b1244c1c,a2=b224d18c1c12b1c+c1,b2=b124d224c2c12b1d112c1213cc113d3.

Suppose that d1=1/2a12b12+2ca1b16a1b1c1. Make a transformation Vξ=aψξb1/2a1, where a and a0. Then, equation (7) can be simplified as(12)ak2ψa2c6c1+3b122a1ψ+2a1a3ψ3=0.

Here, we conclude that system (7) has the same traveling wave solution and topological phase portraits as (1). Equation (7) has the dynamical system(13)dψdξ=y,dydξ=1k22a1a2ψ32c6c1+3b122a1ψ=Aψ3+Bψ,where A=2a1a2/k2 and B=2c/k26c1/k2+3b12/2k2a1. Clearly, (13) has the Hamiltonian system(14)Hψ,y=12y2+14Aψ412Bψ2=h,h.

Note that Gψ=Aψ3+Bψ. Let Eiψi,0i=0,1,2 be the equilibrium points of system (13). By the theory of dynamical system given by Li and Dai , we can easily get that the equilibrium point Eiψi,0 is saddle point (or center point) if Gψi>0 (or Gψi<0); the equilibrium point Eiψi,0 is degraded saddle point if Gψi=0. The phase portraits of (13) are shown in Figure 1. Thus, by the bifurcation theory of planar dynamical system (see ), we can easily construct the exact solution of equation (1) depending on the parameters A and B.

Phase portraits of system (13). (a) A>0, B>0. (b) A<0, B<0.

3. Traveling Wave Solutions of System (<xref ref-type="disp-formula" rid="EEq1">1</xref>)Case 1.

A>0, B>0.

When hB2/4A,0, we can rewrite system (13) as(15)y2=A2ψ4+2BAψ2+4hA=A2ψ2ψ1h2ψ2h2ψ2,

where ψ1h2=B/A1/AB2+4Ah and ψ2h2=B/A+1/AB2+4Ah.

Substituting (15) into system (13), dψ/dξ=y, and we integrate them and obtain(16)ψ1ξ=±ψ2hdnψ2hA2ξξ0,ψ2h2ψ1h2ψ2h.

That is,(17)v1t,x=±aψ2hdnA2kψ2hxββctααA2ψ2hξ0,ψ2h2ψ1h2ψ2hb12a1.

When h=0, we can obtain(18)v2t,x=±a2BAsechkBxββctααBξ0b12a1.

When selecting the appropriate parameters, the bell solitary wave solution v2t,x of equation (1) is drawn as shown in Figure 2.

For h0,+, system (11) can be written as

(19)y2=A2ψ4+2BAψ2+4hA=A2ψ2+ψ3h2ψ4h2ψ2,where ψ3h2=B/A+1/AB2+4Ah and ψ4h2=B/A+1/AB2+4Ah.

Then, we obtain the periodic solutions(20)ψ3ξ=ψ4hcnA2ψ3h2+ψ4h2ξξ0,ψ4hψ3h2+ψ4h2.

That is,(21)v3t,x=aψ4hcnA2ψ3h2+ψ4h2kxββctααξ0,ψ4hψ3h2+ψ4h2b12a1.

The bell solitary wave solution v2t,x at A=1,B=2, a=2, α=1/2, β=1/3, k=2, ξ0=0, b1=3, c=1, and c1=2 (a) Perspective view of the wave. (b) The wave along x.

Case 2.

A<0, B<0.

When h0,B2/4A, we can rewrite system (13) as(22)y2=A2ψ42BAψ24hA=A2ψ5h2ψ2ψ6h2ψ2,

where ψ5h2=B/A1/AB2+4Ah and ψ6h2=B/A+1/AB2+4Ah.

Substituting (22) into dψ/dξ=y, we can obtain(23)ψ4ξ=±ψ5hsnψ6hA2ξξ0,ψ5hψ6h,v4t,x=±aψ5hsnψ6hA2kxββctααξ0,ψ5hψ6hb12a1.

When h=B2/4A, we obtain two kink solitary wave solutions

(24)ψ5ξ=±BAtanhB2ξξ0,(25)v5t,x=±aBAtanhB2kxββctααξ0b12a1.

When selecting the appropriate parameters, the kink solitary wave solution v5t,x of equation (1) is drawn as shown in Figure 3.

The kink solitary wave solution v5t,x at A=B=2, a=2, α=1/2, β=1/3, k=2, ξ0=0, b1=3, c=1, and c1=2. (a) Perspective view of the wave. (b) The wave along x.

Remark 1.

In the above discussion, we have obtained the solution vt,x of system (1). Obviously, we can get the solutions ut,x, wt,x of system (1) from coupling relation (5).

4. Stability Analysis

Consider the perturbed solution of system (2):(26)vt,x=ϖ0+λϖ,where ϖ0 is the steady-state solution.

Substituting (26) into the second equation of (2), we obtain(27)λϖt=12λϖxxx3λϖxa1ϖ0+λϖ2+b1ϖ0+λϖ+c1.

Linearizing equation (27) in λ,(28)ϖt=12ϖxxx3a1ϖ02ϖx3b1ϖ0ϖx3c1ϖx.

Suppose that (28) has a solution(29)ϖ=eiρx+εt,where ρ is the normalized wave number.

Substituting (29) into (28), we get(30)ε=12ρρ26a1ϖ02+b1ϖ0+c1.

From (30), we can see that system (2) is stable when ρ>0 and ρ2>3a1ϖ02+b1ϖ0+c1.

5. Conclusion

In this paper, we have drawn the bifurcations of phase portraits of FGHSCKdV equations, and we have constructed the traveling wave solutions of FGHSCKdV equations which include implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions. Through a fractional traveling wave transformation, the FGHSCKdV equations can be simplified into coupled nonlinear ordinary differential equations. In order to eliminate the coupling relationship, we assume that the solutions U, V, and W of equation (1) satisfy equation (5). Therefore, we can easily obtain the Hamiltonian system of (14) and construct the solutions of equation (1). Compared with the reported literature, the solution of Jacobian functions obtained in the paper has not been published. More importantly, we also give the stability condition of GHSCKdV equations. Therefore, the study of the paper has very important value.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by Institutions of Higher Education of Sichuan Province under grant no. MSSB-2021-13.

GuoS.MeiL.LiY.SunY.The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanicsPhysics Letters A2012376440741110.1016/j.physleta.2011.10.0562-s2.0-84855193408SaberiE.Reza HejaziS.Lie symmetry analysis, conservation laws and exact solutions of the time-fractional generalized Hirota-Satsuma coupled KdV systemPhysica A: Statistical Mechanics and Its Applications20184921529630710.1016/j.physa.2017.09.0922-s2.0-85032513729SirisubtaweeS.KoonprasertS.SungnulS.Some applications of the (G′/G, 1/G)-expansion method for finding exact traveling wave solutions of nonlinear fractional evolution equationsSymmetry201911895298010.3390/sym110809522-s2.0-85070533089AliK.RezazadehH.SenolM.NeiramehA.TasbozanO.EslamiM.MirzazadehM.Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long wavesJournal of Ocean Engineering and Science20194243210.1016/j.joes.2018.12.004ShenJ.XuW.XuY.Travelling wave solutions in the generalized Hirota-Satsuma coupled KdV systemApplied Mathematics and Computation2005161236538310.1016/j.amc.2003.12.0332-s2.0-10044230744HongB.New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV systemApplied Mathematics and Computation2010217247247910.1016/j.amc.2010.05.0792-s2.0-77955654539Abdel RadyA. S.OsmanE. S.KhalfallahM.On soliton solutions for a generalized Hirota-Satsuma coupled KdV equationCommunications in Nonlinear Science and Numerical Simulation201015226427410.1016/j.cnsns.2009.03.0112-s2.0-69049083378FengD.LiK.Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation methodPhysics Letters A2011375232201221010.1016/j.physleta.2011.04.0392-s2.0-79956111293XieM.DingX.A new method for a generalized Hirota-Satsuma coupled KdV equationApplied Mathematics and Computation2011217177117712510.1016/j.amc.2011.01.0482-s2.0-79952772984ChenJ.ChenY.FengB.-F.ZhuH.Multi-component generalizations of the Hirota-Satsuma coupled KdV equationApplied Mathematics Letters201437152110.1016/j.aml.2014.05.0032-s2.0-84901976008YuJ.SunY.WangF.N-soliton solutions and long-time asymptotic analysis for a generalezed complex Hirota-Satsuma coupled KdV equationAppled Mathematics Letters202010610637010.1016/j.aml.2020.106370SunY.MaW.YuJ.N-soliton solutions and dynamic property analysis of a generalized three-component Hirota-Satsuma coupled KdV equationApplied Mathematics Letters202112010722410.1016/j.aml.2021.107224WuY.GengX.HuX.ZhuS.A generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformationsPhysics Letters A1999255425926410.1016/s0375-9601(99)00163-22-s2.0-0345985752HirotaR.SatsumaJ.Soliton solutions of a coupled Korteweg-de Vries equationPhysics Letters A19818510.1016/0375-9601(81)90423-02-s2.0-18944408338KhalilR.Al HoraniM.YousefA.SababhehM.A new definition of fractional derivativeJournal of Computational and Applied Mathematics2014264657010.1016/j.cam.2014.01.0022-s2.0-84893186929LiJ.DaiH.On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach2007Beijing, ChinaScience PressHeB.MengQ.Bifurcations and new exact travelling wave solutions for the Gerdjikov-Ivanov equationCommunications in Nonlinear Science and Numerical Simulation20101571783179010.1016/j.cnsns.2009.07.0192-s2.0-74449087484DuL.SunY.WuD.Bifurcations and solutions for the generalized nonlinear Schrödinger equationPhysics Letters A20193833612602810.1016/j.physleta.2019.1260282-s2.0-85072798818ZhaoLiHanT.HuangC.Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equationAIP Advances2020101111511310.1063/5.0029159ZhangZ.-Y.LiuZ.-H.MiaoX.-J.ChenY.-Z.Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearityPhysics Letters A2011375101275128010.1016/j.physleta.2010.11.0702-s2.0-79951809469ZhaoL.HanT.Bifurcation and exact solutions for the (2 + 1)-dimensional conformable time-fractional Zoomeron equationAdvances in Difference Equations20202020