Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schr¨odinger equations with variable coefficients in nonlinear optical fibers

Wei Yang(杨薇), Xueping Cheng(程雪苹), Guiming Jin(金桂鸣), and Jianan Wang(王佳楠)

1School of Information Engineering,Zhejiang Ocean University,Zhoushan 316022,China

2School of Science,Zhejiang University of Science and Technology,Hangzhou 310023,China

Keywords: nondegenerate solitons,variable coefficients coupled nonlinear Schr¨odinger equations,Hirota bilinear method

1.Introduction

It is well known that one of the dominating application areas of solitons is the optical fiber communication.As the carriers of information, solitons can well solve the problem of nonlinear pulse distortions in the fiber optic channels.In 1973, Hasegawa and Tappert first predicted the existence of solitons in nonlinear optical fibers.[1]Later in 1980, Mollenaueret al.demonstrated the propagation of soliton in optical fibers.[2]Since then, an enormous amount of attention has been paid in this topic due to its versatile applications by both experimental and theoretical physicists.[3-8]In addition to single-hump solitons, it has been reported more recently that dispersion-managed communication systems can also admit double-hump solitons or multi-hump solitons,which may offer potential applications in improving capacity of optical fiber communication.[9,10]Especially, among these multihump solitons,the nondegenerate soliton solutions,[11]which are considered as the solitons in multi-mode fibers traveling with non-identical wave numbers, have been put forward for modeling the propagation of multi-hump solitons in optical fibers.When two nondegenerate solitons collide with each other, their amplitudes may change but their intensities remain unchanged.Later, inspired by nondegenerate solitons, more general nondegenerate Kuznetsov-Ma breathers[12]and nondegenerate Akhmediev breathers[13,14]have also been studied.In recent years, based on several different mathematical physics methods such as the Hirota bilinear method, Kadomtsev-Petviashvili hierarchy reduction and developing Darboux transformation method, nondegenerate solitons have been reported in various multi-mode fiber systems, such as the Manakov systems,[15-17]generalized coupled nonlinear Schr¨odinger systems,[18-22]long-wave shortwave resonant interaction systems,[23]and coupled Fokas-Lenells systems.[24]

Generally speaking,to describe the propagation of beams in two polarized directions, coupled nonlinear Schr¨odinger(CNLS)equations[25]

withuandvreferring to two complex envelopes of the electrical field,zandtrepresenting the longitudinal propagation coordinate and temporal transverse coordinates respectively,are one of the most commonly used models.[26,27]While the beams traveling in higher dimensions are taken into account,(1+1)-dimensional CNLS equations may be extended into(2+1)-dimensional ones[28,29]

whereyrepresents the spatial transverse coordinate.Furthermore,in real fibers,owing to the inhomogeneity of fiber mediums, which may be caused by imperfections of manufacture,variations in the lattice parameters of the fiber media, fluctuations in the fiber diameters, and so on, the fiber mediums,such as dispersion, nonlinear effect and gain (loss) parameters,become the parameters of transmission distance,then the

pulse propagations in such mediums should be described by variable coefficient coupled nonlinear Schr¨odinger(vcCNLS)equations.[30-33]Motivated by the practical significance of vc-CNLS equations and the wide applications of nondegenerate solitons,we intend to investigate the multi-hump nondegenerate solitons in the following system of the(2+1)-dimensional CNLS equations with variable coefficients:

where real analytic functionsβ(z)andρ(z)denote the diffraction and gain (loss) parameters, respectively;R(z) is the coefficient of self-focusing Kerr nonlinearity asR(z)>0, andR(z) is the coefficient of self-defocusing Kerr nonlinearity asR(z)<0.Due to the non-integrability of vcCNLS equations,the studies of these systems were often realized by converting them into integrable constant coefficient CNLS equations.For example, the similarity transformation technique has been brought into vcCNLS equations to obtain the vector rogue wave solutions and bright soliton solutions.[34,35]Wu and Jiang constructed the (2+1)-dimensional breathers for vcCNLS equations through transforming them to the standard NLS equation.[36]However, we would like to construct the nondegenerate solitons for Eq.(3)in the frame of the developing Hirota bilinear method in this paper.

The plan of this work is organized as follows.In Section 2,the bilinear forms of the(2+1)-dimensional vcCNLS equations(3)are derived.The nondegenerate one-soliton and two-soliton solutions of Eq.(3)are obtained by the developing Hirota bilinear method in Section 3.The propagation characteristics of the nondegenerate solitons and the influences of variable coefficients on the shapes of nondegenerate solitons are analyzed in Section 4.Section 5 is allotted for some important conclusions.

2.Bilinear forms of (2+1)-dimensional vcCNLS equations

To seek the nondegenerate solitons for vcCNLS equations (3), we introduce the following dependent variable transformation[37]

Hereεis a formal expansion parameter,gj1andgj2(j=1,3,5,...) are complex functions with respect to (y,z,t), andfj(y,z,t) (j=2,4,6,...) are all real functions.By inserting expansions(8)into the bilinear equations(6)and collecting the terms proportional to the same powers ofε,a system of linear partial differential equations to be solved can be acquired.The multi-hump nondegenerate soliton solution constituting bygj(j=1,2)andffor Eq.(3)can then be determined by solving these linear partial differential equations.

3.Nondegenerate soliton solutions

3.1.Nondegenerate one-soliton solutions for Eq.(3)

In order to explore the nondegenerate one-soliton solutions for(2+1)-dimensional vcCNLS equations(3),we truncate Eq.(8)to

withk1,l1,p1andq1being four complex parameters and the superscript∗denoting the complex conjugation.By substituting Eq.(10) into the bilinear forms (6) and collecting the coefficients of different powers ofε,we have

and the constraint condition amongk1,q1,p1andl1,i.e.,

where∆11,∆12,δ1,δ2andδ11are shown as Eq.(11), anda11anda12are still two arbitrary constants.According to the expression (14), it is evident that the solitons propagating in the two components are represented by two arbitrary functionsβ(z)andR(z),which are associated withρ(z)by formula(5),and five arbitrary complex parametersa11,a12andk1,l1,p1.These wave parameters of different values control the shapes,amplitudes,width and velocities of solitons in Kerr media.

3.2.Nondegenerate two-soliton solutions of Eq.(3)

For investigating the interaction between nondegenerate solitons, we consider constructing nondegenerate two-soliton solutions for Eq.(3), that is, truncating the expansions (8) to the following terms:

Now we take the seed solutions asg11=a11eη1+a21eη2,g12=a12eξ1 +a22eξ2, whereηs=kst+lsy+ws(z),ξs=pst+qsy+rs(z) (s= 1,2),{ks,ls,ps,qs(s= 1,2)}are complex constants, andws(z),rs(z) (s=1,2) are four complex functions ofz.Therefore, similar to the last subsection,by vanishing the coefficients of all different powers ofεand then lettingε=1,the nondegenerate two-soliton solutions of Eq.(3)are given as

Owing to the lengthiness of parametersδ17,δ18,δ19,∆sj,Λsj,∆3j,ρs j,µs j,ρ3j,Θs j,Tsj,∆s4,νs j,φsj,ρs4,∆5j,ρ5j,δsj,σsj,As j,Bsj,Φsj,Ψsj(s,j=1,2), the specific forms of parameters are shown in the Appendix.From Eqs.(16)-(18), it can be discovered that the complicated profiles of the nondegenerate two-soliton solutions are designed by nine arbitrary wave parametersβ(z),R(z),k1,pj,as j(s,j=1,2).

4.Properties of nondegenerate solitons

Based on the nondegenerate soliton solutions (14) and(16) with Eqs.(17) and (18), now we analyze the influences of different parameters, including the nonlinear effectR(z),the diffractionβ(z)and the wave numbers,on the profiles and propagating behaviors of nondegenerate solitons in nonlinear optics.

4.1.Nondegenerate one-soliton solutions of Eq.(3)

First,we discuss how to design different profiles of nondegenerate one-soliton solutions for vcCNLS equations(3)by selecting different diffraction parameterβ(z) and nonlinear coefficientR(z).It can be seen from the nondegenerate onesoliton solutions (14) with Eq.(5) that the function eρ(z)dzmodulating the amplitudes of the solitons that are present in two componentsuandvis determined by the diffraction and nonlinear coefficients.Bringing inρ(z) in different ways,a different dynamical phenomenon can be excited.For instance, when the periodic functions likeR(z) = cos(z) andβ(z)=cos3(z)are settled on the diffraction and nonlinear coefficients,the profiles of the nondegenerate solitons in the two components|u| and|v| are modulated by the periodic function|cos(z)| at the same time, which is determined from the relation (5).At a certain transmission distancez, the wave packets still propagate in thetandydirections in the nondegenerate soliton forms.The corresponding evolution plots of nondegenerate solitons fory=0 andz=0.5 are shown in Figs.1(a),1(b)and 1(c),1(d),respectively,whereas other parameters arek1=l1=1,p1=q1=0.95 anda11=2,a12=4.In addition, if the diffraction and nonlinear coefficientsβ(z)andR(z) are continued to be changed toR(z)=sech(z) andβ(z) = sech3(z), which makes= sech(z), the amplitudes of the nondegenerate solitons decay to zeros quickly along thezandtaxis, as shown in Figs.1(e) and 1(f).The solutions|u| and|v| are localized both intandzdirections.That is to say, under the action of the inhomogeneous coefficients,there appears remarkable difference between the solutions of vcCNLS equations and those of the constant coefficient CNLS equations.[11]In fact, due to the existence of the relation (5) betweenβ(z),R(z) andρ(z), it is not difficult to verify that given different functionsβ(z) andR(z), the same amplitude modulation functionρ(z) can be constructed, that is to say,different diffractions and nonlinear effects may produce the same nondegenerate soliton envelope.For instance,by taking{R(z)=sech(z),β(z)=sech3(z)},{R(z)=cosh(z),β(z)=sech(z)},{R(z)=sinh(z),β(z)=tanh(z)sech(z)}and{R(z)=tanh(z),β(z)=tanh(z)sech2(z)},one can receive the same amplitude modulation function eρ(z)dz=sech(z), but different wave variablesη1andξ1.In contrast to the variable coefficient case, the nondegenerate one-soliton exhibits the shape-preserving propagation as soon as the diffraction parameterβ(z)and the nonlinear coefficientR(z)are selected as some real constants.

Fig.1.Evolutions of the nondegenerate one-solitons with the parameters chosen as [(a), (b)] R(z)=cos(z), β(z)=cos3(z) for y=0; [(c),(d)]R(z)=cos(z),β(z)=cos3(z)for z=0.5;[(e),(f)]R(z)=sech(z),β(z)=sech3(z)for y=0.Other parameters are k1=l1=1, p1=q1=0.95,a11=2,and a12=4.

Moreover, according to the explicit solutions (14) with Eqs.(11), (12) and (13), by selecting different wave parametersk1,p1,l1,a11anda12,the nondegenerate one-solitons with different intensities,propagation velocities and phases can be designed.Particularly, under the special conditionsk1=p1andl1=q1, the nondegenerate one-soliton solutions(14)degenerate into the modulated bright soliton forms

4.2.Nondegenerate two-soliton solutions of Eq.(3)

Now we continue to take into account the effects of varying diffraction and nonlinear parameters on the shapes of the nondegenerate two-soliton solutions.Figure 2 shows the evolutions of two solitons with differentβ(z)andR(z).It can be seen from Figs.2(a)and 2(b)that nondegenerate two-solitons propagate as a tanh function due to the variable coefficients are chosen asβ(z)=tanh(z)andR(z)=sinh(z)cosh(z)(sinh(z)-cosh(z))2,which leads to eρ(z)dz=tanh(z)+1.Other wave parameters arel1=-2,l2= 2.2,p1=-1.2,p2= 1.25,q2=2.5 anda11=5,a12=3,a21=a22=2.Since the diffraction and nonlinear coefficients are given byβ(z) = sin(z)andR(z)=tan(z)sec(z), the wave peaks of two solitons increase and decrease withzat a certain frequency, and the envelopes of the nondegenerate two-solitons vary periodically as eρ(z)dz=|cos(z)|,which are depicted in Figs.2(c)and 2(d).

Fig.2.Evolutions of the nondegenerate two-solitons with y =0 and the parameters chosen as [(a), (b)] β(z) = tanh(z), R(z) =sinh(z)cosh(z)(sinh(z)-cosh(z))2; [(c), (d)] β(z) = sin(z), R(z) =tan(z)sec(z).

In particular,if the variable coefficients are taken as some constants, which makes the gain (loss) parameterρ(z)=0,the nondegenerate two solitons will travel with some specific amplitudes.For example, whenβ(z)=2 andR(z)=1 are selected, the vcCNLS equations (3) have two stable doublehump soliton solutions, which can be found in Figs.3(a)and 3(b),where other parameters are chosen asl1=0.65+1.45i,l2=0.45,p1=0.5+0.6i,p2=0.405-0.55i,q2=0.48i,k2=0.38-0.55i,a11=a22=1+0.8i,a12=a21=1.2+i.The corresponding two-dimensional evolution plots fory=0 andz=±8 are depicted in Figs.3(c) and 3(d), where one may observe that the nondegenerate two-solitons maintain the double-hump structures before and after the collisions,but the energies are redistributed after the inelastic collisions.Moreover, via computing the maximum values of the nondegenerate two-solitons,it can be proved that the total energies of the nondegenerate two-solitons before and after collision remain unchanged,which is just one of the prominent characteristics of the nondegenerate solitons.Sequentially,we find that when the imaginary parts of all wave numbers are taken as the same values, which makes the nondegenerate two solitons of both components propagate at the same speed, the nondegenerate two solitons in this case are bounded into double-hump twosoliton molecules,[39]which can be seen from Fig.4, where we have set the wave parameters asl1=-2.34,l2= 2.2,p1=-1.2,p2=1.25,q2=2.5,a11=a12=a21=a22=2,β(z)=1 andR(z)=2.

Fig.3.The inelastic collisions between nondegenerate two-solitons for y=0.[(a),(b)]Dynamical evolution diagrams of nondegenerate soliton collisions.[(c),(d)]The corresponding strength plots for both components|u|and|v|before(z=-8)and after(z=8)collision.

Fig.4.Stably propagating nondegenerate two-soliton molecules.The wave parameters are chosen as l1 = -2.34, l2 = 2.2, p1 = -1.2,p2=1.25,q2=2.5,a11=a12=a21=a22=2,β(z)=1,and R(z)=2.

5.Conclusions

In summary,we have investigated the nondegenerate onesoliton and two-soliton solutions relating to the Hirota bilinear form for the (2+1)-dimensional CNLS equations (3)with variable coefficients.The results show that the shapes of nondegenerate solitons are controllable by choosing different wave parameters, especially the diffraction coefficientβ(z) and the nonlinearityR(z).Concretely, we have demonstrated how to design the poses and the propagation velocities of nondegenerate solitons by choosing different diffraction coefficients,nonlinear effects and wave numbers,and also analyzed the dynamic behaviors of these nondegenerate solitons based on the analytical expressions(14)with Eqs.(11)-(13), and expressions (16) with Eqs.(17) and (18).For the illustrative purpose, we respectively demonstrate the profiles of nondegenerate one-solitons of cosine function and hyperbolic secant function forms and nondegenerate two-solitons of hyperbolic tangent function and sine function forms.For the nondegenerate two-soliton solutions, it is found that despite the collisions between two double-hump solitons are inelastic and the energies are redistributed after the inelastic collisions,the total energies of the nondegenerate two-solitons remain the same before and after collision.Moreover,as soon as the velocity of one double-hump soliton resonant with that of the other one,the double-hump two-soliton molecule can be constructed,which is shown in Fig.4.Most importantly,we hope that the results obtained in this paper may raise the possibility of relative experiments and potential applications.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos.11975204 and 12075208),the Project of Zhoushan City Science and Technology Bureau(Grant No.2021C21015),and the Training Program for Leading Talents in Universities of Zhejiang Province.

Appendix A