Effect of near PT-symmetric potentials on nonlinear modes for higher-order generalized Ginzburg–Landau model

Da Lin,Kai-Ru Dong,Jia-Rui Zhang and Yu-Jia Shen

College of Science,China Agricultural University,Beijing 100083,China

Abstract In this paper,we study the higher-order generalized Ginzburg–Landau model which contributes to describing the propagation of optical solitons in fibers.By means of the Hirota bilinear method,the analytical solutions are obtained and the effect of relevant parameters is analyzed.Modulated by the near parity-time-symmetric potentials,the nonlinear modes with 5% initial random noise are numerically simulated to possess stable evolution.Furthermore,the evolution of nonlinear modes is displayed through the adiabatical change of some parameters.The investigation of the present work is intended as a contribution to the work for the higher-order generalized Ginzburg–Landau model.

Keywords: generalized Ginzburg–Landau model,parity-time symmetry,stability of soliton solutions

1.Introduction

Optical solitons,which have the unique characteristic that waveform and velocity remain unchanged over long distant propagation,have been paid increasing attention in recent years[1–5].It is found that the formation mechanism of optical solitons during the propagation process is the balance between group velocity dispersion and the self-phase modulation effect in the anomalous dispersion region [1].To describe the propagation of optical solitons in optical fibers,the nonlinear Schr?dinger equation (NLSE) known as an important and universal model has been developed with some generalizations and soliton solutions presented [6–13].Nevertheless,the generalized Ginzburg–Landau equation (GGLE),which is widely applied in such fields as superconductivity,liquid crystal,Bose–Einstein condensate,can be considered as a dissipative generalization of NLSE [14–17].Different analytical and numerical methods have been applied to the GGLE,while various novel solutions including the pulsating,erupting and creeping solitons have been obtained [18–23].By means of numerical simulations,the stability of various solutions has been proved [24,25].For a wider application prospect,the model has been extended to higher-dimension and higher-order cases [26–30].Moreover,parity-time (PT) symmetric potentials have been introduced to the GGLE with several interesting results[25,31,32].Though differentPT-symmetric behaviors have been studied theoretically or observed in experiments[33–38],limited research has been done which is relevant to the higher-order GGLE.In previous work,we have investigated the fourth-order GGLE with quintic nonlinearities and nearPT-symmetric structures [39].

In this paper,we will study the GGLE with third-order dispersion and nonlinear gradient:

whereu(x,t) represents a complex wave envelope,xdenotes the propagation distance andtis the time.The subscripts denote the partial derivative with respect toxortand i represents the imaginary unit.α(x),U(x,t) and β(x,t) are complex functions that can be assumed as α=α1+iα2,U=V+iW,β=β1+iβ2.α(x),U(x,t),β(x,t),σ(t)and γ(t)can describe the variable effect of group velocity dispersion,gain or loss,self-phase modulation,third-order dispersion and nonlinear gradient terms,respectively [18,25,30,40].

There are three special cases that can be reduced by equation (1).

(1) When α2(x)=W(x,t)=β2(x,t)=0,equation(1)turns into the third-order NLSE.It has been used to describe the propagation of ultra-short pulses and optical solitons in fibers in [6,7,41].Some exact solutions and the corresponding abundant structures have been obtained[7],and the linear stability of solitons has been studied[6].

(2) When σ(t)=γ(t)=ρ(t)=0,equation (1) can be reduced to the second-order GGLE.The analytical solutions have been derived by means of the Hirota bilinear method[21,25,42,43].The stability of soliton has been analyzed via numerical simulations in [25,42].

(3) When σ(t)=γ(t)=ρ(t)=0 andU(x,t)isPT-symmetric.Equation (1) is changed into the GGLE withPT-symmetric potential,which has been less investigated so far except [25].The effect of nearPT-symmetric potentials on nonlinear modes has been reported [32].

The rest of this paper is arranged as follows.In section 2,the bilinear form of equation(1)is derived under some constraints.In addition,soliton solutions of equation (1) with constant and variable coefficients are obtained respectively.In section 3,the stable transmission of nonlinear modes is verified through numerical simulations with 5%perturbations.The effect of near PT-symmetric potentials is discussed with relevant figures illustrated,and the adiabatic change of some parameters is considered.Finally,the conclusions are given in section 4.

2.Analytical solutions of equation (1)

The analytical solutions of equation (1) are derived by the Hirota bilinear method.Through variable transformation

with the real functionFand complexG,and the constraint αγ=3σβ,the bilinear equations of equation(1)are written as

The Hirota operator is defined by [44]

We expandGandFin power series of ?as

where ?is a small parameter,Gi(i=1,3,5,…)andFj(j=2,4,6,…) are functions ofxandtto be determined.

In this section,we study two cases of constant and variable coefficients.For the sake of calculation,we set ?=1.The analytical expression of a single soliton solution for equation (1) is

Case 1:

Under the constraints β1=c0α1,β2=c0α2,Wwhere αi(i=1,2),c0,k1are real constants,we substitute equation(5)into equation(3)and collect the coefficients of ?with the same power.Then we can get

i.e.the soliton solution of equation (1) with constant coefficients,wheremi,wi(i=1,2),ρ,σ,V,Ware real constants.

Case 2:

wheremi(i=1,2),k1are real constants.Substituting Expressions(8)into(6),we get the analytical soliton solution likewise.

By modulating dispersion and gain or loss terms,we illustrate their effect of them on the structure and propagation of soliton in figure 1.In figure 1(a),when σ,ρ andW1(t) are chosen as sine functions,the amplitude of the soliton varies with time periodically.Once the dispersion terms are taken as aperiodic functions like exponential functions,the amplitude is still periodic except for a phase shift aroundt=0.Obviously,the periodicity of amplitude is only related to the gain or loss term and the dispersion terms affect the structures.As shown in figure 1(c),the value ofk1is adjusted.When the value ofk1reduces to 0.5,the maximum amplitude decreases and the structure of soliton has changed.

3.Numerical simulations of equation (1)

where σ,γ,ρ,αi,βi(i=1,2) are real constants andV+iWdenotes the nearPT-symmetric potential.

The nonlinear mode of equation (9) can be defined as

where μ is a real propagation constant.We will first study the stability of nonlinear modes with the effect of the last three terms ignored,and reduce equation (9) to the second-order GGLE:

In the last subsection,these parameters will be considered again through adiabatical excitation of them.

3.1.Nonlinear modes under near PT-symmetric Scarf-II potential

We introduce the nearPT-symmetric Scarf-II potential [32]

where the value of real constantsV0,W0andW1can be modulated to obtain stable nonlinear modes.

The power of nonlinear mode is defined asP=Figure 2(a) shows the result that stable evolution does not exist when the value ofW1approaches zero because the potential turns intoPT-symmetric Scarf-II potential.The power decreases obviously with increasing the value of β2,but α2has less effect on the power.The two curves with different values of α2intersect atW1=2.8.Symmetric curves with respect toW0=0 are shown in figure 2(b).Moreover,they attain the lowest power at the pointW0=0 simultaneously.By the change ofV0orW1,the lowest power can be adjusted.

Figure 1.Structures of soliton solution with variable coefficients. Parameters are chosen as: (a) W1 ( t)=0.2 sin (0.5t ) ,ρ( t)=σ( t)=sin t ,(b) W1 ( t)=0.2 sin (0.5t ) ,ρ( t)=σ( t )=,(c) W1 ( t)=0.2 sin (0.5t ) ,ρ( t)=σ( t)=sin t ,k1=0.5and other parameters are fixed as 1.Figure 2.Effect of parameters on the power of nonlinear modes under near PT-symmetric Scarf-II potential.(a)α1=β1=V0=W0=1,(b)α1=β1=β2=1 and α2=?1.

Figure 2.Effect of parameters on the power of nonlinear modes under near PT-symmetric Scarf-II potential.(a)α1=β1=V0=W0=1,(b)α1=β1=β2=1 and α2=?1.

Figure 3.Stable evolution of nonlinear modes under near PT-symmetric Scarf-II potential.(a),(b),(c)W1=2,(d),(e),(f)W1=5.α2=?4 and other parameters are fixed as 1.

The stable nonlinear modes under nearPT-symmetric Scarf-II potential are shown in figures 3(a) and 3(d) with 5%initial perturbations.Increasing the value ofW1to 5,the amplitude becomes larger and the nonlinear mode has a narrower width.That is to say,the energy becomes more concentrated than before.At the same time,the imaginary part of the nonlinear mode takes up a larger proportion.

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3.2.Nonlinear modes under near PT-symmetric δ-signum potential

Equation(11) with nearPT-symmetric δ-signum potential is discussed as follows.The potential can be expressed as

Next,we consider the evolution of nonlinear modes with the potential.In the numerical simulations,5%initial random noise is added likewise.Figures 5(a),(d) and (g) show the stable evolution of peakons,whileW0affects the amplitude and period of oscillation.In contrast to figure 5(a),the peakon maintains a certain value and does not oscillate when the value ofV0increases to 1 in figure 5(g).

Figure 4.Effect of parameters on the power of nonlinear modes under near PT-symmetric δ-signum potential.(a)α1=β1=V0=W0=1,(b) α1=β1=β2=1 and α2=?1.

Figure 5.Stable evolution of nonlinear modes under near PT-symmetric δ-signum potential.(a),(b),(c) V0=0.4,W0=2,(d),(e),(f)V0=0.4,W0=1.5,(g),(h),(i) V0=1,W0=2.Other parameters are α1=β1=β2=1,α2=?1,W1=5.

Figure 6.Adiabatic excitation and evolution of the nonlinear modes under near PT-symmetric Scarf-II potential.(a,b)W0=0,W1=1,(c),(d) W0=0,W1=0.5,(e),(f)and other parameters are α2=?1,α1=β1=β2=V0=1,σ(ini)=γ(ini)=ρ(ini)=0,σ(end)=γ(end)=ρ(end)=1.

3.3.Adiabatic excitation and evolution of the nonlinear modes

The adiabatic change of parameters in equation (9) with near PT-symmetric Scarf-II potential will be considered.The‘switch-on’function in[47]is used so that the parameters can be smoothly adjusted:

The process can be divided into two stages:the excitation stage (0

4.Conclusions

In this paper,we study the higher-order GGLE,i.e.equation (1),with variable parameters and nearPT-symmetric potentials.Under some constraints,the analytical solutions of equation (1) have been derived by the Hirota bilinear method.And several structures of solitons have also been illustrated in figures by the modulation of corresponding parameters.With the nearPT-symmetric Scarf-II and δsignum potentials introduced,stability of the nonlinear modes is proved via numerical simulations.Through the process of adiabatic excitation,stable nonlinear modes are also displayed.The results obtained might advance further investigations on generalized Ginzburg–Landau models by means of analytical and numerical methods.These new findings of nonlinear modes in the generalized Ginzburg–Landau model might be potentially applied to hydrodynamics,optics and matter waves in Bose–Einstein condensates.

Acknowledgments

We express our sincere thanks to all the members of our discussion group for their valuable comments.

Disclosures

The authors declare no conflicts of interest.