# Breathers and Soliton Solutions for a Generalization of the Nonlinear Schrödinger Equation

Authors: Hai-Feng Zhang; Hui-Qin Hao; Jian-Wen Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A generalized nonlinear Schrödinger equation, which describes the propagation of the femtosecond pulse in single mode optical silica fiber, is analytically investigated. By virtue of the Darboux transformation method, some new soliton
solutions are generated: the bright one-soliton solution on the zero background, the dark one-soliton solution on the continuous wave background, the Akhmediev breather which delineates the modulation instability process, and the breather evolving periodically along the straight line with a certain angle ofx-axis and t-axis. Those results might be useful in the study of the femtosecond pulse in single mode optical silica fiber.


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## Body

## 1. Introduction

Investigations on the dynamic features of solitons have attracted certain interest in nonlinear optics [1–4]. Optical solitons have been regarded as a candidate for the optical communication networks [5–8]. On the basis of the balance between the group velocity dispersion and self-phase modulation [9, 10], the propagation of optical soliton is usually governed by the nonlinear Schrödinger (NLS) equation [11–14]:
(1)iEζ+12Eττ+|E|2E=0.
However, when optical pulses are shorter, the NLS equation becomes inadequate, and it is necessary to include additional terms [6, 7]. For example, in single mode optical silica fiber, in order to describe the propagation of femtosecond pulse, the higher order asymptotic terms should be retained [15]; to understand such phenomena, we consider the following generalization of the NLS equation [16]:
(2)iut-uxt+uxx+|u|2u+i|u|2ux=0.
Analogous to the circumstance that the Camassa-Holm equation provides a better approximation of the KdV equation [15], (2) is related to the NLS equation, provided that one retains terms of the next asymptotic order. Under the transformation q=eixu, (2) can be converted into the following equation [16, 17]:
(3)iqxt-iqxx+2qx-|q|2qx+iq=0,
where q denotes the complex field envelope and the subscripts x and t are the longitudinal distance and retarded time, respectively. In recent years, some results have been obtained for (1): (1) Reference [15] has analyzed the dynamic features of the rogue wave solutions; (2) Reference [16] has analyzed the conservation laws, bi-Hamiltonian structure, Lax pair, and initial-value problem; (3) Reference [17] has derived some soliton solutions by using the bilinear method. The aim of this paper is mainly to derive some new soliton solutions for (3) using the Darboux transformation (DT) method and analyze the dynamic features of soliton solutions.This paper will be organized as follows. In Section2, we will give the Lax pair and construct the DT for (3). In Section 3, we will obtain bright one-soliton, dark one-soliton, and breather solutions and analyze the dynamic features of soliton solutions by using some figures. Finally, our conclusions will be addressed in Section 4.


## 2.  Lax Pair and Darboux Transformation

Employing the Ablowitz-Kaup-Newell-Segur formalism [18], [15, 16] has given the Lax pair associated with (3) as (4a)Ψx=UΨ,(4b)Ψt=VΨ,where Ψ=(ψ1,ψ2)T (T denotes the transpose of a matrix), and the matrices U and V have the form (5a)U=λ2U2+λU1,(5b)V=λ2V2+λV1+V0+1λV-1+1λ2V-2,where λ is a spectral parameter and
(6)U2=(-i00i),U1=(0qxqx*0),V2=U2,V1=U1,V-2=14U2,V0=(i-i2|q|200-i+i2|q|2),V-1=(0i2q-i2q*0).Through direct computations, it can be verified that the zero curvature equationUt-Vx+UV-VU=0 exactly gives rise to (3).Next, based on Lax pair (4a) and (4b), we will give the DT [19–22] formalism for (3). Define
(7)Ψ′=(λD1+D0+1λI)Ψ,
where I denotes the identity matrix and
(8)D1=(s100s2),D0=(0s3s40),
with
(9)s1=-(1/λ)|ϕ1|2+(1/λ*)|ϕ2|2λ|ϕ1|2-λ*|ϕ2|2,s2=-(1/λ)|ϕ2|2+(1/λ*)|ϕ1|2λ|ϕ2|2-λ*|ϕ1|2,s3=(λ*/λ-λ/λ*)ϕ1ϕ2*λ|ϕ1|2-λ*|ϕ2|2,s4=(λ*/λ-λ/λ*)ϕ2ϕ1*λ|ϕ2|2-λ*|ϕ1|2.
One can verify that if (ϕ1,ϕ2)T is the solution of Lax pair (4a) and (4b) with λ, then (ϕ2*,ϕ1*)T is also the solution of Lax pair (4a) and (4b) corresponding to λ*.Through direct computation, we can obtain(10)q′=q+(λ*2-λ2)ϕ1ϕ2*λλ*(λ|ϕ1|2-λ*|ϕ2|2).So, ifq is a seed solution of (3), q′ is also a solution of (3).


## 3. One and Breather Solutions (3)

In this section, we will apply the DT constructed to obtain one and breather solutions for (3). Now we take the nonzero continuous wave (cw) solution q=aexpi(At+Bx) as the initial seed for (3), where a, A, and B are all real parameters. Equation (3) requires that the frequency A satisfies the nonlinear dispersion relation:
(11)A=2-a2+B+1B.Solving (4a) and (4b) and setting ϕ1=f1expi(At+Bx),  ϕ2=f2, one can obtain
(12)f1x+iBf1=-iλ2f1+iaBλf2,f2x=-iaBλf1+iλ2f2,f1t+iAf1=αf1+βf2,f2t=-βf1-αf2,
where
(13)α=-iλ2+i-i2a2-i4λ2,β=iaBλ+i2λa.Through tedious computations, one can arrive at(14)ϕ1=(c1expθ1+c2expθ2)expi(At+Bx),ϕ2=c3expθ1+c4expθ2,
where
(15)θ1=x2[-iB+ζ1+iη1]+t2[-iA+ζ2+iη2],θ2=x2[-iB-ζ1-iη1]+t2[-iA-ζ2-iη2],
and c1,  c2,  c3, and c4 are complex constants satisfying:
(16)c2=(L+iM)c4,c3=(L+iM)c1,
with
(17)λ=12(λ1+iλ2),ζ1+iη1=-B2-4(λ2B+λ4-a2B2λ2),ζ2+iη2=-A2-4(β2-α2+iAα),
with λ1,  λ2,  ζ1,  η1,  ζ2, and η2 as real numbers, and we can derive
(18)L=-2ζ1λ2+λ1(λ12+λ22+2B+2η1)2aB(λ12+λ22),M=2ζ1λ1+λ2(λ12+λ22-2B-2η1)2aB(λ12+λ22).Now, substituting (14) into (10), we can obtain the solution for (1) as
(19)q=[a-4iλ1λ2G(λ12+λ22)F]expi(At+Bx),
where
(20)G=2LcoshΘ1-2iMsinhΘ1+(L2+M2+1)cosΘ2+i(1-L2-M2)sinΘ2,F=(1-L2-M2)λ1sinhΘ1+i(L2+M2+1)λ2coshΘ1+2λ1MsinΘ2+2iλ2LcosΘ2,
with
(21)Θ1=ζ1x+ζ2t,Θ2=η1x+η2t.Next according to different values of those parameters in solution (19), we will analyse the novel properties of solitons.


### 3.1. One-Soliton Solutions for (3)

Whena=0, that is to say, the initial seed for (3) is zero, solution (19) reduces to one-soliton solution as
(22)q=λ2-λ*2λλ*2sech(χ1+χ1*)exp(χ1-χ1*),
with
(23)χ1=-iλ2x+(i-i4λ2-iλ2)t,χ1*=iλ*2x+(-i+i4λ*2-iλ*2)t.
Solution (22) represents a bright soliton whose dynamic features are delineated in Figures 1. Through symbolic computation, we can conclude the following physical quantities for solution (22): the maximum amplitude |(λ2-λ*2)/λλ*2|, the width 1/|λ2-λ*2|, the envelope velocity (1/4λ*2-1/4λ2-λ2-λ*2)/(λ2-λ*2), and the energy of the one-soliton solution
(24)E1=∫-∞+∞‍[|q(x,t)|2-|q(±∞,t)|2]dx=128λ1λ2(λ12+λ22)3.
So, the amplitude and envelope velocity will increase when the value of λ1 becomes bigger. As shown in Figures 1, the amplitude is higher and the envelope velocity is bigger in Figure 1(a) than in Figure 1(b).Evolution of the one-soliton solution of (22). The parameters are (a) λ1=2,  λ2=4; (b) λ1=1,  λ2=4.
(a)
(b)


### 3.2. Breather and Dark One-Soliton Solutions for (1)

Whena≠0, the nonzero initial seed q=aexpi(At+Bx) describes the nonvanishing boundary conditions. For simplicity, taking λ1=λ2,  B=2/a2, we have the following relations:(25a)A=2+2a2-a22,ζ1=λ22,η1=2a2,ζ2=-1λ22-λ22,(25b)η2=a22-a22,L=1λ2a,M=-1λ2a+λ2a2.With the previous conclusions, solution (19) can be converted into:
(26)q=[a-4iλ1λ2G′(λ12+λ22)F′]expi[(2+2a2-a22)t+2a2x],
where
(27)G′=D1coshΘ1′+D2sinhΘ1′+D3cosΘ2′+D4sinΘ2′,F=D5coshΘ1′+D6sinhΘ1′+D7cosΘ2′+D8sinΘ2′,
with
(28)Θ1′=λ22x-(λ22+1λ22)t,Θ2′=(a22-2a2)t+2a2x,D1=2aλ2,D2=i(aλ2-D1),D3=2a2λ22+a2λ224,D4=i(2-D3),D5=i(2a2λ2+a2λ234),D6=2λ2+iD5,D7=2ia,D8=aλ22+iD7.
Figure 2 display the propagation characteristics of solitons via solutions (26). Figures 2(a) and 2(b) depict the dynamic features of breathers; as shown in Figure 2(a), the main feature is the propagation of the breather that is periodic in the space coordinate and aperiodic in the time coordinate; that is to say, we can obtain the Akhmediev breather [23] via solutions (26) under suitable parameters chosen. In addition, the Akhmediev breather can be regarded as a modulation instability process. Figure 2(b) portrays the propagation of the breather evolving periodically along the straight line with a certain angle of x-axis and t-axis. Figure 2(c) describes the dynamic feature of the dark one-soliton solutions via solutions (26) on the continuous wave background, which is different with Figures 1 via Solutions (22) on the zero background.Evolution of the soliton solutions of (26). The parameters are (a) a=1, λ2=0.5; (b) a=-2,  λ2=1.1; (c) a=1,  λ2=3.
(a)
(b)
(c)


## 3.1. One-Soliton Solutions for (3)

Whena=0, that is to say, the initial seed for (3) is zero, solution (19) reduces to one-soliton solution as
(22)q=λ2-λ*2λλ*2sech(χ1+χ1*)exp(χ1-χ1*),
with
(23)χ1=-iλ2x+(i-i4λ2-iλ2)t,χ1*=iλ*2x+(-i+i4λ*2-iλ*2)t.
Solution (22) represents a bright soliton whose dynamic features are delineated in Figures 1. Through symbolic computation, we can conclude the following physical quantities for solution (22): the maximum amplitude |(λ2-λ*2)/λλ*2|, the width 1/|λ2-λ*2|, the envelope velocity (1/4λ*2-1/4λ2-λ2-λ*2)/(λ2-λ*2), and the energy of the one-soliton solution
(24)E1=∫-∞+∞‍[|q(x,t)|2-|q(±∞,t)|2]dx=128λ1λ2(λ12+λ22)3.
So, the amplitude and envelope velocity will increase when the value of λ1 becomes bigger. As shown in Figures 1, the amplitude is higher and the envelope velocity is bigger in Figure 1(a) than in Figure 1(b).Evolution of the one-soliton solution of (22). The parameters are (a) λ1=2,  λ2=4; (b) λ1=1,  λ2=4.
(a)
(b)


## 3.2. Breather and Dark One-Soliton Solutions for (1)

Whena≠0, the nonzero initial seed q=aexpi(At+Bx) describes the nonvanishing boundary conditions. For simplicity, taking λ1=λ2,  B=2/a2, we have the following relations:(25a)A=2+2a2-a22,ζ1=λ22,η1=2a2,ζ2=-1λ22-λ22,(25b)η2=a22-a22,L=1λ2a,M=-1λ2a+λ2a2.With the previous conclusions, solution (19) can be converted into:
(26)q=[a-4iλ1λ2G′(λ12+λ22)F′]expi[(2+2a2-a22)t+2a2x],
where
(27)G′=D1coshΘ1′+D2sinhΘ1′+D3cosΘ2′+D4sinΘ2′,F=D5coshΘ1′+D6sinhΘ1′+D7cosΘ2′+D8sinΘ2′,
with
(28)Θ1′=λ22x-(λ22+1λ22)t,Θ2′=(a22-2a2)t+2a2x,D1=2aλ2,D2=i(aλ2-D1),D3=2a2λ22+a2λ224,D4=i(2-D3),D5=i(2a2λ2+a2λ234),D6=2λ2+iD5,D7=2ia,D8=aλ22+iD7.
Figure 2 display the propagation characteristics of solitons via solutions (26). Figures 2(a) and 2(b) depict the dynamic features of breathers; as shown in Figure 2(a), the main feature is the propagation of the breather that is periodic in the space coordinate and aperiodic in the time coordinate; that is to say, we can obtain the Akhmediev breather [23] via solutions (26) under suitable parameters chosen. In addition, the Akhmediev breather can be regarded as a modulation instability process. Figure 2(b) portrays the propagation of the breather evolving periodically along the straight line with a certain angle of x-axis and t-axis. Figure 2(c) describes the dynamic feature of the dark one-soliton solutions via solutions (26) on the continuous wave background, which is different with Figures 1 via Solutions (22) on the zero background.Evolution of the soliton solutions of (26). The parameters are (a) a=1, λ2=0.5; (b) a=-2,  λ2=1.1; (c) a=1,  λ2=3.
(a)
(b)
(c)


## 4. Conclusions

Our main attention has been focused on (3), which can describe the propagation of femtosecond pulse in single mode optical silica fiber. By using the Darboux transformation method, we have obtained (1) bright one soliton on the zero background; (2) two types of breathers: the Akhmediev soliton which delineates the modulation instability process and the breather evolving periodically along the straight line with a certain angle of x-axis and t-axis; (3) the dark one-soliton solution on the continuous wave background.



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