Numerical stability of solitons waves through splices in optical

fibers

Camila F. de Oliveira a,1

, Paulo L. Natti a,2,*

, Eliandro R. Cirilo a,3, Neyva M.

L. Romeiro a,4

and Érica R. Takano Nattib,5

a Universidade Estadual de Londrina, Departamento de Matemática (DMAT/UEL)

C.P. 6001, Londrina, Paraná, Brazil, 86051-990

b Pontifícia Universidade Católica do Paraná

Rua Jóquei Clube, 458, Londrina, Paraná, Brazil, 86067-000

1 E-mail: ca_fogaca@yahoo.com.br.

2 Corresponding author. E-mail: plnatti@uel.br. Web page: http://www.mat.uel.br/plnatti

Tel: 55-43-33714226. Fax: 55-43-33714216

3 E-mail: ercirilo@uel.br.

4 E-mail: nromeiro@uel.br.

5 E-mail: erica.natti@pucpr.br.

ABSTRACT. The propagation of soliton waves is simulated through splices in optical fibers, in which

fluctuations of dielectric parameters occur. The mathematical modeling of these local fluctuations of dielectric

properties of fibers was performed by Gaussian functions. By simulating soliton wave propagation in optical

fibers with Gaussian fluctuations in their dielectric properties, it was observed that the perturbed soliton

numerical solution presented higher sensitivity to fluctuations in the dielectric parameter β, a measure of the

intensity of nonlinearity in the fiber. In order to verify whether the fluctuations of β parameter in the splices of

the optical fiber generate unstable solitons, the propagation of a soliton wave, subject to this perturbation, was

simulated for large time intervals. Considering various geometric configurations and intensities of the

fluctuations of parameter β, it was found that the perturbed soliton wave stabilizes, i.e., the amplitude of the

wave oscillations decreases for increasing values of propagation distance. It is concluded that the propagation of

perturbed soliton wave presents numerical stability when subjected to local Gaussian fluctuations (perturbations)

of the dielectric parameters of the optical fiber.

Keywords: perturbed soliton propagation; optical fiber splice; numerical stability; Gaussian fluctuation; finite

difference method; Gauss-Seidel method.

PACS numbers: 02.60.Lj, 02.70.Bf, 42.65.Tg, 42.81.Dp

1. Introduction

In context of optical communication via solitons, the experiments performed in the late nineties

generated encouraging results. In 1998, Thierry Georges and his team at France Telecom, combining

optical solitons of different wavelengths, demonstrated data transmission of 1 terabit per second [1],

and in 2000, Algety Telecom, then located in Lannion, France, developed undersea telecommunication

equipment for the transmission of optical solitons. However, these promising results were not

translated into actual commercial soliton system deployments, in either terrestrial or submarine

systems, chiefly due to the Gordon–Haus (GH) jitter effect [2]. GH jitter requires a sophisticated and

expensive compensatory solution that ultimately makes the Dense Wavelength Division Multiplexing

(DWDM) soliton transmission unattractive. Consequently, in the last decade the long-haul soliton

transmission has remained as a subject of laboratory research. On the other hand, several solutions

have been proposed to minimize the jitter effect, such as Raman fiber amplifiers [3], tapering

dispersion fiber spans [4], sliding frequency guiding filters [5], in-line synchronous modulation [6],

among others. A review of the basic physics of the dynamics of solitons and futuristic applications of

solitons in optical communication can be found in [7].

In recent years, an increase in the number of theoretical and experimental works on soliton

communications, that aim to overcome the many well-known problems and improve the methods

already proposed, was published. Such studies approach themes related to the new soliton generation

processes [8,9], soliton propagation processes [10,11] and soliton stabilization processes [12-14] in

optical fibers.

This work is about propagation and stability of femtosecond optical solitons in fibers. The

propagation of these waves in optical fibers is affected by several disturbing processes. Usually, the

most important ones are group velocity dispersion (chromatic dispersion) and optical Kerr effect

(intensity dependence of the refractive index). Under certain circumstances, however, the effects of

Kerr nonlinearity and dispersion can just cancel each other, so that the temporal and spectral shapes of

the pulses is preserved even over long propagation distances. Taking only these disturbances into

account, the pulse propagation is a soliton described by a system of coupled nonlinear Schrödinger

differential equations [15-17]. The most remarkable fact about soliton waves is, actually, not the

possibility of balance of dispersion and nonlinearity, but rather that soliton solutions of nonlinear wave

equation are very stable: even for substantial deviations of the initial pulse from the exact soliton

solution, the pulse automatically finds the correct soliton shape.

On the other hand, to describe real-world fiber-optic systems, it is more realistic to include further

disturbing effects such as influence of fusion splice [18], Rayleigh scattering [19], high-order

dispersion and high-order nonlinearities [20], soliton self-steepening, Raman effect and self-frequency

shift [10], polarization-mode dispersion [13], nonlinear phase noise [21], among others [22].

It should be observed that the perturbed coupled nonlinear Schrödinger differential equations

systems, which describe wave propagation in real optical fibers, do not present analytical solution. In

the literature there are several numerical approaches whose objective is to describe the propagation of

perturbed solitons in dielectric environments, most of them using the finite difference method

[14,23,24] or the finite element method [25,26]. On the other hand, to solve numerically the resulting

system of equations, the authors use various methods like Newton's method [26], Crank-Nicolson

method [14], Runge-Kutta Method [27], among others. A review of the several numerical procedures

applied to describe the propagation of solitons in optical fibers is found in [28].

In a previous work [29], we described the propagation of soliton waves in ideal optical fibers

through a procedure based on the finite difference method and relaxation Gauss-Seidel method. By

comparing the obtained numerical results with the known analytical results, the validation of the

developed numerical procedure has been verified.

In this work, aiming to study the propagation and the stability of perturbed soliton waves, through

splices in optical fibers, the general numerical procedure developed in [29] is used. In Section 2 we

present the soliton analytical solutions of the coupled nonlinear differential equations system in case of

ideal )2( optical dielectric fibers. In Section 3, the numerical procedure to study the propagation of

the perturbed solitons in optical fibers is described. In Section 4 the propagation of perturbed soliton

waves is simulated through splices in optical fibers, where fluctuations of the dielectric parameters of

the fiber occur. The mathematical modeling of these local fluctuations of the dielectric properties was

performed by Gaussian functions with various geometric configurations and intensities. In Section 5

the main results of this work are presented.

2. Solitons in ideal dielectric fibers

This section studies the coupled non-linear complex partial differential equations (PDE) system,

obtained from Maxwell’s equations, which describe the longitudinal propagation of two coupled

electromagnetic waves (fundamental and second harmonic modes) in ideal ( )2χ dielectric optical

fibers [20]. The detailed mathematical modeling of this PDE system can be found in [16,17]. This

PDE system is given by

( ) 0-exp2

- 2

*

121

21 =+

∂

∂

∂

∂βξiaa

s

ar

ξ

ai (1)

( ) 0exp2

-- 212

22

22 =+∂

∂

∂

∂

∂

∂βξia

s

aα

s

aδi

ξ

ai ,

where 1-=i is the imaginary unit, ),(1 sξa and ),(2 sξa are complex variables that represent the

normalized amplitudes of the electrical fields of the fundamental and second harmonic waves,

respectively, with ),(*1 sξa and ),(*

2 sξa as their complex conjugates. The independent variable s has

spatial character, whereas the independent variable ξ has temporal character.

The real parameters α , β , δ and r , in (1), are related with the dielectric properties of the optical

fiber. The β quantity is a measure of the intensity of nonlinearity in the optical fiber, or a measure of

the generation rate of the second harmonic. The α quantity measures the relative dispersion of the

group velocity dispersion (GVD) of fundamental and second harmonic waves in the optical fiber. For

values 1>α , the second harmonic wave has higher dispersion than the fundamental wave and for

values 1<α , it is the fundamental wave that has higher dispersion. The r quantity is the signal of the

fundamental GVD wave. When 1r , the fundamental wave is in normal dispersion regime, but if

1-=r , the fundamental wave is in anomalous dispersion regime. Finally, parameter δ measures the

difference of group velocities of fundamental and second harmonic waves, so it accounts for the

presence of Poynting vector walk-off that occurs in birefringent media. It should be noticed that is

possible to choose the characteristics (velocity, width, amplitude, etc.) of the wave to be propagated in

the optical fiber, selecting or proposing materials with the appropriate α , ,β δ and r dielectric

properties [16,17].

The PDE system (1) presents soliton solutions [17], given by

+±= )-2

( )2-(2

3),(

2

1 βrα

δrα

rαsξa ])

-2-()

-2(

)-2(2

1[sech

22 +±× ξ

rα

δrsβ

rα

δ

αr

]} )-2

( -)-2

-)-2()-2(2

)54(( [ {exp 2

2

srα

δξ

αr

βr

αrrα

αrδri× (2)

)-2

()2-(2

3),(

2

2 βrα

δ

rα

rsξa += ])

-2-()

-2(

)-2(2

1[sech

22 +±× ξ

rα

δrsβ

rα

δ

αr

]})-2

(-)2-2

-)-2()-2(2

)54(( [2{exp 2

2

srα

δξ

β

αr

βr

αrrα

αrδri +× . (3)

In [29] a numerical procedure was developed based on the finite difference method and relaxation

Gauss-Seidel method to solve the propagation of soliton waves in optical fibers described by the PDE

system (1). By comparing the obtained numerical results with the known analytical solutions (2-3), the

validation of the developed numerical procedure was verified. This numerical development is

presented in the next section.

3. Numerical model for the propagation of solitons in optical fibers

The numerical scheme developed in [29] to solve the PDEs system (1) consists in approximating the

derivates by finite differences and resolving the algebraic system resulting from the discretization,

implicitly, by means of the relaxation Gauss-Seidel method [30,31].

The system (1) is numerically resolved in domain [ ] [ ]LLTsξ ,-,0 ×=× , where LT , . By

discretizing the variables ( )jk

asξa,111 ,

+≡ and ( )

jkasξa

,122 ,+

≡ , for max,...,1,0 kk = and njj ,...,2,1= ,

where maxk is denominated the last advance in ξ and nj the maximum number of points in s , the

propagation domain of the soliton waves is defined by a discretized computational grid of njk ×max

points, as represented in figure 1.

Fig. 1. Computational domain of the propagation of soliton waves.

Thus, by means of the method of finite differences, approaching the temporal derivates by

progressive differences, and the spatial derivates by central differences [30], the following linear

systems are generated from the differential equations (1), namely,

( )++=++++

]i-exp-[1

,,,1,11-j,1,1 2*11

11

11

111 kpoEW

p

tβaaaAaAaAA

ajkjkjkjkkjk

(4)

( )++=+++++

]iexp)(-[1 2

122

22

22

22 ,1,1,11-j,1,1 kpoEWp

tβaaAaAaAA

ajkjkjkkjk

where

( )21

ΔΔ s

r

ξ

iAp += WE AA 11

21

2 s

rAW

ξ

iApo Δ

1 =

( )22

ΔΔ s

α

ξ

iAp +=

( )22

Δ2Δ2 s

α

s

δiAE +=

( )2

22

Δ2Δ2-

s

α

s

δiAW +=

ξ

iAp Δ0

2 =

and in these approaches 1+k is the current time, k is the previous time, sΔ is the spatial quantity

discretized and ξΔ is the temporal quantity discretized, so that ξktk Δ= .

In this work, the linear system (4) is resolved by means of the relaxation Gauss-Seidel method [30-

32]. Consider this linear system for jk

a,11 +

, given explicitly by

( )++=+++

]i-exp-[1

2,2,2,3,11,12,1 2*11

11

11

111 kpoEW

p

tβaaaAaAaAA

akkkkkk

( )++=+++

]i-exp-[1

3,3,3,4,12,13,1 2*11

11

11

111 kpoEW

p

tβaaaAaAaAA

akkkkkk

( )++=+++

]i-exp-[1

4,4,4,5,13,14,1 2*11

11

11

111 kpoEW

p

tβaaaAaAaAA

akkkkkk

...

( ) ( ) ( ) .]i-exp1-,1-,-[1

2*11

11

11

111 1-,,12-j,11-,1 kpoEW

p

tβnjkanjkaaAaAaAA

anjknjknknjk

++=+++

It can be written in compact form as

,1

11

111

11,11-j,1,

,1

++=

+++

+p

EW

A

aAaABa

jkkjk

jk

where ( )kpo tβiaaaABjkjkjkjk

-exp-,,,, 2

*11

11 = with 1-,...,2 nji = .

From the initial condition j

a,01 , given by soliton solution (2), and imposing the contour conditions

01,11 =

+ka and 0

,11 =+ njk

a , for L sufficiently large, )(1 ,1

ljk

a+

is iteratively calculated by means of the

equations

,1

)1-(1

1)(1

1)(1

)(1

1,11-j,1,

,1

++=

+++

+p

lE

lW

l

l

A

aAaABa

jkkjk

jk (5)

where ...3,2,1=l is the iterative level, ( )klll

pol tβiaaaAB

jkjkjkjk-exp- )(

2)(*

1)(

11)(1

,,,,= , until the stop

criterion is fulfilled, namely,

6-)1-(

1)(

11-j2

10|-|max,1,1

<++≤≤

ll

njjkjk

aa . (6)

This method consists in determining )(1 ,1

ljk

a+

by using the already known components of

)1-(1 1,1

ljk

a++

and )(1 1-,1

ljk

a+

, with the advantage of not requiring the simultaneous storage of the two

vectors )1-(1 1,1

ljk

a++

and )(1 1-j,1

lk

a+

at each step. Likewise, )(2 ,1

ljk

a+

is resolved.

It should be noticed that in equations (5)-(6) the value 0.1=w was used for the parameter of

relaxation [32]. Such value corresponds to the optimal relaxation parameter in relation to the

variations of the dielectric parameters α , β and δ of system (1). Figure 2 presents the flowchart of

the numerical code developed for PDEs system (1).

Fig. 2. Flowchart of the numerical code developed to obtain the numerical soliton solutions.

4. Stability of perturbed soliton waves

In our previous work [29] the dielectric parameters α , β e δ in (1) were considered constant, so that

PDE system (1) presented soliton solutions. On the other hand, in real optical fibers, the dielectric

parameters α , β e δ are not constant. This section will study how localized fluctuations of the

dielectric parameters occurring in the splices of optical fibers affect the propagation of soliton waves.

The fusion splicing process of optical fibers is usually realized by means of an electric arc, but it

can be realized by laser, gas flame, or tungsten filament through which current is passed. The fusion

splicing apparatus consists of two fixtures on which the fibers are mounted. The fibers in the apparatus

are aligned and then fused. In fusion splicing, the splice loss is a direct function of the angles of

alignment and quality of the two fiber-end-faces. A splice loss under 0.1 dB is typical. Alternatives to

fusion splicing includes using optical fiber connectors or mechanical splices, both of which have

higher insertion losses, lower reliability and higher return losses than fusion splicing.

In the fusion splicing process, the local dielectric properties of the optical fiber are modified. To

analyze the stability of the perturbed soliton waves as a function of the fluctuations of the dielectric

parameters in the fiber is our aim. In this case, only numerical solutions are possible, since the

propagated waves are not solitons given by (2-3) anymore.

4.1. A localized perturbation in the optical fiber

Initially, the discretization of the computational grid will be described. For variable s , the interval

9050- << s was established, with discretization 1-100.1Δ ×=s , whereas for variable ξ , the interval

was 500 << ξ , with discretization 3-100.1Δ ×=ξ . The geometry of the computational grid was

adjusted so that the physics of the propagation of the perturbed soliton wave is within the considered

computational domain.

In the sequence, the mathematical modeling of the fluctuation of the dielectric parameters in the

splices of the optical fibers will be considered, as well as such localized fluctuation affect the stability

of the propagation of waves. In the modeling of the optical properties along the dielectric fiber used in

this work, it is supposed that, in average, the dielectric parameters take values 0.1-=r , 4/1-=α ,

2/1-=β and 4/1-=δ . It is also supposed that in the areas surrounding the splices of two optical

fibers, the optical properties α , β e δ are altered according to a Gaussian function. In the following

simulations, Gaussian fluctuations of 5% in the values of the dielectric parameters are considered.

Parameterα : Figure 3 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the

fluctuations in parameter α , around 4/1-=α , with maximum amplitude corresponding to 5% of the

average value of the dielectric parameter. By establishing values 0.1-=r , 2/1-=β , 4/1-=δ and

varying α by means of the Gaussian function )])10-(-exp(05.01[)4/1-( 2ξα ×+= , there is the

occurrence of small variations in the amplitudes of ),(1 sξa and ),(2 sξa , not easily visualized in

figure 3. From the Gaussian function, it is observed that the fluctuations in α occur around 10=ξ .

Fig. 3. Numerical solutions ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)4/1-( 2ξα ×+= with

0.1-=r , 2/1-=β and 4/1-=δ .

Parameter δ : Figure 4 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the

fluctuations in parameter δ , around 4/1-=δ , with maximum amplitude corresponding to 5% of the

average value of the dielectric parameter. By establishing values 0.1-=r , 4/1-=α , 2/1-=β and

varying δ by means of the Gaussian function )])10-(-exp(05.01[)4/1-( 2ξδ ×+= , there is the

occurrence of small variations in the amplitudes of ),(1 sξa and ),(2 sξa , not easily visualized in

figure 4. From the Gaussian function, it is observed that the fluctuations in δ occur around 10=ξ .

Fig. 4. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)4/1-( 2ξδ ×+= with

0.1-=r , 4/1-=α and 2/1-=β .

Parameter β : Figure 5 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the

fluctuation in parameter β , around 2/1-=β , with maximum amplitude corresponding to 5% of the

average value of the dielectric parameter. By establishing values 0.1-=r , 4/1-=α , 4/1-=δ and

varying β by means of the Gaussian function )])10-(-exp(05.01[)2/1-( 2ξβ += , in figure 5 there is

the occurrence of visible variations in the amplitudes of ),(1 sξa and ),(2 sξa . From the Gaussian

function, it is observed that the fluctuations in β occur around 10=ξ .

Fig. 5. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)2/1-( 2ξβ ×+= with

0.1-=r , 4/1-=α and 4/1-=δ .

When figures 3, 4 and 5 are compared, it is verified that the propagation of the soliton wave was

more sensitive to the fluctuations of parameter β .

4.2. Periodic perturbations

In this subsection, the computational grid was constructed by considering, for variable s , the interval

9050- << s , with discretization 1-109.0Δ ×=s , and for variable , the interval 600 , with

discretization 3-100.1Δ ×=ξ . Again, the geometry of the computational grid was adjusted so that the

physics of the propagation of the perturbed soliton wave is completely within the considered

computational domain.

It was also considered that the fluctuations of the dielectric parameters in the splices of the optical

fibers are modeled by means of Gaussian functions. In the simulations conducted, periodic

fluctuations of 1% were considered in the values of the dielectric parameters in kξ 5= with

11,..,1=k . By establishing, for the dielectric parameters, the average values 0.1-=r , 4/1-=α ,

2/1-=β and 4/1-=δ , and considering the described periodic Gaussian perturbations, it is verified

again that ),(1 sξa and ),(2 sξa are more sensitive to the fluctuations of parameter .

Figure 6 shows the simulations of perturbations in ),(1 sξa and ),(2 sξa due to the periodic

fluctuations in parameter β , around 2/1-=β , when 0.1-=r , 4/1-=α , 4/1-=δ . In this case,

significant variations are observed in the amplitudes of ),(1 sξa and ),(2 sξa .

Fig. 6. Numerical solution ),(1 sξa and ),(2 sξa for 1% Gaussian perturbations in the β value, in

kξ 5= , for 11,..,1=k , with 0.1-=r , 4/1-=α , 4/1-=δ .

From the simulations in this subsection, it is observed that, in the context of optical communication

with localized perturbative processes, the propagation of soliton waves is more affected by the

fluctuations in parameter . Therefore, the procedures for optical fiber fusion through voltaic arcs,

for example, should be conceived so that, locally, the dielectric properties related to parameter of

the optical fiber are little affected.

4.3. Parameter and the stability of perturbed solitons

In the preceding sections, by simulating the propagation of soliton waves through localized

perturbations with various geometric configurations and intensities, it was verified that the soliton

wave is more distorted when there are fluctuation in parameter β . With the objective of verifying

whether the perturbed soliton waves achieve stability in function of the fluctuations in the dielectric

parameter β , in the sequence, the evolution of such waves with higher values of ξ will be studied.

Thus a computational grid was constructed, considering for variable s the interval 9050- << s , with

discretization 2-106.5Δ ×=s , and for variable ξ , the interval 1000 <<ξ , with discretization

3-100.1Δ ×=ξ . In all the following simulations, the values considered for the dielectric parameters

were: 0.1-=r , 4/1-=α , 4/1-=δ .

In figure 7, a Gaussian-type perturbation of 1% is considered in parameter β , around 10=ξ . It is

verified that ),(1 sξa and ),(2 sξa evolve towards a stationary situation for higher values of ξ .

Fig. 7. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(01.01[)2/1-( 2ξβ ×+= with

0.1-=r , 4/1-=α and 4/1-=δ , for large values of ξ .

In figure 8, a Gaussian-type perturbation of 5% is considered in parameter β , around 10=ξ . It is

observed again that the perturbed soliton wave evolves towards a stationary situation with damped

fluctuations for increasing values of ξ .

Fig. 8. Numerical solution ),(1 sξa and ),(2 sξa when )])10-(-exp(05.01[)2/1-( 2ξβ ×+= with

0.1-=r , 4/1-=α and 4/1-=δ , for large values of ξ .

5. Conclusions

In this work, the stability of the propagation of soliton waves through optical fiber splices was studied.

In optical fiber splices made, for example, by an electric arc, it is observed that the dielectric

properties are locally altered. In order to simulate the propagation of solitons through such splices,

considered as perturbations, the fluctuations of the dielectric parameters were locally modeled by

means of Gaussian functions. By considering local and periodic configurations for the optical fiber

splices, it was verified that the perturbed soliton wave presents higher sensitivity to parameter ; in

other words, fluctuations in the dielectric parameter generate higher amplitude perturbations

(oscillations) in the soliton waves. With the objective of verifying whether such perturbations generate

unstable solitons, the evolution of the soliton wave was studied for several configurations of

perturbations along of optical fiber, for higher values of . It was verified that, after the end of the

perturbations, the perturbed soliton wave achieves stability, i.e., the amplitude of the oscillations

decrease for increasing values of . It is therefore concluded that soliton waves, subject to Gaussian

perturbations in the dielectric parameters of the optical fiber, present numerical stability.

This work is finalized by suggesting that the procedures and experimental methods utilized in the

process of optical fiber fusion are designed so that, locally, the dielectric properties related to

parameter are less altered during the fusion process. Future works intend to continue the study of

the behavior of perturbed soliton waves in non-ideal optical fibers. Such numerical studies will enable

researchers to propose appropriate materials and procedures in order to make optical communication

via solitons more viable.

Acknowledgements

We express our appreciation to the late Dr. Valdemir Garcia Ferrreira, whose contribution to this work

was of great significance.The author C.F. de Oliveira thanks the State University of Londrina for the

scholarships IC/UEL granted from August/2006 to July/2007 and from August/2008 to February/2009.

The author P.L. Natti thanks the State University of Londrina for the financial support received for this

research through FAEPE/2005 and FAEPE/2009 programs.

References

[1] D. Le Guen, S. Del Burgo, L. Moulinard , D. Grot, M. Henry, F. Favre, T. Georges, Narrow band

1.02 Tbit/s (51×20 Gbit/s) soliton DWDM transmission over 1000 km of standard fiber with 100 km

amplifier spans, in: OFC1999 - Optical Fiber Communication Conference and IOOC1999 -

International Conference on Integrated Optics and Optical Fiber Communication, IEEE Xplore Digital

Library, San Diego, Technical Digest Volume Supplement, 1999, PD4/1-PD4/3.

[2] T. Okamawari, A. Maruta, Y. Kodama, Analysis of Gordon–Haus jitter in a dispersion-

compensated optical transmission system, Optics Letters 23 (1998) 694-696.

[3] M.M.E. El-Halawany, Efficient Raman amplifiers within propagation and multiplexing techniques

for high capacity and ultra long haul transmission systems, International Journal of Computer Science

and Telecommunications 2 (2011) 16-24.

[4] R. Gumenyuk, C. Thür, S. Kivistö, O.G. Okhotnikov, Tapered fiber Bragg gratings for dispersion

compensation in mode-locked Yb-doped fiber laser, IEEE Journal of Quantum Electronics 46 (2010)

769-773.

[5] M.F.S. Ferreira, M.V. Facao, S.C.V. Latas, M.H. Sousa, Optical solitons in fibers for

communication systems, Fiber and Integrated Optics 24 (2005) 287-231.

[6] W.J. Liu, B. Tian, M. Li, Y. Jiang, Q.X. Qu, P. Wang, K. Sun, Symbolic computation of solitons in

the normal dispersion regime of inhomogeneous optical fibers, Quantum Electronics 41 (2011) 545-

551.

[7] A.V. Buryak, P. Trapani, D.V. Skryabin, S. Trillo, Optical solitons due to quadratic nonlinearities:

from basic physics to futuristic applications, Physics Reports 370 (2002) 63-235.

[8] B.A. Malomed, D. Mihalache, F. Wise, L. Torner, Spatiotemporal optical solitons, Journal of

Optics B 7 (2005) R53-R72.

[9] K. Kurokawa, K. Tajima, K. Nakajima, 10-GHz 0.5ps pulse generation in 1000-nm band in PCF

for high-speed optical communication, Journal of Lightware Technology, 25 (2007) 75-78.

[10] S.C.V. Latas, M.F.S. Ferreira, Stable soliton propagation with self-frequency shift, Mathematics

and Computers in Simulation 74 (2007) 379-387.

[11] Y. Tsaraf, B.A. Malomed, Transmission of return-to-zero pulses in a optical split-step system

based on reflecting fiber gratings, Optics Communications 282 (2009) 3957-3963.

[12] A. Biswas, Dispersion-managed solitons in optical fibres, Journal of Optics A 4 (2002) 84–97.

[13] R. Driben, B.A. Malomed, Soliton stability against polarization-mode-dispersion in the split-step

system, Optics Communications 271 (2007) 228-235.

[14] P.Y.P. Chen, B.A. Malomed, Stabilization of spatiotemporal solitons in second-harmonic-

generating media, Optics Communications 282 (2009) 3804-3811.

[15] A.Hasegawa, F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive

dielectric fibers. I. Anomalous dispersion, Applied Physics Letters 23 (1973) 142-144.

[16] C.R. Menyuk, R. Schiek, L. Torner, Solitary waves due to 22 : cascading, Journal of the

Optical Society of America B 11 (1994) 2434-2443.

[17] W. Galleas, L.H. Ymai, P.L. Natti, E.R. Takano Natti, Solitons wave in dieletric optical fibers,

Revista Brasileira de Ensino de Física 25 (2003) 294-304.

http://www.scielo.br/pdf/rbef/v25n3/a07v25n3.pdf

[18] S. Yin, P. Yan, M. Gong, Influence of fusion splice on high power ytterbium-doped fiber laser

with master oscillator multi-stage power amplifiers structure, Optics and Lasers in Engineering 49

(2011) 1054-1059.

[19] M. Böhm, F. Mitschke, Solitons in lossy fibers, Physical Review A 76 (2007) 063822.

[20] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego, 2001.

[21] A.P.T. Lau, M. Kahn, Signal design and detection in presence of nonlinear phase noise, Journal

of Lightwave Technology 25 (2007) 3008-3016.

[22] R. Kohl, A. Biswas, D. Milovic, E. Zerrad, Perturbation of Gaussian optical solitons in

dispersion-managed fibers, Applied Mathematics and Computation 199 (2008) 250–258.

[23] M.S. Ismail, S.Z. Alamri, Highly accurate finite difference method for coupled nonlinear

Schrödinger equation, International Journal of Computer Mathematics 81 (2004) 333-351.

[24] H. Wang, Numerical studies on the split-step finite difference method for nonlinear Schrödinger

equations, Applied Mathematics and Computation 170 (2005) 17-35.

[25] I. Dag, A quadratic B-spline finite element method for solving nonlinear Schrödinger equation,

Computer Methods in Applied Mechanics and Engineering 174 (1999) 247-258.

[26] M.S. Ismail, Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method,

Mathematics and Computers in Simulation 78 (2008) 532-547.

[27] S. Reich, Multi-simpletic Runge-Kutta methods for hamiltonian wave equations, Journal of

Computational Physics 157 (2000) 473-499.

[28] M. Dehghan, A. Taleei, A compact split-step finite difference method for solving the nonlinear

Schrödinger equations with constant and variable coefficients, Computer Physics Communications

181 (2010) 43-51.

[29] E.R. Cirilo, P.L. Natti, N.M.L. Romeiro, C.F. Oliveira, Soliton in ideal optical fibers – a

numerical development, Semina: Ciências Exatas e Tecnológicas 31 (2010) 57-68.

http://arxiv.org/abs/1007.5276

[30] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,

Oxford University Press, New York, 2003.

[31] D. Sperandio, J.T. Mendes, L.H. Monken, Cálculo Numérico: Características Matemáticas e

Computacionais dos Métodos Numéricos, Prentice Hall, São Paulo, 2003.

[32] E.R. Cirilo, P.L. Natti, N.M.L. Romeiro, E.R. Takano Natti, Determination of the optimal

relaxation parameter in a numerical procedure of solitons propagation, Revista Ciências Exatas e

Naturais 10 (2008) 77-94. http://arxiv.org/abs/1007.4817