# The Bifurcations of Traveling Wave Solutions of the Kundu Equation

Authors: Yating Yi; Zhengrong Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We use the bifurcation method of dynamical systems to study the bifurcations of traveling wave solutions for the Kundu equation. Various explicit traveling wave
solutions and their bifurcations are obtained. Via some special phase orbits, we obtain
some new explicit traveling wave solutions. Our work extends some previous results.


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## Body

## 1. Introduction

In this paper, we consider the bifurcations of traveling wave solutions for the Kundu equation with the fifth-order nonlinear term:(1a)iut+uxx+c3|u|2u+c5|u|4u-is2(|u|2u)x-ir(|u|2)xu=0,x∈R,
where c3, c5, s2, and r are real constants. Equation (1a) was derived by Kundu [1] in the study of integrability, and it is an important special case of the generalized complex Ginzburg-Landau [2].For convenience, we denote the Kundu equation by(1b)ut=iuxx+i(c3|u|2+c5|u|4)u+α|u|2ux+βu2u-x,x∈R,where α=2s2+r, β=s2+r.Meanwhile, (1a) and (1b) and their special cases arise in various physical and mechanical applications, such as plasma physics, nonlinear fluid mechanics, nonlinear optics, and quantum physics (see [2–7]). For example, (1b) contains the following particular cases.(1) Ifr=0, then (1a) and (1b) reduce to the derivative Schrödinger equation:
(2)ut=iuxx+i(c3|u|2+c5|u|4)u+s2(|u|2u)x.(2) Ifc3=c5=0, s2=-δ, and r=-s2, then (1a) and (1b) reduce to Chen-Lee-Liu equation [8]:
(3)iut+uxx+iδ|u|2ux=0.(3) Ifc5=2δ2, s2=2δ, and r=-2s2, then (1a) and (1b) reduce to Gerdjikov-Ivanov equation [9]:
(4)iut+uxx+c3|u|2u+2δ2|u|4u+2iδu2u-x=0.In [2], by using primary integral method, van Saarloos and Hohenberg obtained the exact solitary waves of the form(5)u(x,t)=e-iωteiψ(x-ct)φ(x-ct)=e-iωteiψ(ξ)φ(ξ),hhhhhhhhhhhhhhhhhhhhhhhhξ=x-ct,
for (2). Furthermore, Biswas [10] carried out the integration of the generalized Kundu equation to obtain the 1-soliton solution. Feng and Wang [11] gave the explicit solitary wave solutions by the algebraic curve method for (1a) and (1b). Bright/dark solitary wave solutions and triangular periodic wave solutions of (1a) and (1b) were obtained in [12]. Zhang et al. [13] obtained exact solitary waves of (1a) and (1b) by using the undetermined coefficient method, and they also investigated orbital stability of solitary waves. Zhang [14] obtained various exact traveling wave solutions of (1a) and (1b) by the general solutions of a kind of subequation. Besides, Guo and Wu proved that the kind of solitary waves (5) for (2) is stable in [15]. He and Meng [16] investigated the exact traveling waves for (4) by using the bifurcation theory and the method of phase portraits analysis.In this paper, we use the bifurcation method of dynamical systems [17–23] to study the traveling wave solutions and their bifurcations for (1a) and (1b). Through bifurcation phase portraits, we obtain various traveling wave solutions of (1a) and (1b). Via some special phase orbits, we also obtain new explicit traveling wave solutions. Not only the existence of these solutions is proved, but also their concrete expressions are presented. Our work extends some previous results [10–16].In order to derive traveling wave solutions, we assume that (1a) and (1b) have solutions of the form (5). Substituting (5) into (1a) and (1b) and making the real part and imaginary part of (1a) and (1b) equal to zero, we have
(6)ψ′′φ+2ψ′φ′-αφ2φ′-βφ2φ′-cφ′=0,(7)φ′′+[(ψ′)2c3φ2+c5φ4+αψ′φ2hhhhh-βψ′φ2-(ψ′)2+ω+cψ′]φ=0.Let(8)ψ′(ξ)=E+Dφ2(ξ).Substituting (8) into (6) and equating the coefficients of these terms φ, φ′, and φ′′ to zero, we get E=c/2, D=(α+β)/4. Therefore, when
(9)ψ′(ξ)=c2+α+β4φ2(ξ),(6) is identical to zero. Combining (9) and (7), we obtain the following equation:
(10)φ′′-d1φ-2d2φ3-3d4φ5=0,
where
(11)d1=-ω-c24,d2=-c32-(α-β)c4,d4=-13[c5+(α+β)(3α-5β)16].In fact, (10) is equivalent to the system
(12)dφdξ=y,dydξ=d1φ+2d2φ3+3d4φ5,
which has the first integral
(13)H(φ,y)=y2-d1φ2-d2φ4-d4φ6=h.The rest of the paper is organized as follows. Our main results and remarks are arranged in Section2. In Section 3, in order to derive the main results, we show various planar systems and their bifurcation phase portraits of (10). We state the theoretic derivation of our main results by the bifurcation method of dynamical systems in Section 4. Short conclusions are given in Section 5.


## 2. Main Results

In this section, we state our main results and give some corresponding remarks. Our main results are listed in the following two propositions.Proposition 1.
When the parametersd4>0 and d2<0, (10) possesses the following solutions.
(1) Ifd1=d22/3d4, the blowup solution is of expression
(14)φ1(ξ)=(-d2/3d4)((d2/3d4)|ξ|+1)((d2/3d4)|ξ|+1)2-1.
(2) Ifd22/4d4<d1<d22/3d4, the blowup solution is of expression
(15)φ2(ξ)=a1a2tanhη1a1tanh2η1+a22-a12,
and the solitary wave solution possesses the expression
(16)φ3(ξ)=a1a2tanhη2a1tanh2η2+a22-a12,
where
(17)a1=13[-d2d4-d22d42-3d1d4],a2=13[-d2d4+2d22d42-3d1d4],η1=-a1a12-a22d4|ξ|+arctanha12-a22a1,η2=a1a12-a22d4|ξ|+π2.
(3) Ifd1=d22/4d4, the blowup solution is of expression
(18)φ4(ξ)=φ21-e-2φ22d4|ξ|,
the kink wave solution is of expression
(19)φ5(ξ)=φ21+((φ22-φ32)/φ32)e-2φ22d4ξ,
and the periodic blowup solution is of expression
(20)φ6(ξ)=φ32+2φ32(φ32-φ12)φ12sin(c1-η3)-2φ32+φ12,
where
(21)φ1=-2d23d4,φ2=-d22d4,φ3=-d26d4,c1=arcsin2φ32-φ12φ12,η3=2φ3φ12-φ32d4|ξ|.
(4) If0<d1<d22/4d4, the solitary wave solution is of expression
(22)φ7(ξ)=2a3a4a32+a42+(a32-a42)coshη4,
and the kink wave solution is of expression
(23)φ8(ξ)=a5a6tanhη5a52+a62+a52tanh2η5,
where
(24)a3=-d2+d22-4d1d42d4,a4=-d2-d22-4d1d42d4,a5=-d2+d22-4d1d43d4,a6=d2+2d22-4d1d43d4,η4=-2a3a4d4ξ,η5=a5a52+a62d4ξ.
(5) Ifd1=0, the blowup solutions are of expressions
(25)φ9(ξ)=1d2ξ2-2d4ξ,φ10(ξ)=-(2d2/d4)tanhη63-2tanh2η6,
and the kink wave solution is of expression
(26)φ11(ξ)=-(2d2/d4)tanhη73-2tanh2η7,
where
(27)η6=arctanh62+63d2d4|ξ|,η7=-63d2d4ξ.Proposition 2.
When the parametersd4<0 and d2>0, (10) possesses the following solutions.
(1) Ifd1=d22/3d4, the solitary wave solution is of expression
(28)φ12(ξ)=d22ξ227d42-3d22d4ξ2.
(2) Ifd22/3d4<d1<d22/4d4, the solitary wave solution is of expression
(29)φ13(ξ)=a7a8(1+eη8)4a82eη8+a72(1-eη8)2,
the kink wave solution is of expression
(30)φ14(ξ)=a7a8(eη9-1)a72(1+eη9)2-4a82eη9,
and the periodic wave solution is of expression
(31)φ15(ξ)=a9a10tanη10a92-a102+a92tan2η10,
where
(32)a7=-13[d2d4-2d22d42-3d1d4],a8=-13[d2d4-d22d42-3d1d4],a9=13[-d2d4+d22d42-3d1d4],a10=13[-d2d4-2d22d42-3d1d4],η8=-2a8a72-a82-d4|ξ|,η9=2a8a72-a82-d4ξ,η10=a9a92-a102-d4|ξ|.
(3) Ifd1=d22/4d4, the solitary wave solution is of expression
(33)φ16(ξ)=-(d2/3d4)(1+eη11)2eη11+2(1-eη11)2,
and the kink wave solution is of expression
(34)φ17(ξ)=-(d2/3d4)(eη12-1)2(1+eη12)2-2eη12,
where
(35)η11=-d2-3d4|ξ|,η12=d2-3d4ξ.
(4) Ifd22/4d4<d1<0, the periodic wave solution is of expression
(36)φ18(ξ)=2a11a12a112+a122+(a112-a122)cosη13,
where
(37)a11=12[-d2d4-d22-4d1d4d4],a12=12[-d2d4+d22-4d1d4d4],η13=2a11a12-d4|ξ|.
(5) Ifd1=0, the solitary wave solution is of expression
(38)φ19(ξ)=d2-d4+d22ξ2.Remark 1.
The traveling wave solutionsφi(ξ)(i=3,5,6,7,9,11,14,15,18,19) are the same as the ones in [10–14], while the rest of the solutions are new, which cannot be found in the literature.Remark 2.
Choosing(39)d4=116,d2=-14,
we can obtain all the exact solutions reported in [16].Remark 3.
The correctness of the above solutions is tested by using the software Mathematica.


## 3. The Bifurcation Phase Portraits

In order to derive the main results mentioned in Section2, we draw the bifurcation phase portraits of system (12) by the qualitative theory of dynamical systems.First, letting(40)f(φ)=d1φ+2d2φ3+3d4φ5,
we have
(41)f′(φ)=d1+6d2φ2+15d4φ4.
Solving f(φ)=0, we get five roots as follows:
(42)φ*=0,φ1+=-d2+d22-3d1d43d4,φ1-=--d2+d22-3d1d43d4,φ2+=-d2-d22-3d1d43d4,φ2-=--d2-d22-3d1d43d4.Second, let(φ,0) be one of the singular points of system (12). The characteristic values of the linearized system of system (12) at (φ,0) are
(43)λ±(φ,0)=±f′(φ).
From the qualitative theory of dynamical systems, we get the properties of the singular point (φ,0) as follows.(i)
Iff′(φ)>0, then (φ,0) is a saddle point of system (12).(ii)
Iff′(φ)<0, then (φ,0) is a center point of system (12).(iii)
Iff′(φ)=0, then (φ,0) is a degenerate singular point of system (12).By using the property of equilibrium points and bifurcation theory, we obtain four bifurcation curves in the(d2,d1)-parameter plane as follows:
(44)l1:d1=d223d4,l2:d1=d224d4(d4>0),l3:d1=d223d4,l4:d1=d224d4(d4<0).Then, according to the qualitative theory, we obtain the bifurcation phase portraits of system (12) as shown in Figures 1 and 2.Figure 1
The phase portraits of system (12) when d4>0.Figure 2
The phase portraits of system (12) when d4<0.


## 4. The Derivations for Proposition1

In this section, by using the above bifurcation phase portraits (Figure1) and (12) and (13), we will give the exact traveling wave solutions of (10) under the given parameter conditions shown in Section 2.(1) Whend1=d22/3d4, the curve Γ1 possesses the following expression:
(45)y2=d4(φ2-φ02)3,whereφ≥φ0,
where φ0=-d2/3d4. Substituting (45) into dφ/dξ=y and integrating it along Γ1, we have
(46)∫φ+∞‍ds(s2-φ02)s2-φ02=-d4|ξ|.In (46) completing the integral and solving the equation for φ, we get the solution φ1(ξ).(2) Whend22/4d4<d1<d22/3d4, the curves Γ2 and Γ3 have the following expressions:
(47)Γ2:y2=d4(φ2-a12)2(φ2-a22),wherea1<φ,Γ3:y2=d4(a12-φ2)2(φ2-a22),wherea2≤φ<a1.Substituting (47) into dφ/dξ=y and integrating them along Γ2 and Γ3, respectively, it follows that
(48)∫φ+∞‍ds(s2-a12)s2-a22=-d4|ξ|(alongΓ2),∫a2φ‍ds(a12-s2)s2-a22=-d4|ξ|(alongΓ3).Completing the integrals above and solving them forφ, we obtain the solutions φ2(ξ) and φ3(ξ).(3) Whend1=d22/4d4, the curves Γ4, Γ5, and Γ6 have the following expressions:
(49)Γ4:y2=d4(φ2-φ22)2φ2,whereφ2<φ,Γ5:y2=d4(φ22-φ2)2φ2,where0<φ<φ2,Γ6:y2=d4(φ2-φ32)2(φ2-φ12),whereφ1≤φ.Substituting (49) into dφ/dξ=y and integrating them along Γ4, Γ5, and Γ6, respectively, it follows that
(50)∫φ+∞‍ds(s2-φ22)s=d4|ξ|(alongΓ4),∫φ3φ‍ds(φ22-s2)s=d4ξ(alongΓ5),∫φ+∞ds(s2-φ32)s2-φ12=d4|ξ|(alongΓ6).Completing the integrals above, we obtain the solutionsφi(ξ)(i=4,5,6).(4) When0<d1<d22/4d4, the curves Γ7 and Γ8 have the following expressions:
(51)Γ7:y2=d4(a32-φ2)(a42-φ2)φ2,hhhhhhhhwhere0<φ≤a4,Γ8:y2=d4(a52-φ2)2(φ2+a62),hhhhwhere-a5<φ<a5.Substituting (51) into dφ/dξ=y and integrating them along Γ7 and Γ8, respectively, it follows that
(52)∫a4φ‍dss(a32-s2)(a42-s2)=d4|ξ|(alongΓ7),∫0φ‍ds(a52-s2)s2+a62=d4ξ(alongΓ8).Completing the integrals above, we obtain the solutionsφ7(ξ) and φ8(ξ).(5) Ifd1=0, the curves Γ9, Γ10, and Γ11 have the following expressions:
(53)Γ9:y2=d4φ6+d2φ4,where-d2d4≤φ,Γ10:y2=d4(φ2-φ42)2(φ2-d23d4),hhhwhereφ>φ4=-2d23d4,Γ11:y2=d4(φ42-φ2)2(φ2-d23d4),hhhhhhwhere-φ4<φ<φ4.Substituting (53) into dφ/dξ=y and integrating them along Γ9, Γ10, and Γ11, respectively, it follows that
(54)∫φ+∞‍dss2s2+d2/d4=-d4|ξ|(alongΓ9),∫φ+∞‍ds(s2-φ42)s2-d2/3d4=-d4|ξ|(alongΓ10),∫0φ‍ds(φ42-s2)s2-d2/3d4=d4ξ(alongΓ11).Completing the integrals above, we obtain the solutionsφi(ξ)(i=9,10,11) and complete the derivations for Proposition 1.


## 5. The Derivations for Proposition2

Similarly, we will complete the derivations for Proposition2 by using the bifurcation phase portraits (Figure 2) and (12) and (13).(1) Whend1=d22/3d4, the curve Γ12 possesses the following expression:
(55)y2=-d4(φ52-φ2)3,
where φ5=-d2/3d4 and -φ5<φ<φ5. Substituting (55) into dφ/dξ=y and integrating it along Γ12, we have
(56)∫0φ‍ds(φ52-s2)φ52-s2=-d4|ξ|.In (56) completing the integral and solving the equation for φ, we get the solution φ12(ξ).(2) Whend22/3d4<d1<d22/4d4, the curves Γ13, Γ14, and Γ15 have the following expressions:
(57)Γ13:y2=-d4(a72-φ2)(φ2-a82)2,hhhhhhwherea8<φ≤a7,Γ14:y2=-d4(a82-φ2)2(a72-φ2),jjjjjjjjjwhere-a8<φ<a8,Γ15:y2=-d4(a92-φ2)2(a102-φ2),hhhwhere-a10≤φ≤a10.Substituting (57) into dφ/dξ=y and integrating them along Γ13, Γ14, and Γ15, respectively, it follows that
(58)∫a7φ‍ds(s2-a82)a72-s2=-d4|ξ|(alongΓ13),∫0φ‍ds(a82-s2)a72-s2=-d4|ξ|(alongΓ14),∫0φ‍ds(a92-s2)a102-s2=-d4|ξ|(alongΓ15).Completing the integrals above, we obtain the solutionsφi(ξ)(i=13,14,15).(3) Whend1=d22/4d4, the curves Γ16, and Γ17 have the following expressions:
(59)Γ16:y2=-d4(φ2-φ72)2(φ62-φ2),
where φ6=-2d2/3d4, φ7=-d2/6d4 and φ7<φ≤φ6,
(60)Γ17:y2=-d4(a72-φ2)2(a62-φ2),hhhhwhere-a7<φ<a7.Substituting (59) and (60) into dφ/dξ=y and integrating them along Γ16 and Γ17, respectively, it follows that
(61)∫φ6φ‍ds(s2-φ72)φ62-s2=-d4|ξ|(alongΓ16),∫0φ‍ds(φ72-s2)φ62-s2=-d4ξ(alongΓ17).Completing the integrals above, we obtain the solutionsφ16(ξ) and φ17(ξ).(4) Whend22/4d4<d1<0, the curve Γ18 possesses the following expression:
(62)Γ18:y2=-d4(a112-φ2)(φ2-a122)φ2,hhhhhhhwhereφ12≤φ≤φ11.Substituting (62) into dφ/dξ=y and integrating it along Γ18, we have
(63)∫a11φ‍dss(a112-s2)(s2-a122)=-d4|ξ|(alongΓ18).In (63) completing the integral above, we get the solution φ18(ξ).(5) Ifd1=0, the curve Γ19 possesses the following expression:
(64)Γ19:y2=-d4φ4(-d2d4-φ2),where0<φ≤-d2d4.Substituting (64) into dφ/dξ=y and integrating it along Γ19, we get
(65)∫φ-d2/d4‍dss2-d2/d4-s2=-d4|ξ|(alongΓ19).Similar to the process above, we obtainφ19(ξ) and complete the derivations for Proposition 2. Hereto, we have completed the derivations for our main results mentioned in Section 2.


## 6. Conclusions

In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the bifurcations of traveling wave solutions for the Kundu equations (1a) and (1b). In order to derive the traveling wave solutions, we show the bifurcation phase portraits of system (12). Through the phase portraits, we obtain various explicit traveling wave solutions. Also, our work includes some previous results. On the other hand, from some special orbits, we also obtain some new explicit traveling wave solutions.Note that there are two problems waiting to solve. The first one is that we do not give the elliptic periodic wave solutions for the Kundu equations (1a) and (1b). The relationship between the elliptic periodic wave solutions and the solutions given in this paper is not revealed. The second one is that Zhang et al. [13] showed that the solitary wave solutions φi(ξ)(i=3,4) are of orbital stability. However, we do not know whether the other solitary wave solutions obtained in this paper are of orbital stability.



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