Exact Soliton Solutions for the(2+1)-Dimensional Coupled Higher-Order Nonlinear Schrödinger Equations in Birefringent Optical-Fiber Communication∗

Yue-Jin Cai(蔡跃进)Cheng-Lin Bai(白成林)† and Qing-Long Luo(罗清龙)School of Physics Science and Information EngineeringLiaocheng UniversityShandong 5059China

2Shandong Provincial Key Laboratory of Optical Communications Science and Technology,Liaocheng 252059,China

1 Introduction

In the field of nonlinear communications,earlier studies show that optical solitons can propagate over a long distance without the shape change in optical fibers,when the balance is kept between dispersion and nonlinearity.[1−2]To improve the capacity of optical fiber communication systems,the femtosecond pulses has then been applied in the field.[3−4]In consideration of higherorder effects,[5]the dynamics of a femtosecond pulse envelope in optical fibers have been described by the higherorder nonlinear Schrrödinger(HNLS)equation:whereqrepresents the complex envelope amplitudes,k′′describes group velocity dispersion(GVD),k′′′represents the third-order dispersion(TOD),βis the effective nonlinear coefcient,γandγsdescribe self-steepening(SS)and the delayed nonlinear process.The imaginary part ofγsdescribes stimulated Raman scattering(SRS).But in this paper,we only consider the real part ofγs.zis the normalized distance along the direction of the propagation,whiletrepresents the retarded time.

In birefringent optical- fibers,with consideration of components in the two polarized direction,the propagation of pulses can be described as the coupled higher-order nonlinear Schrödinger(CHNLS)equations:whereq1andq2represent the complex envelope amplitudes of components in the two polarized direction,aandbare the coefficients of the self-phase modulation(SPM)and cross-phase modulation(XPM).In our study,we takea=b=1 to simplify the equations.Soliton solutions for CHNLS equations have been obtained in a number of laboratories.[6−10]And with variable coefficients,solitonlike solutions were constructed in recent researches.[11−13]At the special ratio of coefficients,the soliton solution of N-CHNLS equations have been derived.[14−18]

As demonstrated in the literature,the spatiotemporal optical solitons[19]can be extended to the(2+1)-dimensional geometry in transverse directionxand propagation directionz,which are understood as the result of the simultaneous balance of nonlinearity and dispersion.In this paper,we focus on the optical solitons for the(2+1)-dimensional CHNLS equations withxandzdirections.

whereqj=qj(z,t,x),j=1,2,∇1=(∂/∂t,∂/∂x).q1andq2both as the complex-valued functions of the variablesz,tandx,represent the amplitudes of components in the two polarized direction in birefringent optical- fiber communication.(2+1)-and(3+1)-dimensional coupled nonlinear Schrödinger equations have been investigated[20,22]previously.

To our knowledge to date,soliton solutions for(2+1)-dimensional CHNLS equations and their interactions have not been derived yet.The structure of the present paper is as follows.In Sec.2,we will rewrite Eq.(3)in Hirota bilinear forms after using a suitable transformation and parametric condition respectively for both bright and dark soliton solutions.With the aid of symbolic computation,in Secs.3 and 4,bright and dark soliton solutions for(2+1)-dimensional CHNLS equations will be obtained.With coefficients varying,the propagation characteristics and interaction behaviors of the solitons will be discussed in Sec.5.Section 6 is allotted for a conclusion.

2 Bilinear Forms

In order to construct soliton solutions based on Eq.(3),it is rather convenient to introduce the transformation of appropriate variables

whereρ1andρ2are complex functions ofZ,X,andT.The new variablesZ,X,andTare defined as

under the special condition 2k′′γ=βk′′′,we can obtain the following forms of coupled envelope equations corresponding to Eq.(3):

where∇2=(∂/∂T,∂/∂X).Now we make the bilinearizing transformation as:

whereg,h,andfare the complex differentiable functions with respect toZ,X,andT,andfis a real one.When we construct the bright solitons,bilinear forms of(6)can be written as

where∗represents the complex conjugate,mis defined by expressions(13),(18),(23),and(27).WhileDX,DZ,andDTare Hirota bilinear operators defined by

wherea(µ,ν)is a differentiable function of the variablesµandν,b(µ′,ν′)is a differentiable function of the variablesµ′andν′,nandlare positive integers.

While we construct dark solitons,the bilinear forms of(6)could be rewritten as

whereλis a real parameter to be determined.The other characters and operators are defined same as above.

3 Bright Soliton Solutions of System(6)

When we construct the bright solitons,Eq.(6)can be solved by introducing the following power series expansions forg,h,andf.

whereχis the formal expansion parameter.Substituting Expressions(11)into Bilinear Forms(8)and collecting the coefficients of each order ofχ,we can obtain the recursion relations forg,h,andf.Then,the bright soliton solutions of System(6)can be obtained.

3.1 Bright One-Soliton Solution

To obtain the bright one-soliton solution for Eq.(6),we restrict the order series expansion(11)as

Then we assume that

whereη=αX+βT+εZwith,φ1andφ2are real constants,and are complex parameters.Substituting expressions(12)and(13)into Bilinear Forms(8),we can then get

where the asterisk denotes the complex conjugate.Without loss of the generality,we setχ=1,we obtain the bright one-soliton solutions as

3.2 Bright Two-Soliton Solution

We truncate the expressions(11)as

Then we assume that

whereη1=α1X+β1T+ε1Z,η2=α2X+β2T+ε2Zwithα1,β1,ε1,α2,β2,andε2are real constants.Substituting expressions(15)and(16)into Bilinear Forms(8),we can then get

Without lossing generality,settingχ=1,we obtain the bright two-soliton solutions as

4 Dark Soliton Solutions of System(6)

To construct dark solitons,power series expansions forg,h,andfare written as below,

Substituting Expressions(20)into Bilinear Forms(10)and collecting the coefficients of each order ofχ,we can produce the recursion relations forg,h,andf.Then we will obtain the dark soliton solutions of System(6).

4.1 Dark One-Soliton Solution

In order to find the dark one-soliton solution for Eq.(6),we restrict the order series expansion(11)as

Then we assume that

whereθ=αX+βT+εZ+φwithα,β,andεare real constants,σ1,σ2,andφare complex parameters.

Substituting expressions(21)and(22)into Bilinear Forms(10),we will obtain

where the asterisk denotes the complex conjugate.Without loss of the generality,we setχ=1,then our dark one-soliton solution for the case be explicitly expressed as

4.2 Dark Two-Soliton Solution

To predict the dark two-soliton solution for Eq.(6),we restrict the order series expansion(11)as

Then we assume that

whereθ1=α1X+β1T+ε1Z+φ1,θ2=α2X+β2T+ε2Z+φ2,withα1,α2,β1,β2,ε1,andε2,φ1,φ2are real constants,and are complex parameters.Substituting expressions(25)and(26)into Bilinear Forms(10),will obtain

Without loss of generality,settingχ=1 then our dark two-soliton solution for case be explicitly expressed as

5 Discussion on the Soliton Solutions Obtained

In modern communication system,a large number of soliton pulses transmit information in the optical fibers.The interactions between solitons will directly affect the quality and capacity of the system.Therefore,the interactions of solitons are considered as one of the most important issues that need to be addressed,when we design the system.In this section we intend to derive the interactions via the soliton solutions of(2+1)-dimensional CHNLS equations.In addition,we can get the expressions ofq1andq2by combining the inverse transformation of expressions(4a),(4b),(5)with the soliton solution of system(6)constructed in Secs.3 and 4.

Referring the solutions obtained above,we found that the value ofαj(j=1,2)andβj(j=1,2)inuence the velocities and directions of solitons.The interactions between solitons pulses are illustrated in Figs.1–5 where we can directly show the changes in intensities,velocities,and widths during the propagation in thex-z,z-t,andx-tplanes.

Fig.1 Contour plots for the intensity|q1|2via the bright two-soliton solution with k′′′ = −6,k′′ =2,α1=1.2,α2=1,φ1=Ln(0.5),φ2=Ln(0.6),m=1.4,γ =1,γs=1.

Figures 1 and 2 show the elastic interactions between bright two solitons in thex-zplane.From the different time,we can see that the two parallel bright solitons have exchanged their positions.After colliding with each other,two solitons keep their shapes and intensities,which shows that the interactions are elastic.Also we can identify the interactions ofq1andq2are pretty much the same expect the differences in intensities.Therefore we will takeq1for example to illustrate interactions in thez-tandx-tplanes.

Fig.2 Contour plots for the intensity|q2|2via the bright two-soliton solution with k′′′= −6,k′′=2,α1=1.2,α2=1,φ1=Ln(0.5),φ2=Ln(0.6),m=1.4,γ =1,γs=1.

Fig.3 Contour plots for the intensity|q1|2via the bright two-soliton solution with k′′′= −6,k′′=1, α1=0.8,α2=1.5,φ1=Ln(0.5),φ2=Ln(0.6),m= −0.8,γ =1,γs=1.

Fig.4 Contour plots for the intensity|q1|2via the bright two-soliton solution with k′′′= −6,k′′=2,α1=1.2,α2=1,φ1=Ln(0.5),φ2=Ln(0.6),m= −0.8,γ =1,γs=1.

Fig.5 Contour plots for the intensity|q1|2via the bright two-soliton solution with k′′′=6,k′′= −4,α1=0.7,α2=1,φ1=0.1,φ2=0.2,σ1=0.2,σ2=0.3,m=2,γ =0.5,γs=0.4.

In Figs.3 and 4,two kinds of interactions between bright solitons can be seen in thez-tandx-tplanes.The head-on interaction occurs in Fig.3 where two solitons propagate with different directions pass through each other unaffectedly.Figure 4 shows the overtaking interaction between two bright solitons,which means that the faster soliton surpasses the slower one in the same direction.In both Figs.3 and 4,there are no changes in intensities,velocities,and widths except phase shifts in the collisions.As a result,the two kinds of interactions are elastic.However,the interaction inz-tplane has the similar properties to the interaction inx-tplane expect the differences in directions.

In Figs.5,the elastic interactions between dark two solitons in thex-zplane have been identified.We can know that the interactions between the bright solitons are similar to those between the dark solitons.Therefore we will not illustrate the interactions between dark solitons in thez-tandx-tplanes to avoid duplications.

6 Conclusions

In this paper,we have investigated the(2+1)-dimensional coupled nonlinear higher-order nonlinear Schrödinger equations,which describe the propagation of femtosecond soliton pulses comprising of two orthogonally polarized components in birefringent optical- fiber communication.In this study with the aids of the Hirota method and symbolic computation,we are able to construct both bright and dark solitons.Finally the test results indicate that the interactions between the bright and dark solitons are elastic in thex-z,z-t,andx-tplanes referencing to contour figures.

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