# Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

Authors: Zhao Li; Peng Li; Tianyong Han
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.


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## 1. Introduction

The coupled nonlinear Schrödinger (CNLS) equations [1–3] are a very important mathematical physical model in the fields of quantum mechanics, nonlinear optics, optical fiber communication, Bose-Einstein condensate, fluid mechanics, and so on. Because of its importance, many new methods and techniques have been proposed to study the CNLS equation, which include the Bäcklund transformation, the variational iteration method, the extended unified method, and the Hirota method [4–9]. However, the study of CNLS equations with variable coefficients [10–14] has more important theoretical significance and practical value than CNLS equations with constant coefficients because it usually describes the inhomogeneous effects of nonlinear optical pulse propagations in the real optical fiber communication system. The CNLS equations with variable coefficients are expressed as follows:(1)i∂ψ∂x+βx∂2ψ∂t2+2gxψ2+ϕ2ψ=0,i∂ϕ∂x+βx∂2ϕ∂t2+2gxψ2+ϕ2ϕ=0,where ψ=ψt,x and ϕ=ϕt,x denote the slowly varying amplitudes of two orthogonally polarized optical pluses, which are complex-valued functions with the retarded time t and the normalized propagation distance x. t,x∈0,T×a,b, T≥0, −∞≤a≤b≤+∞. The coefficients βx and gx are integrable real functions of x, which represent the group velocity dispersion and the Kerr nonlinearity, respectively. Obviously, the integrability method, the modified sine-Gordon equation method, the unified method, the improved F-expansion method, and the Hirota bilinear method have used to establish the traveling wave solutions of the CNLS equation with variable coefficients in Ref. [10-14], respectively. In this paper, we investigate the dynamical properties and the classification of single traveling wave solutions of the CNLS equation with variable coefficients based on the bifurcation theory and the complete discrimination system method.The synopsis of the article is organized as follows. In Section2, the bifurcation theory is employed to investigate the CNLS equations with variable coefficients; phase portraits and some Jacobian elliptic function solutions are obtained. In Section 3, some single traveling wave solutions of the CNLS equations with variable coefficients are obtained by using the complete discrimination system method. In Section 4, we present some research results.


## 2. Bifurcations of Phase Portraits of System (9)

In order to analyze the dynamic behavior of Equation (1), we first introduce the following traveling wave transformation:(2)ψt,x=Ψξeiη,ϕt,x=Φξeiη,ξ=μt−λx,η=αt−θx,where Ψ and Φ are a real valued function, μ and α are arbitrary constants, and λx and θx are arbitrary function of x.Substituting (2) into (1), we get the coupled nonlinear differential equations(3)βxμ2Ψ′′ξ+iμ2αβx−λ′xΨ′ξ+θ′x−α2βxΨξ+2gxΨ2ξ+Φ2ξΨξ=0,βxμ2Φ′′ξ+iμ2αβx−λ′xΦ′ξ+θ′x−α2βxΦξ+2gxΨ2ξ+Φ2ξΦξ=0.Next, we decompose real parts and imaginary parts of (3) and put zero for imaginary parts of Equations (3), where the imaginary part of Equations (3) is λ′x=2αβx. Then, we set λx=2α∫axβςdς, a≤x≤b. Therefore, ξ=μt−2α∫axβςdς. Using the coefficients of Ψ′′ (or Φ′′) as the normalization coefficient, we have(4)θx=∫axμ2c1+α2βςdς+θ0,gx=c22μ2βx,λx=2α∫axβςdς,where c1 and c2 are constants and θ0 is the integral constant. On the other hand, the real parts of Equations (3) are simplified as(5)Ψ′′ξ+c1Ψξ+c2Ψ2ξ+Φ2ξΨξ=0,Φ′′ξ+c1Φξ+c2Ψ2ξ+Φ2ξΦξ=0.LetdΨ/dξ=p, dΦ/dξ=q; system (5) is transformed into a four-dimensional dynamical system as follows:(6)dΨdξ=p,dpdξ=−c2Ψ3ξ−c2ΨξΦ2ξ−c1Ψξ,dΦdξ=q,dqdξ=−c2Φ3ξ−c2Ψ2ξΦξ−c1Φξ.In order to remove the coupled relationship of (5), we consider the dynamical behavior of system (6) in the submanifold Ψ−lΦ=0(l≠0) of the 4-dimensional phase space Ψ,p,Φ,q. Substituting Ψ=lΦ (or Φ=Ψ/l) into (6), we obtain two completely uncoupled planar dynamical systems:(7)dΨdξ=p,dpdξ=−c21+l2l2Ψ3ξ−c1Ψξ,(8)dΦdξ=q,dqdξ=−c21+l2Φ3ξ−c1Φξ.Systems (7) and (8) are the same if and only if they satisfy l=±1. Then, system (7) can be rewritten as the following planar dynamical system(9)dΨdξ=p,dpdξ=−2c2Ψ3−c1Ψ,which is equivalent to the following Hamiltonian system:(10)Hp,Ψ=12p2+c22Ψ4+c12Ψ2=h,h∈ℝ.SupposefΨ=−2c2Ψ3−c1Ψ, and further assume that MiΨi,0i=0,1,2 is an equilibrium points of system (9), then the eigenvalues of system (9) at the equilibrium points are expressed as λ1,2=±f′Ψ. Thus, we can easily get three zeros of fΨ when c1c2<0, which are Ψ0=0, Ψ1=−−c1/2c2 and Ψ2=−c1/2c2. We can easily obtain one zero of fΨ when c1c2>0, which is Ψ3=0. By the bifurcation theory of planar dynamical systems, the equilibrium point MiΨi,0 is a saddle point when f′Ψi>0, the equilibrium point MiΨi,0 is a degraded saddle point when f′Ψi=0, and the equilibrium point MiΨi,0 is center point when f′Ψi<0. The phase portraits of (9) depending on different parameters c1 and c2 are shown in Figure 1.Lemma 1 (see [15]).
LetΨξ be a continuous solution of system (9) for ξ∈−∞,+∞ and limξ⟶∞Ψξ=m, limξ⟶−∞Ψξ=n:(1)
Ψξ is called a smooth solitary wave solution if m=n. Usually, a solitary wave solution of Equation (1) corresponds to a smooth homoclinic orbit of system (7)(2)
Ψξ is called a smooth kink or antikink solution if m≠n. A smooth kink (or antikink) wave solution of Equation (1) corresponds to a smooth heteroclinic orbit (or a so-called connecting orbit) of system (7)Remark 2.
Obviously (see [16–20]), all traveling wave solutions of (1) can be obtained from the phase orbits of the Hamiltonian system (9) according to Lemma 1.Figure 1
The phase portraits of system (9).
(a)(b)


### 2.1. The Casec1>0 and c2<0

(1)
Forh∈0,−c12/8c2, the system (10) can be written as(11)p2=−c2Ψ4+c1c2Ψ2−2hc2=−c2Ψ1h2−Ψ2Ψ2h2−Ψ2,where Ψ1h2=−c1/2c2−1/2c2c12+8c2h and Ψ2h2=−c1/2c2+1/2c2c12+8c2h.Substituting (11) into the first equation of system (9), integrating them along the periodic orbits, then we obtain(12)Ψ1ξ=±Ψ1hsnΨ2h−c2ξ−ξ0,Ψ1hΨ2h.Namely,(13)ψ1t,x=±Ψ1hsnΨ2h−c2μt−2α∫axβςdς−ξ0,Ψ1hΨ2heiαt−∫axμ2c1+α2βςdς+θ0.(2)
Forh=−c12/8c2, we easily obtain two kink solitary wave solutions(14)Ψ2ξ=±−c12c2tanh−c2ξ−ξ0,(15)ψ2t,x=±−c12c2tanh−c2μt−2α∫axβςdς−ξ0eiαt−∫axμ2c1+α2βςdς+θ0.Whenμ=2, c1=2, c2=−1, α=1/2, and ξ0=0, the dark soliton solution ∣ψ2∣ of Equation (1) is drawn in Figure 2, where βx=1 and βx=sinx, respectively.Figure 2
The dark solitary wave solution∣ψ2t,x∣ of Equation (15) for differential parameter βx.
(a)(b)


### 2.2. The Casec1<0 and c2>0

(1)
Forh∈−c12/8c2,0, the system (10) can be written as(16)p2=c2−Ψ4−c1c2Ψ2+2hc2=c2Ψ2−Φ1h2Ψ2h2−Ψ2,where Ψ1h2=−c1/2c2−1/2c2c12+8c2h, Ψ2h2=−c1/2c2+1/2c21/2c2c12+8c2h.Substituting (16) into dΨ/dξ=p, we can obtain the smooth periodic wave solution(17)Ψ3ξ=±Ψ2hdnΨ2hc2ξ−ξ0,Ψ2h2−Ψ1h2Ψ2h,ψ3t,x=±Ψ2hdnΨ2hc2μt−2α∫axβςdς−ξ0,Ψ2h2−Ψ1h2Ψ2heiαt−∫axμ2c1+α2βςdς+θ0.(2)
Forh=0, we obtain two bell solitary wave solutions:(18)Ψ4ξ=±−c1c2sech−c1ξ−ξ0,(19)ψ4t,x=±−c1c2sech−c1μt−2α∫axβςdς−ξ0eiαt−∫axμ2c1+α2βςdς+θ0.Whenμ=2, c1=−2, c2=1, α=1/2, ξ0=0, the bright solitary wave solution ∣ψ4∣ of Equation (1) is drawn in Figure 3, where βx=ex and βx=ex+2xex2, respectively.(3)
Forh∈0,+∞, the system (10) can be written as(20)p2=c2−Ψ4−c1c2Ψ2+2hc2=c2Ψ3h2+Ψ2Ψ4h2−Ψ2,where Ψ3h2=−c1/2c2+1/2c2c12+8c2h and Ψ4h2=c1/2c2+1/2c2c12+8c2h.Figure 3
The dark solitary wave solution∣ψ4t,x∣ of Equation (19) for differential parameter βx.
(a)(b)Substituting (20) into dΨ/dξ=p and integrating them, we obtain two families of periodic solutions(21)Ψ5ξ=Ψ4hcnc2Ψ3h2+Ψ4h2ξ−ξ0,Ψ4hΨ3h2+Ψ4h2.Namely,(22)ψ5t,x=Ψ4hcnc2Ψ3h2+Ψ4h2μt−2α∫axβςdς−ξ0,Ψ4hΨ3h2+Ψ4h2eiαt−∫axμ2c1+α2βςdς+θ0.Remark 3.
In this section, we mainly discuss the solutionψt,x of Equation (1) when c1c2<0. Moreover, we can easily obtain the solution ϕt,x of Equation (1) through their relationship Ψ=lΦ.


## 2.1. The Casec1>0 and c2<0

(1)
Forh∈0,−c12/8c2, the system (10) can be written as(11)p2=−c2Ψ4+c1c2Ψ2−2hc2=−c2Ψ1h2−Ψ2Ψ2h2−Ψ2,where Ψ1h2=−c1/2c2−1/2c2c12+8c2h and Ψ2h2=−c1/2c2+1/2c2c12+8c2h.Substituting (11) into the first equation of system (9), integrating them along the periodic orbits, then we obtain(12)Ψ1ξ=±Ψ1hsnΨ2h−c2ξ−ξ0,Ψ1hΨ2h.Namely,(13)ψ1t,x=±Ψ1hsnΨ2h−c2μt−2α∫axβςdς−ξ0,Ψ1hΨ2heiαt−∫axμ2c1+α2βςdς+θ0.(2)
Forh=−c12/8c2, we easily obtain two kink solitary wave solutions(14)Ψ2ξ=±−c12c2tanh−c2ξ−ξ0,(15)ψ2t,x=±−c12c2tanh−c2μt−2α∫axβςdς−ξ0eiαt−∫axμ2c1+α2βςdς+θ0.Whenμ=2, c1=2, c2=−1, α=1/2, and ξ0=0, the dark soliton solution ∣ψ2∣ of Equation (1) is drawn in Figure 2, where βx=1 and βx=sinx, respectively.Figure 2
The dark solitary wave solution∣ψ2t,x∣ of Equation (15) for differential parameter βx.
(a)(b)


## 2.2. The Casec1<0 and c2>0

(1)
Forh∈−c12/8c2,0, the system (10) can be written as(16)p2=c2−Ψ4−c1c2Ψ2+2hc2=c2Ψ2−Φ1h2Ψ2h2−Ψ2,where Ψ1h2=−c1/2c2−1/2c2c12+8c2h, Ψ2h2=−c1/2c2+1/2c21/2c2c12+8c2h.Substituting (16) into dΨ/dξ=p, we can obtain the smooth periodic wave solution(17)Ψ3ξ=±Ψ2hdnΨ2hc2ξ−ξ0,Ψ2h2−Ψ1h2Ψ2h,ψ3t,x=±Ψ2hdnΨ2hc2μt−2α∫axβςdς−ξ0,Ψ2h2−Ψ1h2Ψ2heiαt−∫axμ2c1+α2βςdς+θ0.(2)
Forh=0, we obtain two bell solitary wave solutions:(18)Ψ4ξ=±−c1c2sech−c1ξ−ξ0,(19)ψ4t,x=±−c1c2sech−c1μt−2α∫axβςdς−ξ0eiαt−∫axμ2c1+α2βςdς+θ0.Whenμ=2, c1=−2, c2=1, α=1/2, ξ0=0, the bright solitary wave solution ∣ψ4∣ of Equation (1) is drawn in Figure 3, where βx=ex and βx=ex+2xex2, respectively.(3)
Forh∈0,+∞, the system (10) can be written as(20)p2=c2−Ψ4−c1c2Ψ2+2hc2=c2Ψ3h2+Ψ2Ψ4h2−Ψ2,where Ψ3h2=−c1/2c2+1/2c2c12+8c2h and Ψ4h2=c1/2c2+1/2c2c12+8c2h.Figure 3
The dark solitary wave solution∣ψ4t,x∣ of Equation (19) for differential parameter βx.
(a)(b)Substituting (20) into dΨ/dξ=p and integrating them, we obtain two families of periodic solutions(21)Ψ5ξ=Ψ4hcnc2Ψ3h2+Ψ4h2ξ−ξ0,Ψ4hΨ3h2+Ψ4h2.Namely,(22)ψ5t,x=Ψ4hcnc2Ψ3h2+Ψ4h2μt−2α∫axβςdς−ξ0,Ψ4hΨ3h2+Ψ4h2eiαt−∫axμ2c1+α2βςdς+θ0.Remark 3.
In this section, we mainly discuss the solutionψt,x of Equation (1) when c1c2<0. Moreover, we can easily obtain the solution ϕt,x of Equation (1) through their relationship Ψ=lΦ.


## 3. The Classification of Single Traveling Wave Solutions via Complete Discrimination Systems

In order to remove the coupled relationship of (5), we assume Ψ=lΦ (l≠0). Substituting Ψ=lΦ into (5), we have(23)Ψ′′+c1Ψ+c2Ψ2+c1l2Ψ3=0.Multiplying both sides of (23) by Ψ′ and integrating once with respect to ξ, we have(24)Ψ′2=−c22l2Ψ4−2c23Ψ3−c1Ψ2−2c0,where c0 is the integral constant.Make the transformation(25)φ=−c22l21/4Ψ+l23,ξ1=−c22l21/4ξ.Next, substituting (25) into (24), it becomes(26)φξ1′2=φ4+a2φ2+a1φ+a0,where a2=−2c1l/−2c2, a1=−16l4/27c2+c1l2/3−c2/2l2−1/4, and a0=c2l6/54−c1l4/9−2c0. The solution of (26) can be written as the integral form(27)∫dφφ4+a2φ2+a1φ+a0=±ξ1−ξ0,where ξ0 is an arbitrary constant.LetGφ=φ4+a2φ2+a1φ+a0. Thus, we can obtain its complete discrimination system [21–23] as follows:(28)D1=4,D2=−b1,D3=−2b13+8b1b3−9b22,E2=9b12−32b1b3,D4=−b13b22+4b14b3+36b1b22b3−32b12b32−274b24+64b33.Case 1.
WhenD2<0, D3=0, and D4=0. Then, Gφ admits a pair of conjugate complex roots of multiplicities two, namely,(29)Gφ=φ−u2+v22,where v>0. Substituting (29) into (27), we attain(30)ξ1−ξ0=∫dφφ−u2+v2=1varctanφ−uv,then the solution of (23) is expressed as follows:(31)Ψ1ξ=±−c22l2−14vtanv−c22l214ξ−ξ0−l23.
Namely,(32)ψ1t,x=±−c22l2−1/4vtanv−c22l21/4μt−2α∫axβςdς−ξ0−l23eiαt−∫axμ2c1+α2βςdς+θ0.Case 2.
D2=0, D3=0, and D4=0. Gφ has real roots of multiplicities of four, namely, Gφ=φ4. Substituting Gφ=φ4 into (27), we can get(33)ξ1−ξ0=∫dφφ2=−1φ,then the solutions of Equation (1) can be shown as(34)ψ2t,x=∓−c22l2−1/41−c2/2l2−1/4ξ−ξ0−l33eiαt−∫axμ2c1+α2βςdς+θ0.Case 3.
D2>0, D3=0, D4=0, and E2>0. Gφ has two real roots of multiplicities of two, namely,(35)Gφ=φ−u2φ−v2,where u>v. Substituting (35) into (27), we can obtain(36)±ξ1−ξ0=∫dφφ−uφ−v=1u−vlnφ−uφ−v.Whenφ>u or φ<v, we can obtain the solution of Equation (23) as follows:(37)Ψ3ξ1=v−ueu−vξ1−ξ0−1+v=v−u2cothu−vξ1−ξ02−1+v,then we have(38)ψ3t,x=v−u−c2/2l2−1/42cothu−v−c2/2l21/4μt−2α∫axβςdς−ξ02−1+v−l23eiαt−∫axμ2c1+α2βςdς+θ0.Whenv<Φ<u, we can gain the solution of Equation (23) as follows:(39)Ψ4ξ1=v−u−eu−vξ1−ξ0−1+v=v−u2tanhu−vξ1−ξ02−1+v.Similarly, we can have(40)ψ4t,x=v−u−c2/2l2−1/42tanhu−v−c2/2l21/4μt−2α∫axβςdς−ξ02−1+v−l23eiαt−∫axμ2c1+α2βςdς+θ0.Case 4.
D2>0, D3>0, and D4=0. Gφ has two real roots and real roots with multiplicities of two, namely,(41)Gφ=φ−ρ12φ−ρ2φ−ρ3,where ρi (i=1,2,3) are real numbers and ρ2>ρ3.Whenρ1>ρ2 and φ<ρ2, or when ρ1<ρ3 and φ<ρ3, we can have the implicit analytical solution of Equation (26):(42)±ξ1−ξ0=∫dφφ−ρ12φ−ρ2φ−ρ3=1ρ1−ρ2ρ1−ρ3lnφ−ρ2ρ1−ρ3−ρ1−ρ2φ−ρ32∣φ−ρ1∣.Whenρ1>ρ2 and φ<ρ3, or when ρ1<ρ3 and φ<ρ2, we can obtain the implicit analytical solution of Equation (26):(43)±ξ1−ξ0=∫dφφ−ρ12φ−ρ2φ−ρ3=1ρ1−ρ2ρ1−ρ3lnφ−ρ2ρ3−ρ1−ρ2−ρ1φ−ρ32∣φ−ρ1∣.Whenρ2>ρ1>ρ3, we can have the implicit analytical solution of Equation (26):(44)±ξ1−ξ0=∫dφφ−ρ12φ−ρ2φ−ρ3=1ρ2−ρ1ρ1−ρ3arcsinφ−ρ2ρ1−ρ3+ρ1−ρ2φ−ρ3∣φ−ρ1ρ2−ρ3∣.Case 5.
D2>0, D3=0, D4=0, and E2=0. Gφ has real roots of multiplicities of three and real roots with multiplicities of one, namely,(45)Gφ=φ−u3φ−γ,where u and v are real numbers. Substituting (45) into (27), we can yield(46)±ξ1−ξ0=∫dφφ−u3φ−v=2v−uφ−vφ−u.Whenφ>u and φ>v, or when φ<u and φ<v, the solution of Equation (26) is given by(47)φξ1=4u−vv−u2ξ1−ξ02−4+u,then we can obtain a rational solution of Equation (1) as(48)ψ6t,x=4−c2/2l2−1/4u−v−c2/2l21/4μt−2α∫axβςdς−ξ02v−u2−4+u−l23eiαt−∫axμ2c1+α2βςdς+θ0.Case 6.
D2D3<0 and D4=0. Gφ has real roots of multiplicities of two and a pair of conjugate complex roots, namely,(49)Gφ=φ−ρ12φ−ρ22+ρ32,where ρii=1,2,3 are real numbers. Substituting (49) into (27), we can obtain(50)±ξ1−ξ0=∫dφφ−ρ1φ−ρ22+ρ32=1ρ1−ρ22+ρ32lnc1φ+c2−φ−ρ22+ρ32φ−ρ1,where ε1=ρ1−2ρ2/ρ1−ρ22+ρ32, ε2=ρ1−ρ22+ρ32−ρ1ρ1−2ρ2/ρ1−ρ22+ρ32. Thus, we obtain the solution of Equation (1):(51)ψ7t,x=e±ρ1−ρ22+ρ32−c2/2l21/4μt−2α∫axβςdς−ξ0−ε1+ρ1−ρ22+ρ322−ε1−c2/2l21/4e±ρ1−ρ22+ρ32−c2/2l21/4μt−2α∫axβςdς−ξ0−ε12−1−l23eiαt−∫axμ2c1+α2βςdς+θ0,which is a solitary wave solution.Case 7.
D2>0, D3>0, and D4>0. Gφ has four distinct real roots, namely,(52)Gφ=φ−ρ1φ−ρ2φ−ρ3φ−ρ4,where ρ1, ρ2, ρ3, and ρ4 are real numbers, and ρ1>ρ2>ρ3>ρ4.
Whenφ>ρ1 or φ<ρ4, then we make the following transformation:(53)φ=ρ2ρ1−ρ4sin2θ−ρ1ρ2−ρ4ρ1−ρ4sin2θ−ρ2−ρ4.
By using (27), we obtain(54)±ξ1−ξ0=∫dφφ−ρ1φ−ρ2φ−ρ3φ−ρ4=2ρ1−ρ3ρ2−ρ4∫dθ1−m2sin2θ,where m2=ρ1−ρ4ρ2−ρ3/ρ1−ρ3ρ2−ρ4.
From Equation (54) and the definition of Jacobian elliptic sine function, we obtain(55)sinθ=snρ1−ρ3ρ2−ρ42ξ1−ξ0,m.Combining Equation (55) with the expressions (53), we can give the solution of Equation (1):(56)ψ8t,x=ρ2ρ1−ρ4−c2/2l2−1/4sn2ρ1−ρ3ρ2−ρ4/2−c2/2l21/4ξ−ξ0,m−ρ1ρ2−ρ4ρ1−ρ4sn2ρ1−ρ3ρ2−ρ4/2−c2/2l21/4ξ−ξ0,m−ρ2−ρ4−l23eiαt−∫axμ2c1+α2βςdς+θ0,where ξ=μt−2α∫axβςdς.Whenρ3<φ<ρ2, then we make the transformation(57)φ=ρ4ρ2−ρ3sin2θ−ρ3ρ2−ρ4ρ2−ρ3sin2θ−ρ2−ρ4.Similarly, we can obtain the solution of Equation (1):(58)ψ9t,x=ρ4ρ2−ρ3−c2/2l2−1/4sn2ρ1−ρ3ρ2−ρ4/2−c2/2l21/4ξ−ξ0,m−ρ3ρ2−ρ4ρ2−ρ3sn2ρ1−ρ3ρ2−ρ4/2−c2/2l21/4ξ−ξ0,m−ρ2−ρ4−l23eiαt−∫axμ2c1+α2βςdς+θ0.Case 8.
D2D3≥0 and D4<0. Gφ has two different real roots and a pair of conjugate complex roots, namely,(59)Gφ=φ−ρ1φ−ρ2φ−ρ32+ρ42,where ρ1, ρ2, ρ3, and ρ4 are real constants, ρ1>ρ2 and ρ4>0.
We make the following transformation:(60)φ=ε1cosθ+ε2ε2cosθ+c4,where ε1=1/2ρ1+ρ2ε3−1/2ρ1−ρ2ε4, ε2=1/2ρ1+ρ2c4−1/2ρ1−ρ2ε3, ε3=ρ1−ρ3−ρ4/m1, ε4=ρ1−ρ3−ρ4m1, E=ρ42+ρ1−ρ3ρ2−ρ3/ρ4ρ1−ρ2, and m1=E±E2+1.
By using (27), we obtain(61)ξ1−ξ0=∫dφ±φ−ρ1φ−ρ2φ−ρ32+ρ42=2m1m2∓2ρ4m1ρ1−ρ2∫dθ1−m22sin2θ,where m22=2/1+m12.From Equation (61) and the definition of Jacobian elliptic function, we obtain(62)cosθ=cn∓2ρ4m1ρ1−ρ22m1m2ξ1−ξ0,m2.Combining Equation (62) with the expression (60), we can give the solution of Equation (1):(63)ψ10t,x=a4−1/4ε1cn∓2ρ4m1ρ1−ρ2/2m1m2−c2/2l21/4μt−2α∫axβςdς−ξ0,m2+ε2ε3cn∓2ρ4m1ρ1−ρ2/2m1m2−c2/2l21/4μt−2α∫axβςdς−ξ0,m2+ε3eiαt−∫axμ2c1+α2βςdς+θ0,which is an elliptic double periodic function solution.Case 9.
D2D3≤0 and D4>0. Gφ has two pairs of conjugate complex roots, namely,(64)Gφ=φ−ρ12+l22φ−ρ22+l22,where ρ1, ρ2, l1, and l2 are real constants and l1≥l2>0.
We make the following transformation:(65)φ=ε1tanθ+ε2ε3tanθ+ε4,where ε1=ρ1ε3+l1ε4, ε2=ρ1ε4−l1ε3, ε3=−l1−l2/m1, ε4=ρ1−ρ2, E=ρ1−ρ22+l12+l22/2l1l2, and m1=E+E2−1.
By using (27), we obtain(66)ξ1−ξ0=∫dφφ−ρ12+l22φ−ρ22+l22=ε32+ε42l2ε32+ε42m12ε32+ε42∫dθ1−m22sin2θ,where m22=m12−1/m12.From (65) and the definition of Jacobian elliptic function, we obtain(67)sinθ=snl2ε32+ε42m12ε32+ε42ε32+ε42ξ1−ξ0,m2,(68)cosθ=cnl2ε32+ε42m12ε32+ε42ε32+ε42ξ1−ξ0,m2.Combining Equations (67) and (68) with expression (65), we can give the solution of Equation (1)(69)ψ11t,x=−c2/2l2−1/4ε1snμ−c2/2l21/4ξ−ξ0,m2+−c2/2l2−1/4ε2cnμ−c2/2l21/4ξ−ξ0,m2ε3snμ−c2/2l21/4ξ−ξ0,m2+ε4cnμ−c2/2l21/4ξ−ξ0,m2−l23eiαt−∫axμ2c1+α2βςdς+θ0,which is an elliptic double periodic function solution, where ξ=μt−2α∫axβςdς.Remark 4.
In this section, we mainly discuss the solutionψt,x of Equation (1). Similarly, we can easily obtain the solution ϕt,x of Equation (1) through their relationship Ψ=łΦ.


## 4. Conclusion

In the paper, the planar dynamical system method and the complete discrimination system method are applied to the CNLS equations with variable coefficients for constructing different types of traveling wave solutions. The complex periodic wave solutions, implicit analytical solutions, complex hyperbolic function solutions, and complex Jacobian elliptic function solutions of the CNLS equations with variable coefficients are successfully investigated by two different methods. Moreover, phase portraits are drawn with the help of Maple software. New solutions of the CNLS equations with variable coefficients have not been reported in previous literatures, such as complex Jacobian elliptic function solutions and implicit analytical solutions.



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