# New Exact Solutions for High Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

Authors: Yongan Xie; Shengqiang Tang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.


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## Body

## 1. Introduction

Propagation of short pulses in optical fibers is governed by the well-known nonlinear Schrödinger equation (NLS) [1]. In recent years, There have been extensive study and application of NLS. The main purpose of this paper is to discuss the traveling wave solutions for a class of high dispersive cubic-Quintic nonlinear Schrödinger equations describing the ultrashort light pulse propagation as in the following:
(1)Ez=-iβ22Ett+iγ1|E|2E+β36Ettt+iβ424Etttt-iγ2|E|4E,
where E(z,t) is the slowly varying envelope of the electric field, β2 is the parameter of the group velocity dispersion, β3 and β4 are, respectively, the third-order and fourth-order dispersions, and γ1 and γ2 are the nonlinearity coefficients. When the higher order terms are ignored, we obtain the NLS. However, for femtosecond light pulses, whose duration is shorter than 10 fs, the last three terms are not ignored. Equation (1) was derived by Palacios and Fernández-Diáz [2]. Azzouzi et al. [3] by using the extended hyperbolic auxiliary equation method in getting the exact explicit solutions to (1). He et al. [4] find the exact bright, dark, and gray analytical nonautonomous soliton solutions of the generalized CQNLSE with spatially inhomogeneous group velocity dispersion (GVD) and amplification or attenuation by the similarity transformation method under certain parametric conditions.We will study (1) by using the improved Fan subequation method. As a result, more types of exact solutions to (1) are obtained, which include solitons, kink solutions, and Jacobian elliptic function solutions with double periods. The rest of this paper is organized as follows. In Section 2, we give the mathematical framework of the improved method. In Section 3, we apply it to the generalized equation (1) for finding more exact solutions. Finally, some conclusions are given.


## 2. The Ansatz Solution and Fan Subequation Method

The integrability of a nonlinear equation can be studied by applying the Painleve analysis. It is widely believed that possession of the Painleve property is a sufficient criterion for integrability. Moreover, there exists another technique which basically consists of expressing the solution in terms of an amplitude and a phase function as an approach to find exact solutions of nonlinear evolution equations. We will make use of this formalism looking for exact solution of (1) such as
(2)E(z,t)=ei(w0z-wt)φ(ξ),
where φ(ξ) is a real fuction and ξ=v0z-vt. By inserting the expressions (2) into (1), and separating real and imaginary parts, we obtain(3a)l1φ′(ξ)+l3φ′′′(ξ)=0,(3b)l0φ(ξ)+l2φ′′(ξ)+l4φ′′′′(ξ)+γ1φ3(ξ)-γ2φ5(ξ)=0,where
(4)l0=12w2(β2+13β3w+112β4w2)-w0,l1=wv(β2+12β3w+16β4w2)-v0,l2=-v2(12β2+12β3w+14β4w2),l3=-16(β3+β4w)v3,l4=124β4v4.
Let l1=l3=0,and we get
(5)w=-β3β4,v0=vβ3(β32-3β2β4)3β42,l0=-w0-β32(β32-4β2β4)8β43,l2=v2(β32-2β2β4)4β4,l4=124β4v4.
Then, (3a) and (3b) become
(6)l0φ(ξ)+l2φ′′(ξ)+l4φ′′′′(ξ)+γ1φ3(ξ)-γ2φ5(ξ)=0.
We introduce auxiliary equation:
(7)φ′(ξ)=ϵc0+c1φ(ξ)+c2φ2(ξ)+c3φ3(ξ)+c4φ4(ξ),
where ϵ=±1, which is known as Fan subequation method and proposed by Fan in [5]. This method is proposed to seek more types of exact solutions of nonlinear partial differential equations. Obviously, (7) is equivalent to the two-dimensional systems as follows:
(8)dφdξ=y,dydξ=12(c1+2c2φ+3c3φ2+4c4φ3),
which has the Hamiltonian function:
(9)H(φ,y)=y2-(c1φ+c2φ2+c3φ3+c4φ4)=c0.One can easily find thatc0 corresponds to the Hamiltonian constant and (7) is equivalent to the Hamiltonian system (8). Thus, in order to search the exact solutions of (7) we need only to discuss (8). For a fixed c0, (9) determines a set of orbits of (8). As c0 varies, (9) defines different families of orbits of (8) which have different dynamical behavior. Below we will first study the bifurcation of phase portraits of (8) by making use of bifurcation method of dynamical systems and with the aid of the computer symbolic system Mathematica. Then according to the obtained bifurcation and the Hamiltonian function (9), we will gain many new exact solutions of (7) for all possible parameters cj [6].Substituting (7) into (6), we have
(10)(-γ2+24l4c42)φ5+30l4c3c4φ4+(2l2c4+l4(20c2c4+152c32)+γ1)φ3+(32l2c3+l4(152c2c3+15c1c4))φ2+(l0+l2c2+l4(92c1c3+12c0c4+c22))φ+12l2c1+l4(3c0c3+12c1c2)=0.Setting all the coefficients ofφi (i=0,1,…,5) to zero, and solving the obtained algebraic equations, we find the following sets of solutions (I)
(11)c1=c3=0,c2=1v2(-3(β32-2β2β4)5β42-6γ1γ2/β45γ2),c4=1v2γ2β4,c0=((50w0γ2β43-3γ12β43-52γ2β2β32β4γ2β4h+27γ2β22β42+13γ2β34)×γ2β4+12γ1γ2β4(β32-2β2β4))×(25γ22β43v2)-1
and (II)
(12)c1=c3=0,c2=15v2(-3(β32-2β2β4)β42+6γ1γ2/β4γ2),c4=-1v2γ2β4,c0=(-(50w0γ2β43-3γ12β43-52γ2β2β32β4γ2β4+27γ2β22β42+13γ2β34)×γ2β4+12γ1γ2β4(β32-2β2β4))×(25γ22β43v2)-1
and (III), γ2=0:
(13)c0=-((-β32+2β2β4)h×(200β43w0+73β34-292β2β32β4+192β22β42))×(150β45γ1v2)-1,c1=∓200β43w0+73β34-292β2β32β4+192β22β42150β43γ1v2×-5γ1β4,c2=6(-β32+2β2β4)5β42v2,c3=±45v2-5γ1β4,c4=0,
where w0,v are any real number and v≠0.


## 3. Exact Solutions of High Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

### 3.1. Case (I) and Case (II)

In this case, the Hamiltonian system (8) becomes
(14)dφdξ=y,dydξ=c2φ+2c4φ3,
and the Hamiltonian function (9) reduces to
(15)H1(φ,y)=y2-c2φ2-c4φ4=c0.
Now we discuss the bifurcations of phase portraits of (14). Obviously all the equilibrium points of (14) lie in the φ-axis and their abscissas are the real zeros of f(φ)=c2φ+2c4ϕ3. Thus, we have the following proposition on the distribution of the equilibrium points of (14).Proposition 1.
(1)
Forc2c4<0, (14) has three equilibria points at E10(0,0), E11(ψ11,0), and E12(ψ12,0), where ψ11=--c2/(2c4), ψ12=-c2/(2c4).
(2)
Forc2c4≥0, (14) has a unique equilibrium at E10(0,0).Using the bifurcation theory of dynamical systems [7–9], the (c2,c4)-plane was divided into four subregions:
(16)D1:{c2>0,c4>0};D2:{c2<0,c4>0};D3:{c2<0,c4<0};D4:{c2>0,c4<0}.
The phase portraits of (14) are shown in Figure 1.The phase portraits of system (14).
(a)
(
c
2
,
c
4
)
∈
D
1
(b)
(
c
2
,
c
4
)
∈
D
2
(c)
(
c
2
,
c
4
)
∈
D
3
(d)
(
c
2
,
c
4
)
∈
D
4For the function defined by (15), we have
(17)h10=H1(ψ10,0)=0,h11=h12=H1(ψ11,0)=H1(ψ12,0)=c224c4.
Let M(we,ze) be the coefficient matrix of the system (14) at an equilibrium point (we,ze). Then, we have
(18)J(we,0)=det⁡(M(we,0))=-(c2+6c4we2).
By the theory of planar dynamical systems, we know that for an equilibrium point of a planar integrable system, if J<0, then the equilibrium point is a saddle point; if J>0 and Trace (M(we))=0, then it is a center point; if J>0 and (Trace(M(we,ze)))2-4J(we,ze)>0, then it is a node; if J=0 and the index of the equilibrium point is 0, then it is a cusp; otherwise, it is a high order equilibrium point.Below, we will give explicit and exact solutions of (7) (also (6)). We always let Δ=c22-4c0c4 and only pay attention to the bounded solutions of (6).(1) Suppose that(c2,c4)∈D4.(a) Forc0∈(c22/(4c4),0), (7) has periodic solutions:
(19)φ±1=±c2+Δ-2c4+Δc4sn2(c2+Δ2ξ,2Δc2+Δ),
where sn(x,k) and below cn(x,k) are Jacobian elliptic functions with modulus k [10]. The profiles of periodic solutions are shown in Figure 2.The wave profiles of solutionsφ±1 with the parameters values: γ1=γ2=β1=β2=β3=v=1, w0=0.38.
(a)
The 3D wave profiles of solutionsφ±1
(b)
The 2D wave profiles of solutionsφ±1 for t=0Thus, we obtain the following solutions of (1):
(20)E(z,t)=±ei(w0z+β3t/β4)×c2+Δ-2c4+Δc4sn2(c2+Δ2ξ,2Δc2+Δ).(b) Forc0=0, we have solitary wave solutions of (7):
(21)φ±2=±-c2c4sech(c2ξ).Thus, we obtain the following solutions of (1):
(22)E(z,t)=±ei(w0z+β3t/β4)-c2c4sech(c2ξ).
The profiles of solutions of (22) are shown in Figure 3.The wave profile of solutions (22), with the parameters values: γ1=γ2=β1=β2=β3=v=1, w0=0.06.
(a)
The 3D wave profile of real part
(b)
The 2D wave profile of real part
(c)
The 3D wave profile of imaginary part
(d)
The 2D wave profile of imaginary part
(e)
The 3D wave profile of|E|
(f)
The 2D wave profile of|E|Remark 2.
By the expression ofc0, there always exists a w0 such that c0=0 if (c2,c4)∈D4.Because of the limitation of length, we omit the expression ofE(z,t), beginning from here.(c) Forc0>0, (6) has periodic solution:
(23)φ3=c2+Δ-2c4cn(Δ4ξ,c2+Δ2Δ).(2) Suppose that(c2,c4)∈D2.(a)
Forc0∈(0,c22/(4c4)), system (14) has periodic solution:
(24)φ±4=±-c2-Δ-c2+Δsn(-c2+Δ2ξ,-c2-Δ-c2+Δ).(b)
Forc0=c22/(4c4), there exists smooth kink wave solution of (14) as follows:
(25)φ±5=±-c22c4tanh(-c22ξ).(3) If(c2,c4)∈D3, for c0>0, we have the following periodic solution of (14):
(26)φ±6=±-c2-Δ2c4cn(Δ4ξ,c2+Δ2Δ).(4) If(c2,c4)∈D1, it is observed from Figure 1(a) that there is no bounded solution of system (14).


### 3.2. The Case (III)

For this case, the Hamiltonian system (8) becomes
(27)dφdξ=y,dydξ=f(φ)=12(c1+2c2φ+3c3φ2)
and the Hamiltonian function (9) reduces to
(28)H1(φ,y)=y2-c1φ-c2φ2-c3φ3=c0.Similar to the previous discussion, we have the following proposition.Proposition 3.
(1)
Forc22-3c1c3>0, (27) has two equilibria at E21(ψ21,0) and E22(ψ22,0), where
(29)ψ21=-c2-c22-3c1c33c3,ψ22=-c2+c22-3c1c33c3.
(2)
Forc22-3c1c3=0, (27) has a unique equilibrium at E20(ψ20,0),where ψ20=-c2/(3c3).
(3)
Forc22-3c1c3<0, (27) has no equilibrium.Leth2i=H1(ψ2i,0), i=0,1,2, and notice that we need only to consider the case c3≥0 because of the invariance of (27) under the transformations ϕ→-ϕ, y→-y, and c3→-c3.(1)c22-3c1c3>0 and c0∈(h21,h22). In this case,
(30)c3φ3+c2φ2+c1φ+c0=0
has three mutually different real roots φm<φl<φM; thus,
(31)c3φ3+c2φ2+c1φ+c0=c3(φ-φm)(φ-φl)(φ-φM).
Equation (7) has periodic wave solutions as follows:
(32)φ8=φM-((φM-φl)(φM-φm))×(φM-φm-(φl-φm)sn2(c3(φM-φm)2ξ,φl-φmφM-φm)×(c3(φM-φm)2ξ,φl-φmφM-φm))-1,φ9=φm+(φl-φm)sn2(c3(φM-φm)2ξ,φl-φmφM-φm).(2)c22-3c1c3>0 and c0=h22. In this case, φ22 is double root of (30); suppose that φt is other root of the equation, obviously φt<φ22, and we have a solitary wave solution of peak type of (7) as follows:
(33)φ10=φt+(φ22-φt)tanh2⁡(12c3(φ22-φt)ξ).


## 3.1. Case (I) and Case (II)

In this case, the Hamiltonian system (8) becomes
(14)dφdξ=y,dydξ=c2φ+2c4φ3,
and the Hamiltonian function (9) reduces to
(15)H1(φ,y)=y2-c2φ2-c4φ4=c0.
Now we discuss the bifurcations of phase portraits of (14). Obviously all the equilibrium points of (14) lie in the φ-axis and their abscissas are the real zeros of f(φ)=c2φ+2c4ϕ3. Thus, we have the following proposition on the distribution of the equilibrium points of (14).Proposition 1.
(1)
Forc2c4<0, (14) has three equilibria points at E10(0,0), E11(ψ11,0), and E12(ψ12,0), where ψ11=--c2/(2c4), ψ12=-c2/(2c4).
(2)
Forc2c4≥0, (14) has a unique equilibrium at E10(0,0).Using the bifurcation theory of dynamical systems [7–9], the (c2,c4)-plane was divided into four subregions:
(16)D1:{c2>0,c4>0};D2:{c2<0,c4>0};D3:{c2<0,c4<0};D4:{c2>0,c4<0}.
The phase portraits of (14) are shown in Figure 1.The phase portraits of system (14).
(a)
(
c
2
,
c
4
)
∈
D
1
(b)
(
c
2
,
c
4
)
∈
D
2
(c)
(
c
2
,
c
4
)
∈
D
3
(d)
(
c
2
,
c
4
)
∈
D
4For the function defined by (15), we have
(17)h10=H1(ψ10,0)=0,h11=h12=H1(ψ11,0)=H1(ψ12,0)=c224c4.
Let M(we,ze) be the coefficient matrix of the system (14) at an equilibrium point (we,ze). Then, we have
(18)J(we,0)=det⁡(M(we,0))=-(c2+6c4we2).
By the theory of planar dynamical systems, we know that for an equilibrium point of a planar integrable system, if J<0, then the equilibrium point is a saddle point; if J>0 and Trace (M(we))=0, then it is a center point; if J>0 and (Trace(M(we,ze)))2-4J(we,ze)>0, then it is a node; if J=0 and the index of the equilibrium point is 0, then it is a cusp; otherwise, it is a high order equilibrium point.Below, we will give explicit and exact solutions of (7) (also (6)). We always let Δ=c22-4c0c4 and only pay attention to the bounded solutions of (6).(1) Suppose that(c2,c4)∈D4.(a) Forc0∈(c22/(4c4),0), (7) has periodic solutions:
(19)φ±1=±c2+Δ-2c4+Δc4sn2(c2+Δ2ξ,2Δc2+Δ),
where sn(x,k) and below cn(x,k) are Jacobian elliptic functions with modulus k [10]. The profiles of periodic solutions are shown in Figure 2.The wave profiles of solutionsφ±1 with the parameters values: γ1=γ2=β1=β2=β3=v=1, w0=0.38.
(a)
The 3D wave profiles of solutionsφ±1
(b)
The 2D wave profiles of solutionsφ±1 for t=0Thus, we obtain the following solutions of (1):
(20)E(z,t)=±ei(w0z+β3t/β4)×c2+Δ-2c4+Δc4sn2(c2+Δ2ξ,2Δc2+Δ).(b) Forc0=0, we have solitary wave solutions of (7):
(21)φ±2=±-c2c4sech(c2ξ).Thus, we obtain the following solutions of (1):
(22)E(z,t)=±ei(w0z+β3t/β4)-c2c4sech(c2ξ).
The profiles of solutions of (22) are shown in Figure 3.The wave profile of solutions (22), with the parameters values: γ1=γ2=β1=β2=β3=v=1, w0=0.06.
(a)
The 3D wave profile of real part
(b)
The 2D wave profile of real part
(c)
The 3D wave profile of imaginary part
(d)
The 2D wave profile of imaginary part
(e)
The 3D wave profile of|E|
(f)
The 2D wave profile of|E|Remark 2.
By the expression ofc0, there always exists a w0 such that c0=0 if (c2,c4)∈D4.Because of the limitation of length, we omit the expression ofE(z,t), beginning from here.(c) Forc0>0, (6) has periodic solution:
(23)φ3=c2+Δ-2c4cn(Δ4ξ,c2+Δ2Δ).(2) Suppose that(c2,c4)∈D2.(a)
Forc0∈(0,c22/(4c4)), system (14) has periodic solution:
(24)φ±4=±-c2-Δ-c2+Δsn(-c2+Δ2ξ,-c2-Δ-c2+Δ).(b)
Forc0=c22/(4c4), there exists smooth kink wave solution of (14) as follows:
(25)φ±5=±-c22c4tanh(-c22ξ).(3) If(c2,c4)∈D3, for c0>0, we have the following periodic solution of (14):
(26)φ±6=±-c2-Δ2c4cn(Δ4ξ,c2+Δ2Δ).(4) If(c2,c4)∈D1, it is observed from Figure 1(a) that there is no bounded solution of system (14).


## 3.2. The Case (III)

For this case, the Hamiltonian system (8) becomes
(27)dφdξ=y,dydξ=f(φ)=12(c1+2c2φ+3c3φ2)
and the Hamiltonian function (9) reduces to
(28)H1(φ,y)=y2-c1φ-c2φ2-c3φ3=c0.Similar to the previous discussion, we have the following proposition.Proposition 3.
(1)
Forc22-3c1c3>0, (27) has two equilibria at E21(ψ21,0) and E22(ψ22,0), where
(29)ψ21=-c2-c22-3c1c33c3,ψ22=-c2+c22-3c1c33c3.
(2)
Forc22-3c1c3=0, (27) has a unique equilibrium at E20(ψ20,0),where ψ20=-c2/(3c3).
(3)
Forc22-3c1c3<0, (27) has no equilibrium.Leth2i=H1(ψ2i,0), i=0,1,2, and notice that we need only to consider the case c3≥0 because of the invariance of (27) under the transformations ϕ→-ϕ, y→-y, and c3→-c3.(1)c22-3c1c3>0 and c0∈(h21,h22). In this case,
(30)c3φ3+c2φ2+c1φ+c0=0
has three mutually different real roots φm<φl<φM; thus,
(31)c3φ3+c2φ2+c1φ+c0=c3(φ-φm)(φ-φl)(φ-φM).
Equation (7) has periodic wave solutions as follows:
(32)φ8=φM-((φM-φl)(φM-φm))×(φM-φm-(φl-φm)sn2(c3(φM-φm)2ξ,φl-φmφM-φm)×(c3(φM-φm)2ξ,φl-φmφM-φm))-1,φ9=φm+(φl-φm)sn2(c3(φM-φm)2ξ,φl-φmφM-φm).(2)c22-3c1c3>0 and c0=h22. In this case, φ22 is double root of (30); suppose that φt is other root of the equation, obviously φt<φ22, and we have a solitary wave solution of peak type of (7) as follows:
(33)φ10=φt+(φ22-φt)tanh2⁡(12c3(φ22-φt)ξ).


## 4. Conclusions

In this study, we apply bifurcation theory of dynamical systems and the Fan subequation method to investigate (1), and many new exact solutions have been obtained; most importantly, under more general conditions than [3], to the best of our knowledge, these solutions have not been reported in the literature. This method can help us find exact solutions of other types of nonlinear dispersion partial differential equations.



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