Open access peer-reviewed chapter

Nonlinear Schrödinger Equation

By Jing Huang

Submitted: May 9th 2018Reviewed: August 23rd 2018Published: December 10th 2018

DOI: 10.5772/intechopen.81093

Downloaded: 1091


Firstly, based on the small-signal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss is solved. It is also adapted to the NLSE with the high-order dispersion terms. Furthermore, a general theory on cross-phase modulation (XPM) intensity fluctuation which adapted to all kinds of modulation formats (continuous wave, non-return-to-zero wave, and return-zero pulse wave) is presented. Secondly, by the Green function method, the NLSE is directly solved in the time domain. It does not bring any spurious effect compared with the split-step method in which the step size has to be carefully controlled. Additionally, the fourth-order dispersion coefficient of fibers can be estimated by the Green function solution of NLSE. The fourth-order dispersion coefficient varies with distance slightly and is about 0.002 ps4/km, 0.003 ps4/nm, and 0.00032 ps4/nm for SMF, NZDSF, and DCF, respectively. In the zero-dispersion regime, the higher-order nonlinear effect (higher than self-steepening) has a strong impact on the short pulse shape, but this effect degrades rapidly with the increase of β 2. Finally, based on the traveling wave solution of NLSE for ASE noise, the probability density function of ASE by solving the Fokker-Planck equation including the dispersion effect is presented.


  • small-signal analysis
  • Green function
  • traveling wave solution
  • Fokker-Planck equation
  • nonlinear Schrodinger equation

1. Introduction

The numerical simulation and analytical models of nonlinear Schrödinger equation (NLSE) play important roles in the design optimization of optical communication systems. They help to understand the underlying physics phenomena of the ultrashort pulses in the nonlinear and dispersion medium.

The inverse scattering [1], variation, and perturbation methods [2] could obtain the analytical solutions under some special conditions. These included the inverse scattering method for classical solitons [3], the dam-break approximation for the non-return-to-zero pulses with the extremely small chromatic dispersion [4], and the perturbation theory for the multidimensional NLSE in the field of molecular physics [5]. When a large nonlinear phase was accumulated, the Volterra series approach was adopted [6]. With the assumption of the perturbations, the NLSE with varying dispersion, nonlinearity, and gain or absorption parameters was solved in [7]. In [8], the generalized Kantorovitch method was introduced in the extended NLSE. By introducing Rayleigh’s dissipation function in Euler-Lagrange equation, the algebraic modification projected the extended NLSE as a frictional problem and successfully solved the soliton transmission problems [9].

Since the numerical computation of solving NLSE is a huge time-consuming process, the fast algorithms and efficient implementations, focusing on (i) an accurate numerical integration scheme and (ii) an intelligent control of the longitudinal spatial step size, are required.

The finite differential method [10] and the pseudo-spectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step size-controlling method [12] and the predictor-corrector method [13] were proposed to speed up the implementation of split-step Fourier method (SSFM). The cubic (or higher order) B-splines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The Runge-Kutta method in the interaction picture was applied to calculate the effective refractive index, effective area, dispersion, and nonlinear coefficients [15].

Recently, the generalized NLSE, taking into account the dispersion of the transverse field distribution, is derived [16]. By an inhomogeneous quasi-linear first-order hyperbolic system, the accurate simulations of the intensity and phase for the Schrödinger-type pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higher-order NLSE in the presence of non-Kerr quintic nonlinearities [18].

In this chapter, several methods to solve the NLSE will be presented: (1) The small-signal analysis theory and split-step Fourier method to solve the coupled NLSE problem, the MI intensity fluctuation caused by SPM and XPM, can be derived. Furthermore, this procedure is also adapted to NLSE with high-order dispersion terms. The impacts of fiber loss on MI gain spectrum can be discussed. The initial stage of MI can be described, and then the whole evolution of MI can also be discussed in this way; (2) the Green function to solve NLSE in the time domain. By this solution, the second-, third-, and fourth-order dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its probability density function.


2. Small-signal analysis solution of NLSE for MI generation

2.1 Theory for continuous wave

The NLSE governing the field in nonlinear and dispersion medium is


where β1 and β2 are the dispersions, γis the nonlinear coefficient, and αis the fiber loss. In the frequency domain, the solution is


where D̂=i2ω2β2+iωβ1a2and N̂=u2+i2u2[19] ( Figure 1 ).

Figure 1.

Schematic illustration of medium.u(z,t) andu(z + dz,t) correspond to the field amplitudes atzandz + dz, respectively.

Usually, the field amplitudes can be written as


ϕzωis caused by the nonlinear effect, and ϕzω=0zγPzω+2Pzωdz[3].



Assuming: Pzω=Pz+ΔPzω

Pzis the average signal intensity. ΔPzωis the noise or modulation term. There is [20] PzΔPzω

The amplitude Pzωcan be regarded as


The small-signal theory implies that the frequency modulation or noise φ̇z+dzω=dφ̇z+dzωdtis small enough. Finally ([21])


The operation expiωβ1dz+iω2β2dzcan be split into its real and imaginary parts:


The modulation or noise ΔPz+dzωis ΔPz+dzωPz+dzωPz





When only intensity modulation is present and no phase modulation exists, the transfer function cos12β2ω2dzis obtained. The 3 dB cutoff frequency corresponds to 12β2ω2dz=π/4in [22, 23]. This treatment is also adaptable to the case that only the nonlinear phase (frequency) modulation is present; then, the intensity modulation ΔPz+dzωdue to FM-IM conversion is given as


This is in very good agreement with [24] for small-phase modulation index. Even for large modulation index 12β2ω2dz=π/2, the difference is within 0.5 dB. Eq. (10) does not include a Bessel function, so it is simpler than that in [24].

Obviously, the above process can be used to treat NLSE with higher-order dispersion (β3, β4) [25]. Similarly, the result in Eq. (10) will include ω3 and ω4.

The corresponding MI gain gMIin the side bands of ω0 (the frequency of signal) is given by


Figure 2.

MI gain spectra. +++ result of small-signal analysis. –––– result of perturbation approach. The parameters are P0 = 10 dBm, β2 = 15 ps2/km,λ = 1550 nm,a = 0.21 dB/km,γ = 0.015W−1/m, and z = 0 m.

Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case Pz/Pz=1. The maximum frequency modulation index caused by dispersion corresponds to 12β2ω2dz=π[22, 23], and the maximum value of the sideband is ωc=4γPz/β2, so the choice of dzsatisfies 12β2ω2dz=π, which makes Eq. (11) have the same frequency regime as [26]. In Figure 2 , the curves are different but have the same maximum value of gMI. In practice, researchers generally utilize the maximum value of gMIto estimate the amplified noises and SNR [3]. The result of small-signal analysis in Figure 2 has a phase delay of around ω0. Compared with the experiment result of [27], the reason is taking the fiber loss into account, the gain spectrum exhibits a phase delay close to ω0, and the curve descends a little [27]. Fiber loss results in the difference of gMIbetween the small-signal analysis method and the perturbation approach.

2.2 The general theory on cross-phase modulation (XPM) intensity fluctuation

For the general case of two channels, the input optical powers are denoted by Pt,Pt, respectively [28]. Only in the first walk-off length, the nonlinear interaction (XPM) is taken into account; in the remaining fibers, signals are propagated linearly along the fibers, and dispersion acts on the phase-modulated signal resulting in intensity fluctuation. According to [4], the whole length Lis separated into two parts 0 < z < Lwoand Lwo < z < L; Lwois the walk-off length, Lwo=Δt/DΔλ. Δtis the edge duration of the carrier wave, Dis the dispersion coefficient, and Δλis the wavelength spacing between the channels. By the small-signal analysis, the phase modulation in channel 1 originating in dzat zcan be expressed as


This phase shift is converted to an intensity fluctuation through the group velocity dispersion (GVD) from zto the receiver. So, at the fiber output, the intensity fluctuation originating in dzin the frequency domain is given by [29].


representing the convolution operation b=ω2Dλ2/4πc, where cis the speed of light. At the fiber output, the XPM-induced intensity fluctuation is the integral of Eq. (13) with zranging from 0 to L:


The walk-off between co-propagating waves is regulated by the convolution operation.

3. Green function method for the time domain solution of NLSE

3.1 NLSE including the resonant and nonresonant cubic susceptibility tensors

From Maxwell’s equation, the field in fibers satisfies


where Eis the vector field and χ1is the linear susceptibility. PLand PNLrepresent the linear and nonlinear induced fields, respectively [30]. The cubic susceptibility tensor including the resonant and nonresonant terms is


There are


Γand aare the attenuation and absorption coefficients, respectively [31].

Repeating the process of [3]

E=FxyAztexpiβz, there is


k0=ω0/c, where ω0is the center frequency. Aeffis the effective core area. nis the refractive index. The last term is responsible for the Raman scattering, self-frequency shift, and self-steepening originating from the delayed response:


where gω1+ω2+ω3is the Raman gain and fω1+ω2+ω3is the Raman non-gain coefficients.

3.2 The solution by Green function

The solution has the form


Then, there is




and taking the operator V̂tas a perturbation item, we first solve the eigen equation n=2kinn!βnnφTn=.


Assuming E=1, we get the corresponding characteristic equation:


Its characteristic roots are r1,r2,r3. The solution can be represented as


where ϕm=expirmt,m=1,2,3and c1,c2,c3are determined by the initial pulse. The Green function of (30) is


By the construction method, it is


At the point t=t, there are


Let b1=b2=b3=0, then


Finally, the solution of (27) can be written with the eigen function and Green function:


The accuracy can be estimated by the last item of (40). The algorithm is plotted in Figure 3 .

Figure 3.

The Green algorithm for solving NLSE.

3.3 Estimation of the fourth-order dispersion coefficient β4

The NLSE governing the wave’s transmission in fibers is


where sis the self-steepening parameter. In the frequency domain, its solution is


where D̂=i2ω2β2i6ω3β3, N̂=Γexp2αzu2+isu2t+isu2t, and Γrepresents the Fourier transform [32]. Let L̂=zD̂N̂and L̂Gzz'ω=δzz'; we obtain the Green function


Constructing the iteration β3=β30+δβ3, uzω=u0zω+δuzω, then there is


where Zz'ωδβ3z'u0z'ω=i6δβ3z'ω3u0z'ωand u0z'ωβ30is determined by (42).

The minimum value of δuzωsatisfies δuzω/ω=0,R2δuzω/ω2>0, so


Next, we take the higher-order nonlinear effect into account. Constructing another iteration related to δγ:γ=γ0+δγ, uzω=u0zω+δuzωand repeating the above process, we get


Now, we can simulate the pulse shape affected by high-order dispersive and nonlinear effects. Assume LD=t02/β2and u0t=+u0ωexpiωt=u0expt2/t02/2.

Firstly, we see what will be induced by the above items δβ3and δγ. To extrude their impact, we choose the other parameters to be small values in Figures 4 and 5 . The deviation between the red and the black lines in Figure 4(a) indicates the impact of δβ3and δγ; that is, they induce the pulse’s symmetrical split. This split does not belong to the SPM-induced broadening oscillation spectral or β3-induced oscillation in the tailing edge of the pulse, because here γis very small and β3=0[3]. The self-steepening effect attributing to isu2u/tis also shown explicitly in the black line. When we reduce the svalue to 0.0001 in (b), the split pulse’s symmetry is improved.

Figure 4.

The pulse shapes with and withoutδβ3andδγ. The red line: withoutδβ3andδγ; the black line: withδβ3andδγ.ν=ω/2/π,β30=0ps3/km,γ=1.3×102/km/W,t0=80fs,z=3.7×t02/β2,β2=21.7/150ps2/km,u0=β2/γ/t02. (a)s = 0.01 and (b)s = 0.0001.

Figure 5.

The evolutions of pulse. The red line: withoutδγ; the black line: withδβ3andδγ. (a)s=0,γ=1.3×104/km/W; (b)s=0.01,γ=1.3×104/km/W; (c)s=0.01,γ=1.3/km/W. Other parameters are the same asFigure 4.

Is the pulse split in Figure 4(a) caused by δβ3or δγ? The red lines in Figure 5 describe the evolution of pulse affected by the very small second-order dispersion and nonlinear (including self-steepening) coefficients. Here, δβ3induces the pulse’s symmetrical split, and the maximum peaks of split pulse alter and vary from the spectral central to the edge and to the central again. Therefore, its effect is equal to that of the fourth-order dispersion β4[33, 34, 3].

From the deviation between the red and black lines in Figure 5 , we can also detect the impact of δγ. It only accelerates the pulse’s split when the self-steepening effect is ignored (s = 0 in Figure 5(a) ). This is similar to the self-phase modulation-broadening spectral and oscillation. The high nonlinear γaccelerating pulse’s split is validated in [35, 36]. If s ≠ 0 ( Figure 5(b) ), δγsimultaneously leads to the split pulse’s redshift.

Generally, we do not take δγinto account, so we should clarify in which case it creates impact. Compared (c) with (b) in Figure 5 , the red lines change little means that δβ3has a tiny relationship with γ. But with the increase of γ( Figure 5(c) ), the split pulse’s redshift is strengthened, so δγhas a relationship with γ. In Figure 6 , the pulse is not split until z = 9 LD, and the black line with δγis completely overlapped by the red line without δγ, so the high second-order dispersion β2 results in the impact of δγcovered and the impact of δβ3weakened. Therefore, only in the zero-dispersion regime, δγshould be taken into account in the simulation of pulse shape.

Figure 6.

The pulse shapes with and withoutδγ.β2=21.7ps2/km,s=0.01,γ=1.3/km/W. Other parameters are the same asFigure 5.

So, we can utilize δβ3to determine the fourth-order dispersion coefficient β4. Fiber parameters are listed in Table 1 . The process is shown in Figure 7 , and the dispersion operator including β4is D̂=i2ω2β2i6ω3β3+i24ω4β4.


Table 1.

Fiber parameters.

Figure 7.

The process of calculatingβ4.

Table 2 is the average of β4. They are different from those determined by FWM or MI where β4is related to power and broadening frequency [35, 36]. By our method, the fourth-order dispersion is also a function of distance, and every type of fibers has its special average β4which reveals the characteristic of fibers. These values are similar to those experiment results in highly nonlinear fibers [35, 36]. Although we take the higher-order nonlinear effect δγinto account which upgrades the pulse’s symmetrical split and redshift, the items isδγu2u/tand iδγexp2αzu2uhave a very tiny contribution to β4, only 10−26 ps4/km quantity order for the typical SMF. Here, the impact of δγis hidden by the relative strong β2.

Z = 1.5LDZ = 5LDZ = 50LD

Table 2.

The average.

Units (ps4/km).

4. Traveling wave solution of NLSE for ASE noise

4.1 The in-phase and quadrature components of ASE noise

The field including the complex envelopes of signal and ASE noise is:


where ulztand Alztare the complex envelopes of signal and ASE noise, respectively [37, 38]. Nis the channel number. ASE noise generated in erbium-doped fiber amplifiers (EDFAs) is Al0t=AlR0t+iAlI0t, AlR0tand AlI0tare statistically real independent stationary white Gaussian processes, and AlR0t+τAlR0t=AlI0t+τAlI0t=nsphvlGl1Δvlδτ. In the complete inversion case, nsp=1. his the Planck constant. Glis the gain for channel l.

Substituting Eq. (47) into (1), we can get the equation that Alztsatisfies:


So, the in-phase and quadrature components of ASE noise obey:


We now seek their traveling wave solution by taking [37] AlR=ϕξ,AlI=φξ, and ξ=tcz.

Then, (49) and (50) are converted into


(52) is differentiated to ξ


Replacing ϕand ϕ'''in (53) with (51) and the differential of (51), there are


From (51) and (54), we can easily obtain




In the above calculation process, B, c, and kshould be regarded as constants, and AlR,AlIare the functions of the solo variable ξ, respectively.

4.2 Probability density function of ASE noise

Because AlRand AlIhave been solved, the time differentials of (49) and (50) can be calculated. Thus, the stochastic differential equations (ITO forms) around AlRand AlIare




Now, they can be regarded as the stationary equations, and we can gain their probabilities according to Sections (7.3) and (7.4) in [39]. By solving the corresponding Fokker-Planck equations of (60) and (61), the probabilities of ASE noise are


C,Care determined by +pdp=1. Compared with [40], these probabilities of ASE noise take dispersion effect into account. This is the first time that the p.d.f. of ASE noise simultaneously including dispersion and nonlinear effects is presented.

(66) and (67) are efficient in the models of Gaussian and correlated non-Gaussian processes as our (49) and (50). Obviously, the Gaussian distribution has been distorted. They are no longer symmetrical distributions, and both have phase shifts consistent with [40], and as its authors have expected that “if the dispersion effect was taken into account, the asymmetric modulation side bands occur.” The reasons are that item iβ2ωltAlztin (48) brings the phase shift and item β222t2Alztbrings the expansion and induces the side bands, the self-phase modulation effects, and the cross-phase modulation effects. Their synthesis impact is amplified by (66) and (67) and results in the complete non-Gaussian distributions.

5. Conclusion

NLSE is solved with small-signal analyses for the analyses of MI, and it can be broadened to all signal formats. The equation can be solved by introducing the Green function in the time domain, and it is used as the tool for the estimations of high-order dispersion and nonlinear coefficients. For the conventional fibers, SMF, NZDSF, and DCF, the higher-order nonlinear effect contribution to β4 can be neglected. This can be deduced that each effect has less impact for another coefficient’s estimation. The Green function can also be used for the solving of 3 + 1 dimension NLSE.

By the traveling wave methods, the p.d.f. of ASE noise can be obtained, and it provides a method for the calculation of ASE noise in WDM systems. So, the properties of MI, pulse fission, coefficient value, and ASE noise’s probability density function are also discussed for demonstrations of the theories.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Jing Huang (December 10th 2018). Nonlinear Schrödinger Equation, Nonlinear Optics - Novel Results in Theory and Applications, Boris I. Lembrikov, IntechOpen, DOI: 10.5772/intechopen.81093. Available from:

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