## Uncertainty relation for the optimization of optical-fiber transmission systems simulations

Optics Express, Vol. 13, Issue 10, pp. 3822-3834 (2005)

http://dx.doi.org/10.1364/OPEX.13.003822

Acrobat PDF (416 KB)

### Abstract

The mathematical inequality which in quantum mechanics gives rise to the uncertainty principle between two non commuting operators is used to develop a spatial step-size selection algorithm for the Split-Step Fourier Method (SSFM) for solving Generalized Non-Linear Schrödinger Equations (G-NLSEs). Numerical experiments are performed to analyze the efficiency of the method in modeling optical-fiber communications systems, showing its advantages relative to other algorithms.

© 2005 Optical Society of America

## 1. Introduction

2. Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. **21**, 61–68 (2003). [CrossRef]

## 2. Presentation of the step-size selection algorithm

*et al.*[2

2. Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. **21**, 61–68 (2003). [CrossRef]

## 2.1 Standard SSFM for G-NLSE without background losses

*A*=

*A*(

*z*,

*t*) is the slowly varying field amplitude,

*β*and

_{2}*β*are, respectively, the second and third order dispersion coefficients,

_{3}*γ*is the nonlinear coefficient (optical Kerr effect),

*z*is the position along the fiber, and t is the local time, i.e., in a reference frame that travels with the average group-velocity of the pulse. (Note that we are not considering here higher order dispersion coefficients

*β*,

_{4}*β*,… although high order dispersion and attenuation can be included in the NLSE, see below). Equation (1) can be written as

_{5}**D**and

**N**defined as

*h*(

*h*can vary along the fiber) and the field is then propagated at each step assuming first no dispersion (

*β*=

_{2}*β*=0) and then no nonlinearity (

_{3}*γ*=0). The G-NLSE with no dispersion is directly solved in the time domain, whereas the G-NLSE without the nonlinear term is solved in the frequency domain using a Fast Fourier Transform (FFT) routine:

**F**is the FFT (

**F**

^{-1}is the IFFT or inverse FFT) and

**D**̃(

*ω*) is the dispersion operator in the frequency domain, given by replacing the differential operator ∂/∂t in Eq. (3) by

*i*ω(ω is the frequency). Observe that

**N**is a multiplicative factor in the time domain, whereas

**D**̃ is also a multiplicative factor, but in the Fourier domain.

*h*and

*dt*, respectively. Here we concentrate on the algorithm for the choice of

*h*, which, as pointed out in reference [2

2. Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. **21**, 61–68 (2003). [CrossRef]

*dt*is changed.

*h*is small enough such that variation in

*N*is negligible in an interval from

*z*to

*z*+

*h*, is

**D**,

**N**]=

**DN**-

**ND**is the commutator between

**D**and

**N**. Thus, from (4), (5) and (6), to the lowest order in

*h*, the error in the SSFM at each sample time point

*t*is -(1/2)

*h*

^{2}[

**D**,

**N**]

*A*(

*t*).

*relative local error*, given by

*A*is the numerical result after one step numerical propagation and

_{n}*A*is the analytical result, supposed to be somehow known or estimated, and ‖

_{a}*A*‖=(∫|

*A*(

*t*)|

^{2}

*dt*)

^{1/2}. Using Eqs. (4), (5), and (6) it is easy to show that

4. Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. **16**, 1304–1306 (2004). [CrossRef]

*relative local error δ*value to determine the longitudinal step size at each given desired target precision. In the method we are presenting here, we use another parameter to the adaptive control of the step size, based in the averaged error,

**D**,

**N**]〉, in “state”

*A*as given in Quantum Mechanics (QM). Thus, the field amplitude

*A*plays the analogous role of the wave function in quantum mechanics. We will show trough this article that this error

*ε*, defined with the help of QM, is also a good parameter to determine how much the numerical results differs from the exact result. Since the error at each sample time point

*t*is, as mentioned before, - (1/2)

*h*

^{2}[

**D**,

**N**]

*A*(

*t*), then ε tends to zero when the error at all sample times goes to zero and, moreover, ε gives some kind of “averaged” value of the error. In Section 4.2 we return to discuss theoretically the validity of our quantum mechanically defined error ε as a good parameter to guarantee the convergence of the simulations, comparing it with the

*relative local error δ*(Eqs. (7) and (8)) defined in Ref. [2

**21**, 61–68 (2003). [CrossRef]

*δ*, the simulations converge if the quantum mechanically defined error ε is bounded by some small value. We discuss this point theoretically in Section 4.2, but, before, in Section 3, we perform numerical experiments showing the usefulness of ε as a parameter to guarantee convergence in three cases of practical interest. Now we focus on how the uncertainty principle can be used to establish an upper bound to the value of ε, enabling the optimization of the SSFM through a correct choose of

*h*.

*it also applies to the operators*

**D**

*and*

**N**(

*since they are hermitian*) and takes the form [8

8. Although * N* is a non-linear operator, it involves only a multiplication operation and is considered constant in each interval. Eq. (10) follows from applying the Schwartz inequality to the functions [

*-<*

_{D}*>]*

**D***A*and [

*-<*

**N***>]*

**N***A*. Actually, this rigorous derivation determines an even smaller upper bound than that stated by eq. (10):

**C**〉

^{2}, the “quantum covariance,” is given by 〈C〉

*ΔD*and

*ΔN*and are standard deviation as defined in QM, calculated as

*ε*=(1/2)

*h*

^{2}|<[

**D**,

**N**]>|≤(1/2)

*h*

^{2}2Δ

*D*Δ

*N*,

*h*according to

*A*(

*z*,

*t*) as the wave-function of QM. After this point the analogy breaks down. In physics, the Heisenberg Uncertainty Principle expresses a limitation on accuracy of simultaneous measurement of canonically conjugated observables such as the position and momentum of a particle. There is no such an interpretation for Δ

*D*and Δ

*N*. Dispersion and nonlinearity can be simultaneously measured (or computed) to any degree of accuracy. Our method could be called, perhaps more precisely, the

*Schwartz inequality method*, but for the reasons just exposed, we prefer the UPM name.

*D*and Δ

*N*that are used in Eq. (12) to determine the step size

*h*for a chosen error ε (typically

*ε*=10

^{-3}). When modeling communication systems it is not necessary to re-calculate

*h*on every step. Normally, we call this routine on every 20 steps.

*N*can be calculated directly in the time domain using Eq. (9), but for the dispersion operator Δ

*D*, which involves a second order time-derivative, this can be computationally slow. Fortunately, we can do better if we realize that we can calculate QM averages (and thus QM variances) in the frequency domain with exactly the same result [9], i.e.

*A*̃=

*A*̃(

*z*,ω)=

**F**

*A*(

*z*,

*t*) is the Fourier transform of the field amplitude. For instance, the QM variance of the dispersion operator defined in Eq. (3) is

## 2.2 Inclusion of background loss

**L**and

**N**are such that Eq. (15) is easily integrated in the time-domain if

**N**acts alone and in the Fourier space if

**L**acts alone. There are many problems in physics where the SSF is used. Examples from optics are the wave equations describing spatial solitons or linear propagation in gradient index media [10

10. J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am. , **71**, 808–810 (1981). [CrossRef]

*t*is substituted by transverse coordinates, and the Fourier transform operates in the bi-dimensional wave-vector space), nonlinear surface waves [11] (where

*t*is the coordinate transverse to the surface), as well as several problems in Beam Propagation Methods. In many of these problems (including optical fibers) the physical system is not conservative on account for losses. In these cases the operators may not be hermitian and the UPM cannot be applied directly. We discuss now how to handle these situations.

**D**is hermitian but L is not. If the loss coefficient can be considered frequency independent over the field spectrum, α(

*z*,

*ω*)=

*α*(

*z*), then it commutes with

**N**and the UPM as given by Eq. (12) can be used. If this is not the case, then the quantum mechanically defined error ε is given by

*h*as

*D*and Eq. (12) can be used. However, if the field spectrum is broad and close to some resonance (for example, the water peak near 1.4 µm of conventional silica fibers), or in the cases of doped fibers (such as a distributed amplifier) or

*dispersion flattened*fibers with β

_{2}=β

_{3}=0, then Eq. (19) can be safely used.

## 2.3 Symmetrized SSFM

*h*/2, is used as the input field to perform the non-linear propagation [1]. Mathematically,

*h*, which can be easily shown applying Eq. (6) twice in Eq. (20). Because of the symmetric form of the above equation, this version of the SSFM is called Symetrized SSFM (S-SSFM). It is also known as the second-order SSFM [3] (the first order is the standard SSFM discussed in Section 2.1).

*h*

^{3}(1/6){(

**N**+2

**D**)[

**D**,

**N**]+[

**D**,

**N**](

**D**+

**2**N)}, the Schwartz inequality can be applied to establish an upper limit to the QM average of this operator. We do not show here explicitly the result because it is rather cumbersome, but its derivation is straightforward using Eq. (10).

*h*, even when the S-SSFM is used. This is what we do in our program, i.e., we calculate

*h*using Eq. (12) even though our algorithm uses the symmetrized SSFM. Although this procedure is conservative, we show below that it is more efficient than other methods in a wide range of situations. In principle, we could improve the efficiency of our program using an UPM criterion extended to case of the S-SSFM, but it is not clear that calculating the step in a more complicated expression than Eq. (12) could be faster than just using a few more FFT’s. This is an open question for future works.

## 3. Numerical results

**21**, 61–68 (2003). [CrossRef]

12. B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys. , **155**, 456–467 (1999). [CrossRef]

**21**, 61–68 (2003). [CrossRef]

5. Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004). [CrossRef]

*global relative error δ*as an indicator of the accuracy. This is defined by Eq. (7) but with

_{G}*A*(

_{n}*t*) being the numerical field and

*A*(

_{a}*t*) the analytical solution at the

*end of the propagation*(and not after a single step, as in the definition of

*δ*). In the case of isolated solitons, for instance, an exact solution actually exist, and this is used for

*A*(

_{a}*t*). In the general case, however,

*A*(

_{a}*t*) is computed numerically using the S-SSFM with a very small

*h*. This is done running several simulations with decreasing constant

*h*sizes until the results from two successive simulations are equal at all sample points within machine precision. We have chosen to use

*δ*as an accuracy indicator because it has been used in two recent publications on the efficiency of the SSFM in modeling optical communication systems [2

_{G}**21**, 61–68 (2003). [CrossRef]

5. Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004). [CrossRef]

*δ*.

_{G}## 3.1 Second-order soliton

*A*(

*t*)=2

*η*(-β

_{2}/

*γ*)

^{1/2}sech(η

*t*) with η=0.44 ps

^{-1}, β

^{2}=-0.1 ps

^{2}/km and

*γ*=2.2 W

^{-1}/km. For these parameters, the peak power is 35 mW and the FWHM pulse duration is 4 ps [2

**21**, 61–68 (2003). [CrossRef]

^{10}=1024 sample points and a simulation time windows of 50 ps. In Fig. 1 we show the results. For completeness, we show the results not only for the UPM and LEM, but also for the Non-Linear Phase-Rotation Method (NPRM), which has been extensively used (and its still used in commercially available software packages). The NPRM is thoroughly discussed in Ref. [2

**21**, 61–68 (2003). [CrossRef]

^{-3}the LEM is more efficient. This happens because the LEM warrants accuracy to third order in

*h*, while the implemented UPM warrants accuracy to second order.

## 3.2 Soliton collision

*A*(

*t*)=η(-β

_{2}/

*γ*)

^{1/2}sech(η

*t*), where the values of η, β

_{2}, and

*γ*are the same as in the previous section, giving a pulse duration of 4 ps and a peak power of 8.8 mW [2

**21**, 61–68 (2003). [CrossRef]

^{-4}, while the LEM is more efficient if extreme accuracy is desired.

## 3.3 WDM system

**21**, 61–68 (2003). [CrossRef]

^{-5}, far from the region of practical interest in system design (which is between 10

^{-1}and 10

^{-3}[2

**21**, 61–68 (2003). [CrossRef]

^{-1}and 10

^{-3}. Moreover, the LEM curve crosses the others, becoming more efficient, at global accuracies of ~10

^{-4}, closer to the region of practical interest. This results can be generalized so that, for any given input field, the longer the distance to be simulated, the greater the region of accuracies in which the LEM becomes the most efficient method.

## 4. Discussion

### 4.1 Variation of method parameters

**21**, 61–68 (2003). [CrossRef]

*ε*in the UPM case and

*δ*in the LEM case) for different systems.

^{-3}the method parameter varies in both cases (UPM and LEM) over one order of magnitude for the systems considered. Thus both methods are equally robust in the sense that no matter the system to be simulated, the method parameter value to achieve this desired target accuracy are not significantly different. As a comparison, the WOM exhibits variations of 2 to 3 order of magnitude larger than the LEM [2

**21**, 61–68 (2003). [CrossRef]

^{-4}the UPM becomes more sensitive to the specific system. Also, it is clear that the curves corresponding to the 10- and 50- km WDM propagation are more separated in the UPM than in the LEM graphs. We attribute this observed larger variance on the method parameter to the average error ε used in the UPM.

## 4.2 Quantum mechanically defined error versus relative local error (ε vs. *δ*)

*h*in the LEM is determined by a recursive algorithm that is time consuming, whereas in the UPM

*h*is determined using an analytical expression, Eq. (12), which is very fast to be calculated. This explains why the UPM is more efficient for moderate accuracy.

*N*or Δ

*D*are very small and Eq. (12) gives an excessively large step, leading to errors at some particular time points. This might happen because Eq. (12) warrants only that the time integrated error is bounded. In the LEM, δ is defined by an integral of the squared absolute values at each sample point, thus warranting that if δ goes to zero, then the error is bounded at all time points. This is not assured in the UPM. Since in Eq. (9) we first integrate the error and then take the absolute value of the integral, if the QM averaged error goes to zero this does not ensure that the error is bounded at all time points. Because of that the UPM will show a large sensitivity to the system condition to achieve a given target accuracy, as said in Section 4.1 and shown in Fig. 4.

*ε*works even better than

*δ*as an indicator of accuracy and as a parameter to decide the step size to be taken in a wide range of situations of practical interest. Second, the cases in which the real and imaginary parts of the argument in the integral in Eq. (9) are perfectly canceled during the integration seem to be very rare. During one step propagation the error at half of the points must have exactly the same and opposite value than in the other half. But generally the errors at different sample points do not cancel each other and, moreover, in many cases they have the same sign. In a lossy fiber, for instance, if the step is too large, the effects of the attenuation will be over-estimated, reducing the field magnitude at all sample points.

*N*and Δ

*D*do not vanish unless there is no nonlinearity or no dispersion at all. In both cases the UPM gives quickly the correct answer: you can take

*h*as being the total fiber length (in both cases there is an exact solution either in the time domain or in the frequency domain). The UPM was not intended to solve problems such as supercontinuum generation, self-steppening, or the propagation of a cw laser; although we feel that it could be easily extended to deal accurately with these problems.

## 4.3 Number of FFTs as a measure of computational cost

*D*and Δ

*N*are not mayor time-consuming computer tasks and, then, if this operation take longer than performing an FFT. In such case, the number of FFTs would not be a good parameter to quantify the computational cost of the UPM.

*h*at each step, which, as shown by Eq. (7), involves the calculus of two integrals. Roughly speaking, 5N arithmetical operations (N is the number of sample points) are necessary to calculate δ (N subtractions in the numerator, 2N multiplication to calculate the square values of the field at each sample point at the numerator and at the denominator and 2N sums to integrate them). Similarly, 15N arithmetical operations are necessary to compute Δ

*D*and Δ

*N*in the UPM. On the other hand, it is well-known that each FFT performs between 4Nlog

_{2}N and 5Nlog

_{2}N operations, depending on the implemented algorithm.

^{15}sample points are necessary (here, for instance, we used 2

^{16}to simulate the simple WDM system of Section 3.3). Using these numbers in the estimates above, the number of arithmetical operations performed to the calculus of

*h*in the UPM is less than 10% of that used by the FFT. Of course, care must be taken when simulating systems with very small number of sample points because in this case the time consumed to calculate

*h*can be of the order of the time consumed by the FFT algorithm.

## 5. Conclusions

**21**, 61–68 (2003). [CrossRef]

13. Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys. , **148**, 397–415 (1999). [CrossRef]

*h*but we implemented the S-SSFM, which enables the use of an even larger

*h*to obtain the same accuracy. As said in Section 2.3, how much improvement it would be obtained if we used the UPM extended to the S-SSFM case is left as an open question to be addressed in future works. Second, because we use inequality (10) in a conservative way, using always the worst case, i.e., the equal case, with gives the smallest

*h*. If the right hand side of inequality (10) is much larger than the left hand side, the UPM is being conservative. It is well known that the equal sign holds only for gaussian wave-packets [7]. So, unless the field is gaussian and retains a gaussian shape during the propagation, the UPM is being conservative. The challenge to improve the UPM is to find situations in which one can be sure that (10) is far from the equality establishing how much the step can be increased in relation to the equal sign case.

**21**, 61–68 (2003). [CrossRef]

5. Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004). [CrossRef]

## Acknowledgments

## References and links

1. | G. P. Agrawal, |

2. | Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems” IEEE J. of Lightwave Technol. |

3. | G.M. Muslu and H.A. Erbay,, “Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation” Mathematics and Computers in Simulation (In press). |

4. | Malin Premaratne, “Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems” IEEE Photon. Technol. Lett. |

5. | Xueming Liu and Byoungho Lee, “A Fast Method for Nonlinear Schrödinger Equation,” IEEE Photon. Technol. Lett.15, 1549–1551 (2003). See also Xueming Liu and Byoungho Lee, “Effective Algorithms and Their Applications in Fiber Transmission Systems” Japanese Journal of Applied Physics, 43, 2492–2500, (2004). [CrossRef] |

6. | Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss, A.A. Maraudin, and J. Math. Phys, 3, 771–777 (1962). |

7. | E. Merzbacher, |

8. | Although -<_{D}>]DA and [-<N>]NA. Actually, this rigorous derivation determines an even smaller upper bound than that stated by eq. (10): C〉^{2}, the “quantum covariance,” is given by 〈C〉 |

9. | This is a straightforward consequence of Parseval’s theorem. In quantum mechanics the Fourier transform of an operator is nothing but the same operator expressed in the conjugate representation. Of course, the QM average value of an observable operator can not depend on the representation. |

10. | J. Van Roey, J. van der Donk, and P.E. Lagasse, “Beam propagation method: analysis and assessment” J. Opt. Soc. Am. , |

11. | N.N. Akhmediev, V.I. Korneev, and Yu.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams” Sov. Phys. JETP , |

12. | B. Fornberg and T.A. Driscoll, “A fast spectral algorithm for nonlinear wave equations with linear dispersion” J. Comp. Phys. , |

13. | Q. Chang, E. Jia, and W. Suny, “Difference schemes for solving the generalized nonlinear Schrödinger equation” J. Comp. Phys. , |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 4, 2005

Revised Manuscript: May 3, 2005

Published: May 16, 2005

**Citation**

A. Rieznik, T. Tolisano, F. A. Callegari, D. Grosz, and H. Fragnito, "Uncertainty relation for the optimization of optical-fiber transmission systems simulations," Opt. Express **13**, 3822-3834 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3822

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### References

- G. P. Agrawal, Nonlinear Fiber Optics (London, U.K. Academic, 1995).
- Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis Menyuk, �??Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems�?? IEEE J. of Lightwave Technol. 21, 61-68 (2003). [CrossRef]
- G.M. Muslu and H.A. Erbay, �??Higher-order split-step Fourier schemes for generalized nonlinear Schrödinger equation�?? Mathematics and Computers in Simulation (In press).
- Malin Premaratne, "Split-Step Spline Method fpr Modeling Optical Fiber Communications Systems�?? IEEE Photon. Technol. Lett. 16, 1304-1306 (2004). [CrossRef]
- Xueming Liu and Byoungho Lee, �??A Fast Method for Nonlinear Schrödinger Equation,�?? IEEE Photon. Technol. Lett. 15, 1549-1551 (2003). See also Xueming Liu and Byoungho Lee, �??Effective Algorithms and Their Applications in Fiber Transmission Systems�?? Japanese Journal of Applied Physics, 43, 2492-2500, (2004). [CrossRef]
- Eq. (6) is a variant of the so-called Baker-Hausdorff formula. See, for example, G.H. Weiss and A.A. Maraudin, J. Math. Phys, 3, 771-777 (1962).
- E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
- Although N is a non-linear operator, it involves only a multiplication operation and is considered constant in each interval. Eq. (10) follows from applying the Schwartz inequality to the functions [D �?? <D>]A and [N �?? <N>]A. Actually, this rigorous derivation determines an even smaller upper bound than that stated by eq. (10): where <C>2, the �??quantum covariance,�?? is given by ...
- This is a straightforward consequence of Parseval�??s theorem. In quantum mechanics the Fourier transform of an operator is nothing but the same operator expressed in the conjugate representation. Of course, the QM average value of an observable operator can not depend on the representation.
- J. Van Roey, J. van der Donk, and P. E. Lagasse, �??Beam propagation method: analysis and assessment�?? J. Opt. Soc. Am., 71, 808-810 (1981). [CrossRef]
- N. N. Akhmediev, V. I. Korneev, and Yu.V. Kuz�??menko, �??Excitation of nonlinear surface waves by Gaussian light beams,�?? Sov. Phys. JETP 61, 62-67 (1985).
- B. Fornberg and T. A. Driscoll, �??A fast spectral algorithm for nonlinear wave equations with linear dispersion,�?? J. Comp. Phys. 155, 456-467 (1999). [CrossRef]
- Q. Chang, E. Jia, and W. Suny, �??Difference schemes for solving the generalized nonlinear Schrödinger equation,�?? J. Comp. Phys. 148, 397-415 (1999). [CrossRef]

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