# Application of theG'/G Expansion Method in Ultrashort Pulses in Nonlinear Optical Fibers

Authors: Jiang Xing-Fang; Wang Jun; Wei Jian-Ping; Hua Ping
Journal: Advances in Optical Technologies
 (2013)

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## Abstract

With the increasing input power in optical fibers, the dispersion problem is becoming a severe restriction on wavelength division multiplexing (WDM). With the aid of solitons, in which the shape and speed can remain constant during propagation, it is expected that the transmission of nonlinear ultrashort pulses in optical fibers can effectively control the dispersion. The propagation of a nonlinear ultrashort laser pulse in an optical fiber, which fits the high-order nonlinear Schrödinger equation (NLSE), has been solved using theG'/G expansion method.  Group velocity dispersion, self-phase modulation, the fourth-order dispersion, and the fifth-order nonlinearity of the high-order NLSE were taken into consideration. A series of solutions has been obtained such as the solitary wave solutions of kink, inverse kink, the tangent trigonometric function, and the cotangent trigonometric function. The results have shown that the G'/G expansion method is an effective way to obtain the exact solutions for the high-order NLSE, and it provides a theoretical basis for the transmission of ultrashort pulses in nonlinear optical fibers.


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## Body

## 1. Introduction

It is understood that a soliton is excited by a nonlinear field, and its energy is relatively concentrated in a small area. The elastic scattering phenomenon occurs during the interaction between two solitons. The energy is nondispersed, so that the shape and speed can remain unchanged during its propagation. The history of the studies of solitons can be traced back to 1834, when James Scott Russell, a British scientist, accidentally observed that a bulge of water in Edinburgh-Glasgow Canal was propagating undistorted over several kilometers [1]. In 1973, Hasegawa and Tappert [2] first proposed the idea of applying “optical solitons” to photonic communication. After rigorous theoretical and mathematical deduction, he predicted that both bright and dark soliton pulses were present in an optical fiber. He also proved that any nondestructive optical pulse could travel as stable as a soliton during its transmission in an optical fiber. As an intrinsically nonlinear phenomenon, a soliton is the product of the optical fiber dispersion and the nonlinear interaction. It obeys the NLSE and is controlled by the optical fiber dispersion and the nonlinear effects of self-phase modulation. In 1980, bright solitons were observed for the first time in an optical fiber by Mollenaure [3] and his colleagues during their experiments at the Bell Laboratory of the United States. In 1987, Emplit observed dark solitons in a fiber by using an amplitude technique and a phase filtering technique. In mathematics, the progress of the soliton theory has been achieved mainly by finding a large number of soliton solutions for nonlinear partial differential equations, so as to gradually establish a systematic theory of solitons. The special properties of a soliton, the related mathematical description of its rich structure, and the deep physical origins have profoundly expanded the applications of soliton theory and methods. The study of the exact solutions for the NLSEs has become an important research and development direction. The G′/G expansion method [4], one of the methods for obtaining exact solutions for NLSE, has attracted much attention recently [4–6]. In particular, it plays an important role in deriving the propagation wave solution of NLSE. Therefore, a good understanding of the G′/G expansion method is of great significance to the transmission application of ultrashort pulses in nonlinear optical fibers. This paper presents the method in detail with examples for obtaining multiple soliton solutions.


## 2.G′/G Expansion Method

The nonlinear equation with two variablesx and t could be written as
(1)P(u(x,t),∂u∂x,∂u∂t,∂2u∂x2,∂2u∂x∂t,∂2u∂t2,…)=0.The first step of theG′/G expansion method is to assume a variable ξ=x-vt; thus, u=u(x,t)=u(ξ); it can then be expressed as a G′/G power polynomial:
(2)u(ξ)=αm(G′G)m+αm-1(G′G)m-1+⋯+α0,
in which G=G(ξ) satisfies the second-order linear ordinary differential equation:
(3)G′′+λG′+μG=0,
where αm,αm-1,…,α0,λ,μ in (2) and (3) are undetermined parameters, in which, αm≠0, and the general solution of (3) is expressed as
(4)G(ξ)=Ae(-λ+λ2-4μ)ξ/2+Be(-λ-λ2-4μ)ξ/2.The derivative ofG(ξ) can be derived from (4):
(5)G′=(-λ2)e(-λ/2)ξ×[(A+B)cosh(λ2-4μ2ξ)+(A-B)sinh(λ2-4μ2ξ)]+e(-λ/2)ξ[(A+B)sinh(λ2-4μ2ξ)+(A-B)cosh(λ2-4μ2ξ)]×λ2-4μ2.The ratio ofG(ξ) and G′(ξ) is expressed as [6]
(6)G′G=-λ2+λ2-4μ2×(A+B)sinh((λ2-4μ/2)ξ)+(A-B)cosh((λ2-4μ/2)ξ)(A+B)cosh((λ2-4μ/2)ξ)+(A-B)sinh((λ2-4μ/2)ξ).


## 3. Applying theG′/GExpansion Method to Solve NLSE for Ultrashort Pulses in Optical Fibers

Many problems in the study of natural science can be related to nonlinear evolution equations, and the optical soliton is one of them. It is always problematic to find soliton solutions for nonlinear evolution equations by physicists and mathematicians. With the development of modern computers, many methods have been developed, such as the homogeneous balance method and the hyperbolic function method [7–9]. This research uses the G′/G expansion method to solve the high-order NLSE.The generalized NLSE is given by(7)∑l=0∞ilαl∂l∂xlψ+∑m=0∞imβm∂mm!∂tmψ+∑n=0∞γ2n|ψ|2nψ=0.
Considering the case when l=1, m=2,3,4, and n=1,2, the corresponding coefficients are nonzero and the rest of the coefficients are zero. Assuming the coefficient α1=1 and allowing the variable x to be replaced by z and the function ψ to be replaced by q, the high-order NLSE (with items included, namely, the group velocity dispersion, the self-phase modulation effect, the fourth-order dispersion, and the fifth-order nonlinear term) can then be expressed as
(8)i∂q∂z-β22∂2q∂t2+γ2|q|2q=iβ36∂3q∂t3+β424∂4q∂t4-γ4|q|4q,
in which β2, β3, and β4 are the second-, third-, and fourth-order dispersion coefficients and γ2 and γ4 are the third- and fifth-order nonlinear coefficients. The second-order dispersion coefficient β2 is also known as the group velocity dispersion coefficient. With increasing optical power in a fiber, it is necessary to take the third-order nonlinear term into account, which represents the phase equilibrium between the group velocity dispersion and the nonlinear effect. This will allow the maintaining of unchanged properties for soliton pulses in an optical fiber, which is necessary for long distance transmission, and is also beneficial to WDM.Other methods, such as the extended tanh-function method, the subsidiary ordinary differential method, and the extended hyperbolic auxiliary equation method, have been used in order to find the exact solutions for (8) [10–12]. It has been found that there is no solution for (8) when β3-β4ν≠0. In order to obtain the exact solution, ν=β3/β4 is assumed and (8) is then converted to a real equation after its real part and imaginary part are separated. The G′/G expansion method is then used for solving the equation.


### 3.1. Reduction Travelling Wave

Assume the form of the exact solution is(9)q(z,t)=u(ξ)ei(Lz-vt),ξ=kz-ωt.The parameters ofk,L,ω,v are undetermined parameters. A series of expressions can be obtained based on (9). The key equation (10) can be obtained after the real part and the imaginary part of (8) are separated:
(10)β4ω4u′′′′+(6β3ω2β3β4+12β2ω2)u′′+(-12β2β32β42+24L-3β34β43)u-24γ2u3-24γ4u5=0.


### 3.2. The Exact Solution Obtained Using theG′/G Expansion Method

TheG′/G expansion method was proposed for the exact solutions by Wang et al. [4]. The method is effective to solve the exact solutions for the nonlinear partial differential equation.The form of the exact solution for (10) is
(11)u(ξ)=∑i=0nαi(G′G)i,
where G=G(ξ) and it satisfies (3).The result ofn=1 can be obtained by balancing the highest derivative u′′′′ and the highest derivative with high nonlinearity u5 in (10). The general expression for (10) can be expressed as
(12)u(ξ)=α1(G′G)+α0.Based on (12) and (10), and assuming the coefficients of (G′/G)i, i=0,1,2,3,4,5 are zero, a set of six algebraic equations can be obtained:
(13)α1(β4ω4-γ4α14)=0,(14)α1(β4ω4λ-2γ4α13α0)=0,(15)α1[β4ω4(25λ2+20μ)+(6β3ω2β3β4+12β2ω2)-12γ2α12-120γ4α12α02(6β3ω2β3β4+12β2ω2)]=0,(16)α1[β4ω4(5λ3+20λμ)+(6β3ω2β3β4+12β2ω2)λ-24γ2α1α0-80γ4α1α03(6β3ω2β3β4+12β2ω2)]=0,(17)α1[β3β4β4ω4(λ4+22λ2μ+16μ2)+(6β3ω2β3β4+12β2ω2)(λ2+2μ)-(12β2β32β42-24L+3β34β43)-72γ2α02-120γ4α04β3β4]=0,(18)β4ω4(μλ3+8λμ2)α1+(6β3ω2β3β4+12β2ω2)λμα1+(-12β2β32β42+24L-3β34β43)α0-24γ2α03-24γ4α05=0.The parameterα1 can be obtained from (13):
(19)α1=β4γ44ω.It was mistakenly written asα1=(β4/γ4)5ω in [6].


#### 3.2.1. The First Type Exact Solution

From (14) when λ=0, the result is α0=0:
(20)μ=310β4ω3[2γ2β4γ4-(β32β4+2β2)ω],L=[2γ2β4γ4-(β32β4+2β2)ω]×[-3γ225β4ω2β4γ4+350β4ωβ32β4+3β225β4ω-320ω(β3β4)2-3β210β4ωβ4γ4]+12β2β32β42+18β34β43.The exact solution of (21) based on (6) is
(21)q1=β4γ44ω-μ·(A+B)sinh((-4μ/2)ξ)+(A-B)cosh((-4μ/2)ξ)(A+B)cosh((-4μ/2)ξ)+(A-B)sinh((-4μ/2)ξ)×ei(Lz-vt).Consider the following.(a)
Whenμ<0 and A=B=1, the kink wave solution is
(22)q11=β4γ44ω-μtanh(-4μ2ξ)ei(Lz-vt).(b)
Whenμ<0 and A=-B=1, the inverse kink wave solution is
(23)q12=β4γ44ω-μcoth(-4μ2ξ)ei(Lz-vt).(c)
Whenμ≥0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(24)q13=-ωμβ4γ44tan(μξ)ei(Lz-vt).(d)
Whenμ≥0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(25)q14=μβ4γ44ωcot(μξ)ei(Lz-vt).


#### 3.2.2. The Second Type Exact Solution

From (14) when λ≠0, the result is α1=(β4/γ4)4ω, α0=(β4ω4λ/2γ4α13)=(λ/2)(β4/γ4)4ω=(λ/2)α1,
(26)μ=-310(β3β4ω)2-35β2(β4ω)2+3γ25ω21β4γ4+λ24;(27)L=β2β322β42+β348β43+γ23λ24ω2β4γ4+5λ416β4-β4ω4λ424-14β3ω2β3β4λ2-12β2ω2λ2-{1112β4ω4λ2+12β3ω2β3β4+β2ω2}μ-23β4ω4μ2.The exact solution of (26) based on (6) is
(28)q2=[β4γ44ω×(-λ2+λ2-4μ2·((A+B)sinh(λ2-4μ2ξ)+(A-B)cosh(λ2-4μ2ξ))×((A+B)cosh(λ2-4μ2ξ)+(A-B)sinh(λ2-4μ2ξ))-1λ2-4μ2)+λ2β4γ44ω]ei(Lz-vt).The item of-(λ/2) in [6] was lost and the item of α0 was not compensated.Consider the following.(a)
Whenλ2-4μ≥0 and A=B=1, the kink wave solution is
(29)q21=β4γ44ωλ2-4μ2tanh(λ2-4μ2ξ)ei(Lz-vt).(b)
Whenλ2-4μ≥0 and A=-B=1, the inverse kink wave solution is
(30)q22=β4γ44ωλ2-4μ2coth(λ2-4μ2ξ)ei(Lz-vt).(c)
Whenλ2-4μ<0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(31)q23=-ωβ4γ444μ-λ22tan(4μ-λ22ξ)ei(Lz-vt).(d)
Whenλ2-4μ<0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(32)q24=β4γ44ω4μ-λ22cot(4μ-λ22ξ)ei(Lz-vt).The application of the feature equations can be found in [13–17].


## 3.1. Reduction Travelling Wave

Assume the form of the exact solution is(9)q(z,t)=u(ξ)ei(Lz-vt),ξ=kz-ωt.The parameters ofk,L,ω,v are undetermined parameters. A series of expressions can be obtained based on (9). The key equation (10) can be obtained after the real part and the imaginary part of (8) are separated:
(10)β4ω4u′′′′+(6β3ω2β3β4+12β2ω2)u′′+(-12β2β32β42+24L-3β34β43)u-24γ2u3-24γ4u5=0.


## 3.2. The Exact Solution Obtained Using theG′/G Expansion Method

TheG′/G expansion method was proposed for the exact solutions by Wang et al. [4]. The method is effective to solve the exact solutions for the nonlinear partial differential equation.The form of the exact solution for (10) is
(11)u(ξ)=∑i=0nαi(G′G)i,
where G=G(ξ) and it satisfies (3).The result ofn=1 can be obtained by balancing the highest derivative u′′′′ and the highest derivative with high nonlinearity u5 in (10). The general expression for (10) can be expressed as
(12)u(ξ)=α1(G′G)+α0.Based on (12) and (10), and assuming the coefficients of (G′/G)i, i=0,1,2,3,4,5 are zero, a set of six algebraic equations can be obtained:
(13)α1(β4ω4-γ4α14)=0,(14)α1(β4ω4λ-2γ4α13α0)=0,(15)α1[β4ω4(25λ2+20μ)+(6β3ω2β3β4+12β2ω2)-12γ2α12-120γ4α12α02(6β3ω2β3β4+12β2ω2)]=0,(16)α1[β4ω4(5λ3+20λμ)+(6β3ω2β3β4+12β2ω2)λ-24γ2α1α0-80γ4α1α03(6β3ω2β3β4+12β2ω2)]=0,(17)α1[β3β4β4ω4(λ4+22λ2μ+16μ2)+(6β3ω2β3β4+12β2ω2)(λ2+2μ)-(12β2β32β42-24L+3β34β43)-72γ2α02-120γ4α04β3β4]=0,(18)β4ω4(μλ3+8λμ2)α1+(6β3ω2β3β4+12β2ω2)λμα1+(-12β2β32β42+24L-3β34β43)α0-24γ2α03-24γ4α05=0.The parameterα1 can be obtained from (13):
(19)α1=β4γ44ω.It was mistakenly written asα1=(β4/γ4)5ω in [6].


### 3.2.1. The First Type Exact Solution

From (14) when λ=0, the result is α0=0:
(20)μ=310β4ω3[2γ2β4γ4-(β32β4+2β2)ω],L=[2γ2β4γ4-(β32β4+2β2)ω]×[-3γ225β4ω2β4γ4+350β4ωβ32β4+3β225β4ω-320ω(β3β4)2-3β210β4ωβ4γ4]+12β2β32β42+18β34β43.The exact solution of (21) based on (6) is
(21)q1=β4γ44ω-μ·(A+B)sinh((-4μ/2)ξ)+(A-B)cosh((-4μ/2)ξ)(A+B)cosh((-4μ/2)ξ)+(A-B)sinh((-4μ/2)ξ)×ei(Lz-vt).Consider the following.(a)
Whenμ<0 and A=B=1, the kink wave solution is
(22)q11=β4γ44ω-μtanh(-4μ2ξ)ei(Lz-vt).(b)
Whenμ<0 and A=-B=1, the inverse kink wave solution is
(23)q12=β4γ44ω-μcoth(-4μ2ξ)ei(Lz-vt).(c)
Whenμ≥0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(24)q13=-ωμβ4γ44tan(μξ)ei(Lz-vt).(d)
Whenμ≥0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(25)q14=μβ4γ44ωcot(μξ)ei(Lz-vt).


### 3.2.2. The Second Type Exact Solution

From (14) when λ≠0, the result is α1=(β4/γ4)4ω, α0=(β4ω4λ/2γ4α13)=(λ/2)(β4/γ4)4ω=(λ/2)α1,
(26)μ=-310(β3β4ω)2-35β2(β4ω)2+3γ25ω21β4γ4+λ24;(27)L=β2β322β42+β348β43+γ23λ24ω2β4γ4+5λ416β4-β4ω4λ424-14β3ω2β3β4λ2-12β2ω2λ2-{1112β4ω4λ2+12β3ω2β3β4+β2ω2}μ-23β4ω4μ2.The exact solution of (26) based on (6) is
(28)q2=[β4γ44ω×(-λ2+λ2-4μ2·((A+B)sinh(λ2-4μ2ξ)+(A-B)cosh(λ2-4μ2ξ))×((A+B)cosh(λ2-4μ2ξ)+(A-B)sinh(λ2-4μ2ξ))-1λ2-4μ2)+λ2β4γ44ω]ei(Lz-vt).The item of-(λ/2) in [6] was lost and the item of α0 was not compensated.Consider the following.(a)
Whenλ2-4μ≥0 and A=B=1, the kink wave solution is
(29)q21=β4γ44ωλ2-4μ2tanh(λ2-4μ2ξ)ei(Lz-vt).(b)
Whenλ2-4μ≥0 and A=-B=1, the inverse kink wave solution is
(30)q22=β4γ44ωλ2-4μ2coth(λ2-4μ2ξ)ei(Lz-vt).(c)
Whenλ2-4μ<0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(31)q23=-ωβ4γ444μ-λ22tan(4μ-λ22ξ)ei(Lz-vt).(d)
Whenλ2-4μ<0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(32)q24=β4γ44ω4μ-λ22cot(4μ-λ22ξ)ei(Lz-vt).The application of the feature equations can be found in [13–17].


## 3.2.1. The First Type Exact Solution

From (14) when λ=0, the result is α0=0:
(20)μ=310β4ω3[2γ2β4γ4-(β32β4+2β2)ω],L=[2γ2β4γ4-(β32β4+2β2)ω]×[-3γ225β4ω2β4γ4+350β4ωβ32β4+3β225β4ω-320ω(β3β4)2-3β210β4ωβ4γ4]+12β2β32β42+18β34β43.The exact solution of (21) based on (6) is
(21)q1=β4γ44ω-μ·(A+B)sinh((-4μ/2)ξ)+(A-B)cosh((-4μ/2)ξ)(A+B)cosh((-4μ/2)ξ)+(A-B)sinh((-4μ/2)ξ)×ei(Lz-vt).Consider the following.(a)
Whenμ<0 and A=B=1, the kink wave solution is
(22)q11=β4γ44ω-μtanh(-4μ2ξ)ei(Lz-vt).(b)
Whenμ<0 and A=-B=1, the inverse kink wave solution is
(23)q12=β4γ44ω-μcoth(-4μ2ξ)ei(Lz-vt).(c)
Whenμ≥0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(24)q13=-ωμβ4γ44tan(μξ)ei(Lz-vt).(d)
Whenμ≥0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(25)q14=μβ4γ44ωcot(μξ)ei(Lz-vt).


## 3.2.2. The Second Type Exact Solution

From (14) when λ≠0, the result is α1=(β4/γ4)4ω, α0=(β4ω4λ/2γ4α13)=(λ/2)(β4/γ4)4ω=(λ/2)α1,
(26)μ=-310(β3β4ω)2-35β2(β4ω)2+3γ25ω21β4γ4+λ24;(27)L=β2β322β42+β348β43+γ23λ24ω2β4γ4+5λ416β4-β4ω4λ424-14β3ω2β3β4λ2-12β2ω2λ2-{1112β4ω4λ2+12β3ω2β3β4+β2ω2}μ-23β4ω4μ2.The exact solution of (26) based on (6) is
(28)q2=[β4γ44ω×(-λ2+λ2-4μ2·((A+B)sinh(λ2-4μ2ξ)+(A-B)cosh(λ2-4μ2ξ))×((A+B)cosh(λ2-4μ2ξ)+(A-B)sinh(λ2-4μ2ξ))-1λ2-4μ2)+λ2β4γ44ω]ei(Lz-vt).The item of-(λ/2) in [6] was lost and the item of α0 was not compensated.Consider the following.(a)
Whenλ2-4μ≥0 and A=B=1, the kink wave solution is
(29)q21=β4γ44ωλ2-4μ2tanh(λ2-4μ2ξ)ei(Lz-vt).(b)
Whenλ2-4μ≥0 and A=-B=1, the inverse kink wave solution is
(30)q22=β4γ44ωλ2-4μ2coth(λ2-4μ2ξ)ei(Lz-vt).(c)
Whenλ2-4μ<0 and A=B=1, the solitary wave solution of the tangent trigonometric function is
(31)q23=-ωβ4γ444μ-λ22tan(4μ-λ22ξ)ei(Lz-vt).(d)
Whenλ2-4μ<0 and A=-B=1, the solitary wave solution of the cotangent trigonometric function is
(32)q24=β4γ44ω4μ-λ22cot(4μ-λ22ξ)ei(Lz-vt).The application of the feature equations can be found in [13–17].


## 4. Conclusions

AG′/G expansion method was applied to obtain the exact solutions for NLSE which is the governing equation for ultrashort pulse transmission in a nonlinear fiber. The items of the group velocity dispersion, self-phase modulation effect, the fourth-order dispersion, and the fifth-order nonlinearity were considered. Two kinds of the kink wave solution, inverse kink wave solutions, the solitary wave solution of the tangent trigonometric function, and the solitary wave solution of the cotangent trigonometric function were found by using the G′/G expansion method.These results have important significance in optical fiber communication and its applications.



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