Cheng-Lin Bai(白成林),Yue-Jin Cai(蔡躍進), and Qing-Long Luo(羅清龍)

1School of Physics Science and Information Engineering,Liaocheng University,Liaocheng 252059,China

2School of Computer and Information,Anqing Normal University,Anqing 246133,China

3Shandong Province Key Laboratory of Optical Communications Science&Technology,Liaocheng 252059,China

AbstractIn this paper,an extended multi-dimensional N-coupled higher-order nonlinear Schr?dinger equation(NCHNLSE),which can describe the propagation of the ultrashort pulses in wavelength division multiplexing(WDM)systems,is investigated.By the bilinear method,we construct the breather solutions for the extended(1+1),(2+1)and(3+1)-dimensional N-CHNLSE.The rogue waves are derived as a limiting form of breathers with the aid of symbolic computation.The effect of group velocity dispersion(GVD),third-order dispersion(TOD)and nonlinearity on breathers and rogue waves solutions are discussed in the optical communication systems.

Key words:N-coupled higher-order nonlinear Schr?dinger equations,multi-dimensional,rogue waves,breathers,bilinear transformation

1 Introduction

Rogue waves, first emerged in the studies of oceans,are also called freak waves or extreme waves.It has been ten years since rogue waves were introduced into optics by Solli et al.[1]Then in recent years,the study of breathers and rogue waves have attracted much interests in the researches of nonlinear optics,[2?4]fluid mechanics,[5]Bose-Einstein condensate(BEC)[6]and so on.Various mathematical methods such as the Darboux transformation[7?9]and the bilinear method[10?13]were applied to the construction of the explicit expressions of rogue wave solutions.

The propagation of nonlinear pulse can be well described by the nonlinear Schr?dinger equation(NLSE):

where q represents the complex envelope amplitudes,? and κ are the normalized distance along the direction of the propagation and retarded time.Breathers and rogue waves for NLSE were reported in many literatures.[14?17]However in modern optical systems,in order to increase bit rates,the width optical pulses have already come to be femtosecond.[18?19]When we consider the higher-order effects in the high speed communication systems,the ultrashort pulses can be governed by the higher-order nonlinear Schr?dinger equation(HNLSE).[20?23]Also the technology of multiplexing has become a significant topic,which draws much attention of researchers.[24?25]

In wavelength division multiplexing(WDM)systems,there are more fields need to be considered simultaneously.Therefore we intend to investigate an extended multidimensional N-coupled higher-order nonlinear Schr?dinger equations(N-CHNLSE),which can describe the propagation of optical breathers and rogues waves in optical fibers.

where qj(j=1,2,...,N)are the complex envelope amplitudes of the channel j in all N transmission channels,N is the number of all multiplexing channels in WDM system,k′′and k′′′describe the effect of group velocity dispersion(GVD)and third-order dispersion(TOD)respectively,β is the effective nonlinear coefficient,γ and γsdescribe self-steepening(SS)and the low frequency component of stimulated Raman scattering(SRS).The parameter c is the coefficient of the group velocity,which we have extended into the normal CHNLSE in order to make a balance between TOD and nonlinearity(see expression(9))when we construct breathers solutions.When?is defined as(?/?x),(?/?x,?/?y),or(?/?x,?/?y,?/?z),Eq.(2)respectively means the(1+1),(2+1)and(3+1)-dimensional extended N-CHNLSE.Here,x,y,z,t are nor-malized distances along the each propagation direction in the space and retarded time.

Recently,breathers and rogue waves have been discovered for coupled nonlinear Schr?dinger equations(CNLSE),[26?28]even in multi-dimensions.[29]Nonautonomous multi-peak solitons,breathers and rogue waves for the inhomogeneous nonlinear Schr?dinger equations have been studied in Refs.[30–31]including the dynamics of Peregrine combs and Peregrine walls.[32]Breathers and rogue waves have been obtained for CHNLSE.[33?34]To our knowledge,breathers and rogue waves for the extended multi-dimensional N-coupled higher-order nonlinear Schr?dinger Eq.(2)have not been derived yet.The structure of this paper is organized as follows.In Sec.2,comprehensive bilinear forms and breathers solutions for Eq.(2)are derived with the help of symbolic computation.Discussion on features of breathers and rogue waves solutions is arranged in Sec.3.Section 4 is allotted for a conclusion.

2 Bilinear Forms and Solutions

The bilinear method used to obtain the soliton solutions can also be applied to get the breathers and rogue waves for nonlinear Schr?dinger equations as shown in this paper.Here we set the bilinear transformation as:

where σj(j=1,2,...,N)are real amplitude of background waves,ω is real angular frequency,g is complex,while f is real.We construct g and f as:

where ξ1and ξ2are complex parameters,M is a real parameter,and the star denotes complex conjugation.For the convenience of description,we choose m+1(m=1,2,3)to represent(1+1),(2+1)and(3+1)dimensions of equation.Here,we define m as the number of space coordinate axes in this study.In expression(4),θ has the different definitions,in(m+1)-dimensional equations

where φ represents the phase factors.Without loss of generality while creating the comprehensive bilinear forms for the extended multi-dimensional N-CHNLSE,we intend to set ξ1= ξ2and p=p1=p2=p3.In addition,p and δ are real parameters.Combining expressions(3),(4),(5)with Eq.(2),the bilinear forms can then be written as

with γ + γs=0,Pm=m.λ is a real parameter to be determined.Also the following provision is established in our study:

While Dzand Dtare Hirota bilinear operators in Eq.(6)defined

According to the conditions we employed above plus substituting expressions(4),(5),(7)into bilinear forms(6),we will obtain

Now the solutions of the extended multi-dimensional N-CHNLSE are obtained,when we combine all the conditions in this section.The solutions for(m+1)-dimensional Eq.(2)can be written as follows:

The breathers and rogue waves solutions for the(1+1),(2+1)and(3+1)-dimensional extended N-CHNLSE can be described as combinations of expressions(10a)with(10b),(10c),or(10d)respectively.

3 Discussions on Breathers and Rogue Waves for Multi-dimensional Eq.(2)

In this section,we will discuss the characters of the solutions(10)with the effects of GVD,TOD,and nonlinearity.Due to space limitations in this paper,we just take(1+1)and(2+1)-dimensional Eq.(2)as examples to discuss.

3.1 Breathers and Rogue Waves for(1+1)-Dimensional Eq.(2)

According to solutions we got in expression(10),the breathers solutions for(1+1)-dimensional 2-CHNLSE are shown in Fig.1,where breathers of|q1|and|q2|are displayed.It is noticed that the breathers are localized in time and periodic in the transmission direction,which were termed as“Akhmediev” breather in Ref.[35].Furthermore,the breathers we got in this paper are all considered to be Akhmediev breathers.Figure 1(c)is the pro files of breathers in two channels,which shows the differences in amplitudes with the same period between each transport channel.It shows that each signal propagates with same speed but different amplitudes.Besides,we found there will be rogue waves solutions,when the parameters come to limit in our study.

To generate rogue waves,we restrict the parameter p tending to zero with the following provision:

Then the rogue waves solution for(1+1)-dimensional 2-CHNLSE can be defined as:

In Figs.2(a)and 2(b),we will see rogue waves of|q1|and|q2|,which shows that they are completely symmetric about x=0 and t=0.The rogue waves appear as localized large amplitude waves on a rough background,which have two obvious characteristics.They not only appear from nowhere and disappear without a trace,but also exhibit a dominant peak.[36]The amplitudes of rogue waves are obtained as follows on the axis of symmetry,with the condition of t=0:

Fig.1(Color online)Breathers for(1+1)-dimensional 2-CHNLSE via solution(10)with σ1=0.6, σ2=0.8,k′′′=?1,kk′′=1,p=1,γ =1,φ =0.(a)|q1|,(b)|q2|,(c)The pro files of breathers q1and q2with t=0.

Fig.2(Color online)Rogue waves for(1+1)-dimensional 2-CHNLSE via solution(12)with σ1=0.6,σ2=0.8,k′′′= ?1,k′′=1,γ =1.(a)|q1|,(b)|q2|,(c)The pro files of rogue waves q1and q2with t=0.

Fig.3(Color online)The evolution of width of rogue waves(a)via expression(14)with via σ1=0.6,σ2=0.8,(b)via expression(15)with k′′′= ?1,γ =1.σn=0.6,n=1,2,...,N.

In Fig.2(c),one can observe that the amplitudes of rogue waves solutions increases to a maximum value at x=0,along the propagation direction. After more accurate calculations,the peak of this solution can be represented as|q1|max=3σ1,|q2|max=3σ2.According to those results,it is proved that rogue waves create possibilities for generating highly energetic optical pulses for optical systems.[1]In addition,it also can be used as another alternative to the existing techniques such as passively mode-locked lasers.The widths of rogue waves are described as the distances between two valleys,which arein Fig.2(c).As a result,the widths of rogue waves(see Fig.3)are

The effects of TOD and nonlinearity on the widths of rogue waves are explicated in Fig.3(a).Here,we just take k′′′γ <0,to keep the parameters δ we have defined before real.Furthermore,the width of rogue waves for(1+1)-dimensional N-CHNLSE are generated as

where σ has been defined in Sec.2.In Fig.3(b),the width of rogue waves decrease rapidly with increasing number of channels when we set k′′′= ?1,γ =1.

The localized characters including length and width of the rogue wave have been studied for other equations.[37]Here we define the length of rogue waves at height 2σn.So the length of the rogue waves can be written as

By changing the value of GVD k′′in Fig.4(a)we shall discover the changes in the length of rogue wave comparing with Fig.2(a).When there is no group velocity dispersion in optical communication system,the rogue wave turns to be a W-shaped soliton wave in Fig.4(b)with an in finite length.The amplitude of W-shaped soliton wave have been depicted in Fig.4(c)at t=0.Comparing Fig.4(c)with Fig.2(c),there is no difference in amplitude of q1.It proves that the value of GVD does not affect the amplitudes of signals in optical system.

Fig.4(Color online)Evolution of length via solution(12a)with σ1=0.6,σ2=0.8,k′′′= ?1,γ =1.(a)Rogue wave with k′′=0.2,(b)W-shaped soliton wave with k′′=0,(c)The pro files of W-shaped soliton wave with k′′=0,t=0.

Fig.5(Color online)Evolution of breathers for(1+1)-dimensional N-CHNLSE via expression(10)with k′′′= ?1,k′′=1,p=1,γ =1.σn=0.6,n=1,2,...,N.(a)N=4,(b)N=8,(c)N=16,(d)The pro files of a period of breather with t=0.

In Figs.5(a),5(b),and 5(c),the solution status of breathers for(1+1)-dimensional N-CHNLSE are changing with increasing number of channels.Figure 5(d)describes the pro files of a period of breather illustrates that the peaks,valleys and width of breathers are different depending on the number of channels in multiplexing systems.Exactly speaking,the number increased with the rising peaks,descending valleys and narrowing width.

3.2 Breathers and Rogue Waves for(2+1)-Dimensional Eq.(2)

Fig.6(Color online)The solutions of q1for(2+1)-dimensional 2-CHNLSE via expression(10)with σ1=0.6,σ2=0.8,k′′′= ?1,k′′=1,p=0.8,γ =1.φ =0.(a)Breather in the x-t plane,(b)Breather in the y-t plane,(c)The periodic linear waves in the x-y plane.

Fig.8(Color online)The solutions of q1for(2+1)-dimensional 2-CHNLSE via solutions(16)with σ1=0.6,σ2=0.8,k′′′= ?1,k′′= ?1,γ =1. φ =0.(a)Rogue wave in the x-t plane,(b)Rogue wave in the y-t plane,(c)The W-shapedsoliton waves in the x-y plane.

Fig.9(Color online)(a)Amplitudes ofrogue waves in the x-t plane,(b)The pro files of W-shaped soliton waves in the x-y plane,(c)The rogue wave formed in the continuous-time.

The breather solutions for(2+1)-dimensional N-CHNLSE are described by Eqs.(10a)and(10c).Figures 6(a)and 6(b)display the breather solutions of q1in the x-t and y-t planes.In both x,y directions,signal transmission form are basically the same.In Fig.6(c),it is noticed that there are periodic linear wave solutions in x-y plane.In addition,the pro files of q1will be shown in Fig.7(a)along y=x.In Fig.7(b),the amplitude of q1is formed as breathers with the continuous-time.To generate the rogue waves,by the means mentioned above,the rogue wave solution for(2+1)-dimensional 2-CHNLSE can be written as follows.

In Figs.8(a)and 8(b),the rogue wave solutions are shown in x-t and y-t planes.The signals travel as rogue waves in different derections.However,it is noticed that there is W-shaped soliton wave in x-y plane depicted in Fig.8(a).The amplitudes of rogue waves in x-t and y-t planes are obtained as follows with the condition t=0:

According to Eq.(18),we get different amplitudes in Fig.9(a),when we change the values of y.In x-y plane,when we choose y=x,the pro files of the W-shaped soliton wave in different time are displayed in Fig.9(b).Similarly,the amplitude is formed as rogue wave with respect to t and x in Fig.9(c)with the continuous-time.The widths of rogue waves are expressed as

which are all the same in x-t,y-t,and x-y planes.In the same way,the width of rogue waves for(2+1)-dimensional N-CHNLSE is described as

From expressions(14),(15),(19),and(20),it is indicated the additional dimension does not affect widths of rogue wave solutions.

4 Conclusions

In this study,our main objective has focused on an extended multi-dimensional N-CHNLSE,which governs the propagation properties of ultrashort pulses in N optical fields.Based on a series of settings,we have constructed the comprehensive bilinear forms(6)and the so-lutions(10)for(1+1),(2+1),and(3+1)-dimensional NCHNLSE.According to the solutions,this paper puts emphasis on discussing the features of breather and rogue waves for(1+1)and(2+1)-dimensional N-CHNLSE.The features of the solutions are principally amplitude,peak,valley,width and length which are revealed the relations to GVD,TOD,nonlinearity and the number of space dimensions and transmission channels.In addition,as for improving the comprehensive degree of bilinear forms and solutions,there are more interesting special cases we have not shown in this paper and will be researched later.

Acknowledgment

The authors would like to express their sincere thanks to the anonymous referees for their helpful advice and kind suggestions.