Quasi-particle theory of optical soliton interaction

Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228 www.elsevier.com/locate/cnsns

Quasi-particle theory of optical soliton interaction Anjan Biswas

a,b,*

, Swapan Konar

c

a

Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA b Center of Excellence in ISEM, Tennessee State University, Nashville, TN 37209-1561, USA c Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi 835215, India Received 29 November 2005; accepted 29 November 2005 Available online 2 February 2006

Abstract The intra-channel collision of optical solitons, with non-Kerr law nonlinearities, is studied in this paper by the aid of quasi-particle theory. The perturbations terms that are considered in this paper are both of Hamiltonian as well as nonHamiltonian type. The suppression of soliton–soliton interaction, in presence of these perturbation terms, is achieved. The nonlinearities that are studied in this paper are Kerr, power, parabolic and dual-power laws. The numerical simulations support the quasi-particle theory. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Ik; 02.30.Jr; 02.60.Cb; 42.65.Tg; 42.81.Dp Keywords: Optical solitons; Quasi-particle theory; Soliton–soliton interaction; Soliton perturbation

1. Introduction The theoretical possibility of existence of optical solitons in a dielectric dispersive fiber was first predicted by Hasegawa and Kodama [12]. A couple of years later Mollenauer et al. successfully performed the famous experiment to verify this prediction. Important characteristic properties of these solitons are that they posses a localized waveform which remains intact upon interaction with another soliton. Because of their remarkable robustness, they attracted enormous interest in optical and telecommunication community. At present optical solitons are regarded as the natural data bits for transmission and processing of information in future, and an important alternative for the next generation of ultra high speed optical communication systems. The fundamental mechanism of soliton formation namely the balanced interplay of linear group velocity dispersion (GVD) and nonlinearity induced self-phase modulation (SPM) is well understood. In the pico second regime, the nonlinear evolution equation that takes into account this interplay of GVD and SPM and which describes the dynamics of soliton is the well known nonlinear Schro¨dinger’s equation (NLSE). * Corresponding author. Address: Department of Applied Mathematics and Theoretical Physics, Delaware State University, 1200 N. DuPont Highway, Dover, DE 19901-2277, USA. Tel.: +1 302 659 0169; fax: +1 302 857 7517. E-mail address: [email protected] (A. Biswas).

1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.11.010

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The NLSE, which is the ideal equation in an ideal Kerr media, is in its original form found to be completely integrable by the method of inverse scattering transform (IST) and tremendous success has been achieved in the development of soliton theory in the framework of the NLSE model. However, communication grade optical fibers or as a matter of fact any optical transmitting medium does posses finite attenuation coefficient, thus optical loss is inevitable and the pulse is often deteriorated by this loss. Therefore, optical amplifiers have to be employed to compensate for this loss. When the gain bandwidth of the amplifier is comparable to the spectral width of the ultrashort optical pulse, the frequency and intensity dependent gain must be considered. Another hindrance to the stable propagation in a practical system is the noise induced Gordon–Hauss timing jitter. An important aspect that has not been addressed with proper perspective is the fact that due to its nonsaturable nature, Kerr nonlinearity is inadequate to describe the soliton dynamics in the ultrahigh bit rate transmission. For example, when transmission bit rate is very high, for soliton formation the peak power of the incident field accordingly becomes very large. On the other hand higher order nonlinearities may become significant even at moderate intensities in certain materials such as semiconductor doped glass fibers. Under circumstances, as mentioned above, non-Kerr law nonlinearities come into play changing essentially the physical features of optical soliton propagation. Therefore when very high bit rate transmission or transmission through materials with higher nonlinear coefficients are considered, it is necessary to take into account higher order nonlinearities. This problem can be addressed by incorporating various non-Kerr law nonlinearities in the NLSE. It has been realized that the Gordon–Hauss timing jitter can be reduced by introducing bandpass filtering. Stabilization of soliton propagation with the aid of nonlinear gain or under combined operation of gain and saturable absorption was recommended by Kodama et al. [20–24]. Thus, in order to model these features in the soliton dynamics, in a practical situation, the NLSE should be modified by incorporating additional terms. Thus, the concept of control of soliton propagation described by the NLSE with non-Kerr law nonlinearities is new and important developments in the application of solitons for optical communication systems. Because the NLSE, with non-Kerr laws, is not integrable, perturbation methods or numerical techniques have to be applied. Therefore, the control of soliton and interaction of two neighboring solitons incorporating perturbation terms like nonlinear gain, saturable amplification, filtering, higher order dispersion, self-steepening and nonlinear dispersions are going to be addressed in this paper. 2. Mathematical model The dimensionless form of the generalized NLSE is given by 1 iqZ þ qTT þ F ðjqj2 Þq ¼ 0 2

ð1Þ

where Z represents the non-dimensional distance along the fiber while, T represents time in dimensionless form. Also, in (1), F is a real-valued algebraic function and it is necessary to have the smoothness of the complex function F(jqj2)q : C # C. Considering the complex plane C as a two-dimensional linear space R2, the function F(jqj2)q is k times continuously differentiable, so that 1 [ 2 C k ððn; nÞ  ðm; mÞ; R2 Þ ð2Þ F ðjqj Þq 2 m;n¼1

Eq. (1) is a nonlinear partial differential equation (PDE) of parabolic type that is not integrable, in general. The special case, F(s) = s, also known as Kerr law of nonlinearity, is integrable by IST [12]. The IST is the nonlinear analog of Fourier transform that is used for solving the linear partial differential equations. Schematically, the IST and the technique of Fourier transform are similar [12]. The solutions are known as solitons. The general case F(s) 5 s takes (1) away from the IST picture as it is not integrable. In the anomalous dispersion regime [12], the particularly relevant solutions to (1) are called solitons, or non-topological solitons. In a rigorous sense, the pulses of the non-integrable systems are not solitons. However, the term solitons has been used broadly for the solutions of the non-integrable system as well, and this has become common. So, in this paper, the pulses shall be referred to as ‘solitons’. Although stationary pulses exist, and some solutions can be written in the analytic form, their behaviour is different from that of the solutions of the cubic NLSE.

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In most cases, the interest is confined to a single pulse described by the 1-soliton solution of the NLSE. However, in this paper, the effects of the perturbation terms in NLSE on 2-soliton interaction will be studied. It is necessary to launch the solitons close to each other for enhancing the information carrying capacity of the fiber. If two solitons are placed close to each other then it can lead to their mutual interaction thus providing a very serious hindrance to the performance of the soliton transmission system. However, the presence of the perturbation terms of the NLSE can lead to the suppression of the two soliton interaction thus solving the problem. The perturbed NLSE that is going to be studied in this paper for the SSI is i

oq 1 o2 q 2 þ þ F ðjqj Þq ¼ iR½q; q  oZ 2 oT 2

ð3Þ

where 2m

R ¼ djqj q þ nq

Z

T

1

2

jqj dt  a

oq o2 q o3 q o4 q o o 2 2 þ b 2  c 3  ir 4 þ k ðjqj qÞ þ lq ðjqj Þ oT oT oT oT oT oT

ð4Þ

In fiber optics  is called the relative width of the spectrum, that arises due to quasi-monochromaticity, and is assumed that 0 <   1. For the perturbation terms given by R in (4), d represents the coefficient of nonlinear gain, while r is the coefficient of saturable amplifiers. Also, in (4), m could be 0, 1 or 2. For m = 0, there is linear gain, while for m = 1 it is called quadratic gain and for m = 2, it is quintic gain or gain saturation. The coefficient of b is called the bandpass filtering term. Also, in (4), a is the frequency separation between the soliton carrier and the frequency at the peak of EDFA gain. Moreover, k is the self-steepening coefficient for short pulses [12] (typically 6100 fs), l is the higher order dispersion coefficient [12] and c is the coefficient of the third order dispersion [12]. Finally, n represents the coefficient of fourth order dispersion. It needs to be noted that the coefficients of d, b and n represents the non-Hamiltonian perturbation while the remaining terms in (4) represents the Hamiltonian type perturbation. It is known that the NLSE, as given by (1), does not give correct prediction for pulse widths smaller than 1 ps. For example, in solid state solitary lasers, where pulses as short as 10 fs are generated, the approximation breaks down. Thus, quasi-monochromaticity is no longer valid and higher order dispersion terms come in. If the group velocity dispersion is close to zero, one needs to consider the third order dispersion for performance enhancement along trans-oceanic distances. Also, for short pulse widths where group velocity dispersion changes within the spectral bandwidth of the signal can no longer be neglected, one needs to take into account the presence of the third order dispersion. Eq. (1), unlike the Kerr law case, does not have infinitely many conserved quantities. In fact, it has as few as three integrals of motion [4]. They are the energy (E) also known as the wave power, linear momentum (M) and the Hamiltonian (H) that are respectively given by Z 1 2 E¼ jqj dT ð5Þ 1 Z 1 i M¼ ðq qT  qqT Þ dT ð6Þ 2 1  Z 1 1 2 jqT j  f ðIÞ dT ð7Þ H¼ 1 2 where Z I F ðnÞ dn ð8Þ f ðIÞ ¼ 0

and the intensity I is given by I = jqj2. One can see that (1) can be written in a canonical form dH dq dH iqZ ¼  dq

iqZ ¼

ð9Þ ð10Þ

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This defines a Hamiltonian dynamical system on an infinite dimensional phase space. It can be analyzed using the theory of Hamiltonian systems. This means that a behaviour of the solution is defined, to a large extent, by the singular points of the system, namely the stationary solutions of (1) and depends on the nature of these points as determined by the stability of its stationary solutions [3]. The quasi-particle theory (QPT) of soliton–soliton interaction (SSI) is investigated [3,17–20] and it is proved by virtue of it that the interaction can be suppressed due to nonlinear gain and filters. Also it has been proved that the sliding frequency guiding filters [23–28] leads to the suppression of the SSI. Here, by virtue of the QPT, it will be proved that SSI can be suppressed due to the NLSE given by (1) in presence of the perturbation terms in (2), for Kerr, power, parabolic and dual-power laws. The soliton solution of (3), for  = 0, although not integrable, for any law of nonlinearity, is assumed to be given in the form qðZ; T Þ ¼ gðZÞg½fðZÞðT  vZ  T 0 ÞeðijT þixZþir0 Þ

ð11Þ

where j ¼ v

ð12Þ

fðZÞ ¼ vðgðZÞÞ xðZÞ ¼ wðgðZÞ; jðZÞÞ

ð13Þ ð14Þ

In (11), g represents the shape of the soliton described by the GNLSE and it depends on the type of nonlinearity in (1). The parameters g(Z) and f(Z), in (11), respectively represent the soliton amplitude and width, while j(Z) and x(Z) are the frequency and wave number of the soliton respectively while v is the velocity. Also T0 and r0 respectively represent the center of the soliton and center of soliton phase. In (13) and (14), the functional forms, v and w, depend on the type of nonlinearity in (1). The 2-soliton solution of the NLSE, given by (1), takes the asymptotic form 2 X gl ðZÞg½fl ðZÞðT  vl Z  T l Þeðijl T þixl Zþirl Þ ð15Þ qðZ; T Þ ¼ l¼1

with jl ¼ vl fl ðZÞ ¼ vðgl ðZÞÞ

ð16Þ ð17Þ

xl ðZÞ ¼ wðgl ðZÞ; jl ðZÞÞ

ð18Þ

where l = 1, 2. In the study of SSI, the initial pulse waveform is taken to be of the form       T0 T0 ð19Þ qð0; T Þ ¼ g1 g f1 T  ei/1 þ g2 g f2 T þ ei/2 2 2 which represents the injection of 2-soliton like pulses into a fiber. Here T0 represents the initial separation of the solitons namely the center-to-center soliton separation. It is to be noted that for T0 ! 1 (19) represents exact soliton solutions, but for T0  O(1) it does not represent an exact 2-soliton solution. The initial pulse form will modify depending on the type of perturbation considered as seen below. 1. Non-Hamiltonian perturbations For studying the SSI with non-Hamiltonian type perturbations, the case of in-phase injection of solitons with equal amplitudes will be considered. So, without any loss of generality, g1 = g2 = 1 and /1 = /2 = 0 is chosen so that (19) modifies to       T0 T0 qð0; T Þ ¼ g f1 T  þ g f2 T þ ð20Þ 2 2 2. Hamiltonian perturbations For studying the SSI with Hamiltonian type perturbations, the case of in-phase injection of solitons with unequal amplitudes will be considered. So without any loss of generality g1 = g0, g2 = 1 and /1 = /2 = 0 is chosen so that (19) modifies to

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      T0 T0 þg f T þ qð0; T Þ ¼ g0 g f0 T  2 2

ð21Þ

where f0 ¼ vðg0 Þ

ð22Þ

and f is given by (13). The special cases with regards to the four laws of nonlinearity will now be individually discussed in the following four subsections. 2.1. Kerr law For the case of Kerr law of nonlinearity F(s) = s so that Eq. (3) becomes oq 1 o2 q 2 þ þ jqj q ¼ iR½q; q  oZ 2 oT 2 The 1-soliton solution of (23), for  = 0, that can be obtained by IST has the form [3,4,12] g eiðjT þxZþr0 Þ qðZ; T Þ ¼ cosh½fðT  vZ  T 0 Þ i

ð23Þ

ð24Þ

where f  vðgÞ ¼ g

ð25Þ 2

2

g j 2 Also, the 2-soliton solution of the NLSE (23), for  = 0 takes the asymptotic form [3,12] x  wðg; jÞ ¼

qðZ; T Þ ¼

2 X l¼1

gl eiðjl T þxl Zþr0l Þ cosh½gl ðT  vl Z  T 0l Þ

ð26Þ

ð27Þ

where fl  vðgl Þ ¼ gl xl  wðgl ; jl Þ ¼

ð28Þ g2l

 2

j2l

and l = 1, 2. The modification of the initial pulse form (19) is given by g g  1  ei/1 þ  2  ei/2 qð0; T Þ ¼ cosh g1 T  T20 cosh g2 T þ T20

ð29Þ

ð30Þ

The initial pulse form for the two types of perturbations are as follows: 1. Non-Hamiltonian perturbations In this case, the choice g1 = g2 = 1 and /1 = /2 = 0, gives 1 1   þ   qð0; T Þ ¼ T0 cosh T  2 cosh T þ T20 which represents an in-phase injection of pulses with equal amplitudes. 2. Hamiltonian perturbations Here, the choice g1 = g0, g2 = 1 and /1 = /2 = 0, gives g 1  0  þ   qð0; T Þ ¼ cosh T þ T20 cosh g0 T  T20 which represents an in-phase injection of pulses with unequal amplitudes.

ð31Þ

ð32Þ

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2.2. Power law For the case of power law nonlinearity, F(s) = sp where 0 < p < 2 to prevent wave collapse [3]. So Eq. (3) modifies to oq 1 o2 q þ þ jqj2p q ¼ iR½q; q  ð33Þ oZ 2 oT 2 In this case, Eq. (33) with  = 0 is not integrable by IST. However, (33), for  = 0, supports solitons of the form [3] g qðZ; T Þ ¼ ð34Þ eiðjT þxZþr0 Þ 1 coshp ½fðT  vZ  T 0 Þ i

where f  vðgÞ ¼ gp



2p2 1þp

12 ð35Þ

f2 j2  2p2 2

x  wðg; jÞ ¼

ð36Þ

The 2-soliton solution of the NLSE (33) with  = 0 takes the asymptotic form qðZ; T Þ ¼

2 X l¼1

gl 1 p

cosh ½fl ðT  vl Z  T 0l Þ

eiðjl T þxl Zþr0l Þ

ð37Þ

where fl  vðgl Þ ¼

gpl



xl  wðgl ; jl Þ ¼

2p2 1þp

12

f2l j2l  2p2 2

In the study of SSI for power law nonlinearity, the initial pulse waveform is assumed to be g1 g2 i/ i/ qð0; T Þ ¼   e 1 þ   e 2 1 1 T 0 coshp f1 T  2 coshp f2 T þ T20

ð38Þ ð39Þ

ð40Þ

The initial pulse form modifies as follows: 1. Non-Hamiltonian perturbation Here, the choice g1 = g2 = 1 and /1 = /2 = 0, gives qð0; T Þ ¼

1 1    þ  1  T0 p cosh f T  2 cosh f T þ T20 1 p

ð41Þ

where sffiffiffiffiffiffiffiffiffiffiffi 2p2 f¼ 1þp

ð42Þ

which represents an in-phase injection of pulses with equal amplitudes. 2. Hamiltonian perturbations For this case, the choice g1 = g0, g2 = 1 and /1 = /2 = 0, gives qð0; T Þ ¼

g0 1    þ  1  T0 coshp f T þ T20 cosh f0 T  2 1 p

ð43Þ

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where, f0 ¼

gp0



2p2 1þp

12 ð44Þ

which represents in-phase injection of solitons with unequal amplitudes.

2.3. Parabolic law For the parabolic law of nonlinearity, F(s) = s + ms2, where the parameter m > 0, and so the NLSE is oq 1 o2 q þ þ ðjqj2 þ mjqj4 Þq ¼ iR½q; q  oZ 2 oT 2 Eq. (45) is not integrable by the IST. However, (45), for  = 0 supports solitons of the form [3] g iðjT þxZþr0 Þ qðZ; T Þ ¼ 1 e ½1 þ a coshffðT  vZ  T 0 Þg2 i

ð45Þ

ð46Þ

where pffiffiffi f  vðgÞ ¼ g 2

ð47Þ 2

2

g j ð48Þ x  wðg; jÞ ¼  4 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a ¼ 1 þ mg2 ð49Þ 3 In this paper it will be assumed that m > 0 although m could be negative as well. In fact, it is to be noted that for (45), for  = 0, solitons exist for m 2 (3/4A2, 1). Also, the 2-soliton solution of the parabolic law takes the asymptotic form [2] qðZ; T Þ ¼

2 X l¼1

gl ½1 þ al coshffl ðT  vZ  T l Þg

1 2

eiðjl T þxl Zþr0l Þ

ð50Þ

with pffiffiffi fl  vðgl Þ ¼ gl 2 g2  2j2l xl  wðgl ; jl Þ ¼ l 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 al ¼ 1 þ mgl 3

ð51Þ ð52Þ ð53Þ

In the study of SSI, with parabolic law nonlinearity, the initial pulse waveform is taken to be of the form g1 g2 i/1 i/2 ð54Þ qð0; T Þ ¼     12 e þ     12 e T0 T0 1 þ a1 cosh f1 T  2 1 þ a2 cosh f2 T þ 2 This initial pulse form modifies as follows: 1. Non-Hamiltonian perturbation Here, the choice g1 = g2 = 1 and /1 = /2 = 0, gives 1 1 qð0; T Þ ¼ h qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi i12 þ h p ffiffi ffi    pffiffiffi  i12 2 T  T20 2 T þ T20 1 þ 1 þ 43 m cosh 1 þ 1 þ 43 m cosh which represents an in-phase injection of solitons with equal amplitudes.

ð55Þ

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2. Hamiltonian perturbations In this case, on setting g1 = g0, g2 = 1 and /1 = /2 = 0, gives qð0; T Þ ¼ h

g0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi i12 þ h p ffiffi ffi    pffiffiffi  i12 1 þ 1 þ 43 mg0 cosh g0 2 T  T20 2 T þ T20 1 þ 1 þ 43 m cosh

ð56Þ

which represents an in-phase injection of pulses with unequal amplitudes.

2.4. Dual-power law For the dual-power law nonlinearity, F(s) = sp + ms2p where, in this case, m < 0 so that the NLSE is i

oq 1 o2 q 2p 4p þ þ ðjqj þ mjqj Þq ¼ iR½q; q  oZ 2 oT 2

ð57Þ

Eq. (57), for  = 0, is not integrable by IST. However, it supports solitary waves of the form g iðjT þxZþr0 Þ qðZ; T Þ ¼ 1 e 2p ½1 þ a coshffðT  vZ  T 0 Þg

ð58Þ

where 1 2p2 2p f  vðgÞ ¼ g 1þp g2p j2  x  wðg; jÞ ¼ 2p þ 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 mf ð1 þ pÞ a¼ 1þ 2 2p 1 þ 2p p



ð59Þ ð60Þ ð61Þ

For dual-power law nonlinearity, solitons exist for 

2p2 1 þ 2p
ð62Þ

In this case, the 2-soliton solution of the NLSE (1) takes the asymptotic form qðZ; T Þ ¼

2 X

gl

l¼1

½1 þ al coshffl ðT  vZ  T l Þg2p

1

eiðjl T þxl Zþr0l Þ

ð63Þ

with fl  vðgl Þ ¼

gpl



2p2 1þp

2p1

g2p j2 l  l xl  wðgl ; jl Þ ¼ 2p þ 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 mf ð1 þ pÞ2 al ¼ 1 þ l2 2p 1 þ 2p

ð64Þ ð65Þ ð66Þ

In the study of SSI, with dual-power law nonlinearity, the initial pulse waveform is taken to be of the form qð0; T Þ ¼ 

g1 g2 i/1 i/2    2p1 e þ     2p1 e T0 T0 1 þ a1 cosh f1 T  2 1 þ a2 cosh f2 T þ 2

ð67Þ

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The initial pulse form simplifies as follows: 1. Non-Hamiltonian perturbation In this case, on choosing g1 = g2 = 1 and /1 = /2 = 0, gives 1 1 qð0; T Þ ¼  1 þ        1 1 þ a1 cosh f T  T20 2p 1 þ a2 cosh f T þ T20 2p where f is given by (59). Eq. (68) represents an in-phase injection of solitons with equal amplitudes. 2. Hamiltonian perturbations Here, the choice g1 = g0, g2 = 1 and /1 = /2 = 0, leads to g0 1 qð0; T Þ ¼     2p1 þ     1 T0 1 þ a1 cosh f0 T  2 1 þ a2 cosh f T þ T20 2p where

1 2p2 2p f0 ¼ 1þp which represents in-phase injection of pulses with unequal amplitudes. gp0

ð68Þ

ð69Þ



ð70Þ

3. Quasi-particle theory The QPT dates back to 1981 since the appearance of the paper by Karpman and Solov’ev [14]. The mathematical approach to SSI will be studied using the QPT. Here, the solitons are treated as particles. If two pulses are separated and each of them is close to a soliton they can be written as the linear superposition of 2-soliton like pulses as [2] qðZ; T Þ ¼ q1 ðZ; T Þ þ q2 ðZ; T Þ ð71Þ with ð72Þ ql ðZ; T Þ ¼ Al g½Dl ðT  T l ÞeifBl ðT T l Þdl g where Dl ¼ vðAl Þ ð73Þ and l = 1, 2 while Al, Bl, Dl, Tl and dl are functions of Z. Here, Al, Dl and Bl do not represent the amplitude, width and the frequency of the full wave form. However, they approach the amplitude, width and frequency respectively for large separation namely if DT = T1  T2 ! 1, then Al ! gl, Dl ! fl and Bl ! jl. Since the waveform is assumed to remain in the form of two pulses, the method is called the quasi-particle approach. First, the equations for Al, Bl, Tl and dl using the soliton perturbation theory (SPT) [3,12] will be derived. Substituting (71) into (3) yields oql 1 o2 ql þ ¼ iR½ql ; ql   F ðjql þ ql j2 Þjql þ ql j ð74Þ oZ 2 oT 2 where l = 1, 2 and l ¼ 3  l. By SPT, the evolution equations are dAl ðlÞ ¼ F l ðA; DT ; D/Þ þ M l ð75Þ dZ dBl ðlÞ ¼ F 2 ðA; DT ; D/Þ þ N l ð76Þ dZ dT l ¼ Bl  F 3 ðA; DT ; D/Þ þ Ql ð77Þ dZ ddl ¼ wðAl ; Bl Þ þ F 4 ðA; DT ; D/Þ þ P l ð78Þ dZ ðlÞ ðlÞ where, the functions F 1 ; F 2 ; F 3 and F4 formulate on using the SPT in (74), with the right side being treated as perturbation terms. The exact form of these functions can be obtained when a specific law of nonlinearity is considered. Also, i

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

Z

M l ¼ h1 ðAl Þ N l ¼ h2 ðAl Þ Ql ¼ h3 ðAl Þ P l ¼ h4 ðAl Þ

1211

1

^ l ; q ei/l ggðsl Þ dsl RfR½q l

ð79Þ

^ l ; q ei/l gg0 ðsl Þ dsl IfR½q l

ð80Þ

^ l ; q ei/l gsl gðsl Þ dsl RfR½q l

ð81Þ

^ l ; q ei/l g½gðsl Þ  sl g0 ðsl Þ dsl IfR½q l

ð82Þ

Z 1 1 Z 1 1 Z 1 1 1

where the functions hj(Al) for 1 6 j 6 4 are by virtue of (75)–(78) and the type of nonlinearity that is considered. Also, in (79)–(82), R and I stands for the real and imaginary parts respectively. Moreover, the following notations are used:  ^ ql ; q ¼ R½ql ; q   F ðjql þ ql j2 Þjql þ ql j R ð83Þ l l sl ¼ Al ðT  T l Þ /l ¼ Bl ðT  T l Þ  dl

ð84Þ ð85Þ

D/ ¼ BDT þ Dd

ð86Þ

DT ¼ T 1  T 2 Dd ¼ d1  d2 1 A ¼ ðA1 þ A2 Þ 2 1 B ¼ ðB1 þ B2 Þ 2 DA ¼ A1  A2

ð87Þ ð88Þ ð89Þ

DB ¼ B1  B2

ð92Þ

ð90Þ ð91Þ

Finally, it is assumed that jDAj  A

ð93Þ

jDBj  1

ð94Þ

jDDj  D ADT 1

ð95Þ ð96Þ

DDT 1 jDAjDT  1

ð97Þ ð98Þ

jDDjDT  1

ð99Þ

From (84)–(92), one can now obtain dA ¼ M dZ dB ¼ N dZ dðDAÞ ð1Þ ð2Þ ¼ F 1 ðA; DT ; D/Þ  F 1 ðA; DT ; D/Þ þ DM dZ dðDBÞ ð1Þ ð2Þ ¼ F 2 ðA; DT ; D/Þ  F 2 ðA; DT ; D/Þ þ DN dZ dðDT Þ ¼ DB þ DQ dZ  dðD/Þ DT ð1Þ ð2Þ ¼ wðA1 ; B1 Þ  wðA2 ; B2 Þ  BDB þ F 2 þ F 2 þ DP þ BDQ dZ 2

ð100Þ ð101Þ ð102Þ ð103Þ ð104Þ ð105Þ

1212

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

where 1 M ¼ ðM 1 þ M 2 Þ 2 1 N ¼ ðN 1 þ N 2 Þ 2 and DM, DN, DQ and DP are the variations of M, N, Q and P which are written as, for example DM ¼

oM oM DA þ DB oA oB

ð106Þ ð107Þ

ð108Þ

assuming that they are functions of A and B only, which is, in fact, true for most of the cases of interest, otherwise, the equations for 1 T ¼ ðT 1 þ T 2 Þ 2

ð109Þ

and 1 / ¼ ð/1 þ /2 Þ ð110Þ 2 would have been necessary. The results in this section, that are derived so far, will now be utilized to show that the SSI can indeed be suppressed in presence of the perturbation terms given by (4) for the four cases of nonlinearity. In all the four types of nonlinearities, the initial conditions for the Kerr law nonlinearity, corresponding to the initial waveform (19), are A ¼ 1;

B ¼ 0;

DA0 ¼ 0;

DB0 ¼ 0;

DT 0 ¼ T 0

and

D/0 ¼ 0

The four laws of nonlinearity are now studied in the following subsections. 3.1. Kerr law In this case, (72) reduces to ql ðZ; T Þ ¼

Al eiBl ðT T l Þþidl cosh½Dl ðT  T l Þ

ð111Þ

where Dl  vðAl Þ ¼ Al

ð112Þ

so that (74) transforms to i

oql 1 o2 ql 2 2 þ þ jql j ql ¼ iR½ql ; ql   ðq2l ql þ 2jql j ql Þ oZ 2 oT 2

ð113Þ

where l = 1, 2 and l ¼ 3  l and the separation jqj2 q ¼ ðjq1 j2 q1 þ q21 q2 þ 2jq1 j2 q2 Þ þ ðjq2 j2 q2 þ q22 q1 þ 2jq2 j2 q1 Þ

ð114Þ

was used based on the degree of overlapping. By SPT, the evolution equations are dAl lþ1 ¼ ð1Þ 4A3 eADT sinðD/Þ þ M l dZ dBl lþ1 ¼ ð1Þ 4A3 eADT cosðD/Þ þ N l dZ dT l ¼ Bl  2AeADT sinðD/Þ þ Ql dZ ddl 1 2 ¼ ðA þ B2l Þ  2ABeADT sinðD/Þ þ 6A2 eADT cosðD/Þ þ P l dZ 2 l

ð115Þ ð116Þ ð117Þ ð118Þ

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

where

Z

1

 ^ l ; q ei/l R R½q l

1 dsl cosh sl 1 Z 1   ^ ql ; q ei/l tanh sl dsl Nl ¼  I R l cosh sl Z11 1 ^ l ; q ei/l g sl dsl Ql ¼ 2 RfR½q l cosh sl Al 1 Z 1 1 ^ l ; q ei/l g ð1  sl tanh sl Þ dsl Pl ¼ IfR½q l Al 1 cosh tl

Ml ¼

and

 ^ l ; q  ¼ R½ql ; q   i q2 q þ 2jql j2 q R½q l l l l l

1213

ð119Þ ð120Þ ð121Þ ð122Þ

ð123Þ

The study for the Kerr law case will now be split into the following two subsections. 3.1.1. Non-Hamiltonian perturbations In presence of non-Hamiltonian perturbation terms, given in, (4), the dynamical system of the soliton parameters, by SPT are [1,5–12]  C 12 Cðm þ 1Þ 2mþ1 dA 2 2mþ3 A ¼ d þ 2rA2  bAðA2 þ 3B2 Þ ð124Þ dZ 3 C 2 dB 4 ¼  bA2 B ð125Þ dZ 3 so that by virtue of (102)–(105)

  mþ1  2m þ 1 dðDAÞ d C 12 Cðm þ 1Þ 2X 2rþ1 2m2r 3 ADT 2mþ3 ¼ 8A e sinðD/Þ þ 2m ð2AÞ ðDAÞ dZ C 2 2r þ 1 2 r¼0 þ 4rADA  2bðA2 þ B2 ÞDA  4bABDB

dðDBÞ 8 4 ¼ 8A3 eADT cosðD/Þ  bABDA  bA2 DB dZ 3 3 dðDT Þ ¼ DB þ r dZ dðD/Þ 4 ¼ ADA  bA2 BDT dZ 3 where in (126)   n nðn  1Þ ðn  r þ 1Þ ¼ rðr  1Þ 3 2 1 r

ð126Þ ð127Þ ð128Þ ð129Þ

ð130Þ

For the fixed point of the dynamical system, given by (126) and (127), with A = 1 and B = 0, one recovers  3d C 12 Cðm þ 1Þ   b ¼ 3r þ ð131Þ 2 C 2mþ3 2 From (128) and (129), one has the coupled system of equations for D/, the phase difference, and DT, the soliton separation, with the fixed point A = 1 and B = 0 as follows: d2 ðDT Þ 4 dðDT Þ þ 8eDT cosðD/Þ ¼ 0 þ b 3 dZ dZ 2  2m2r  mþ1  2m þ 1 d2 ðD/Þ dðD/Þ d C 12 Cðm þ 1Þ 2X dðD/Þ    þ 2ðb  2rÞ  8eDT sinðD/Þ ¼ 0 dZ dZ C 2mþ3 dZ 2 2r þ 1 22m r¼0 2

ð132Þ ð133Þ

1214

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

where in (132) and (133) b is given by (131). Eqs. (132) and (133) shows that inserting filters produces a damping in both pulse separation and phase difference [12,13,15,16,29] as seen in Fig. 1(a)–(c).

Fig. 1. (a) m = 0, r = d = 0.005; (b) m = 1, r = d = 0.005 and (c) m = 2, r = d = 0.005.

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

1215

3.1.2. Hamiltonian perturbations In case of Hamiltonian perturbations of (4), the dynamical system of the soliton parameters is dA ¼0 dZ dB ¼0 dZ dT 0  ¼ B  3cB2  fA2 ð3k þ 2l þ 3cÞ þ 3ag 3 dZ

ð134Þ ð135Þ ð136Þ

so that by virtue of (102)–(105) dðDAÞ ¼ 8A3 eADT sinðD/Þ dZ dðDBÞ ¼ 8A3 eADT cosðD/Þ dZ dðDT Þ 3  ¼ DB  cBDB  ð3k þ 2l þ 3cÞADA dZ 2 6 dðD/Þ ¼ ADA dZ

ð137Þ ð138Þ ð139Þ ð140Þ

Now, 1 A ¼ ðA0 þ 1Þ 2 B¼0 DA0 ¼ A0  1

ð141Þ ð142Þ ð143Þ

DB0 ¼ 0

ð144Þ

DT 0 ¼ T 0 D/0 ¼ 0

ð145Þ ð146Þ

D/ ¼ Dd

ð147Þ

so that dðDT Þ  dðD/Þ þ ð3k þ 2l þ 3cÞ ¼ DB dZ 6 dZ

ð148Þ

For DB = 0  DT ¼ T 0  ð3k þ 2l þ 3cÞDd 6

ð149Þ

Now, T0  O(1) so that DT 9 0 and thus the pulses do not collide during the transmission [30]. This is observed in the following numerical simulation (Fig. 2). 3.2. Power law In this case, (72) is ql ðZ; T Þ ¼

Al 1 p

cosh ½Dl ðT  T l Þ

eiBl ðT T l Þþidl

ð150Þ

where Dl  vðAl Þ ¼

Apl



2p2 1þp

12 ð151Þ

1216

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

Fig. 2.  = 0.1, c = 0.14.

so that (74) gives

" #" # p   p   X p pr r X p oql 1 o2 ql   pr  r þ i ¼ iR½ql ; ql   q q ðq1 Þ ðq2 Þ ðq1 þ q2 Þ oZ 2 oT 2 r 1 2 r¼0 r r¼0

Here, the separation " #" # p   p   X p pr r X p 2p  pr  r jqj q ¼ q q ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r 1 2 r¼0 r r¼0

ð152Þ

ð153Þ

was used based on the degree of overlapping. By SPT, the evolution equations are dAl ðlÞ ¼ F 1 ðA; DT ; D/; pÞ þ M l dZ dBl ðlÞ ¼ F 2 ðA; DT ; D/; pÞ þ N l dZ dT l ¼ Bl  F 3 ðA; DT ; D/; pÞ þ Ql dZ ddl B2l A2p ¼ þ l þ F 4 ðA; DT ; D/; pÞ þ P l dZ 2 pþ1

ð154Þ ð155Þ ð156Þ ð157Þ

where, for power law

  2 p3 1 þ 12 Z 1   2p C 1 p 1 2p ^ ql ; q ei/l

  Ml ¼ R R dsl 1 l 2p 1þp coshp sl C 1p C 12 1

  2 p1 1 1 Z 1 þ 2p C   p 2 2 p1 2p ^ ql ; q ei/l tanh1sl dsl

  I R N l ¼ Al l 1 1 p 1þp coshp sl C p C 2 1

  pþ2 1 1 Z 1 þ 2p C   sl p 2 1 pþ1 ^ ql ; q ei/l

 R R dsl Ql ¼ pþ1   1 l 2 2p Al coshp sl C 1p C 12 1

  2 pþ1 1 þ 12 Z 1   2p C p 1 2p sl Þ ^ ql ; q ei/l ð1  sl tanh

  I R dsl P l ¼ pþ1 1 l 1 1 pþ1 Al 1 coshp sl C C p

2

ð158Þ

ð159Þ

ð160Þ

ð161Þ

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

In addition, the following notations are used: " #" # p   p   X X p p pr r 2p pr ^ l ; q  ¼ R½ql ; q   i R½q q1 qr2 ðq1 Þ ðq2 Þ ðq1 þ q2 Þ þ ijql j ql l l r r r¼0 r¼0

1217

ð162Þ

For the power law, the study will now be split into the following two subsections. 3.2.1. Non-Hamiltonian perturbations In presence of non-Hamiltonian perturbation terms, given in (4), the dynamical system of the soliton parameters, by virtue of SPT, are



  2p1 C 1 þ 1 mþ1 C p 2 p dA 2d 1 þ p



 A2mþ1 ¼ 1 mþ1 1 dZ 2  p 2p2 C p C p þ2

   2p Z s  1 1 Z 1 r 1 þ p pþ1 C p þ 2 1 ds

  þ dsA3p 2 2 2  p 2p2 C 1p C 12 1 coshp s 1 coshp s



 pþ1  2 2p1 1 1 þ C 2p C p 2 p 2b 2p



 A2pþ1 þ 2 pþ1 1 1 p ð2  pÞ p þ 1 C C þ p



2

p

2p1



2b 2p2 2b 2p2 AB2  2 2p pþ1 p ð2  pÞ p þ 1    2 3p2 2p dB 4b 2p p2 ¼ 2 BA2p dZ p pþ2 pþ1

2p1 2p



A2pþ1

ð163Þ ð164Þ

so that by virtue of (102)–(105) dðDAÞ ð1Þ ð2Þ ¼ F 1 ðA; DT ; D/Þ  F 1 ðA; DT ; D/Þ dZ



   2p1 C 1 þ 1 mþ1 2X mþ1  C 2m þ 1 p 2 p d 1þp



 þ ð2AÞ2rþ1 ðDAÞ2m2r 1 mþ1 1 2  p 2p2 2r þ 1 C p C p þ 2 r¼0

   2p  Z s  2X 1 1 Z 1 mþ1  3p r 1 þ p pþ1 C p þ 2 1 ds 2pþ2r 2rþ1

 þ ðDAÞ ð2AÞ ds 1 2 2 1 2  p 2p2 ps ps 2r þ 1 1 1 cosh cosh C p C 2 r¼0



  2 2p1 1 2mþ1 þ 12 C pþ1 2p C X  2m þ 1  p p b 2p 2rþ1 2m2r



 þ 2 ð2AÞ ðDAÞ pþ1 1 1 p ð2  pÞ p þ 1 2r þ 1 C C þ r¼0 p

2

p

 2p1 mþ1  2p 2X 2m þ 1 b 2p2 2rþ1 2m2r ABDB  2 ð2AÞ ðDAÞ p ð2  pÞ p þ 1 2r þ 1 r¼0    2 3p2 2p dðDBÞ 4b 2p p2 ð1Þ ð2Þ ¼ F 2 ðA; DT ; D/Þ  F 2 ðA; DT ; D/Þ þ 2 ½DBA2p þ 2pBA2pþ1 DA dZ p pþ2 pþ1

 ! Z s  1 1 1 Z 1 dðDT Þ A1pþ2 A2pþ2 2p2 2 C p þ 2 s ds

 ¼ DB þ 2r  ds dZ 1 þ p C 1 C1 1 cosh2p s 1 cosh2p s D31 D32

ð166Þ

   3p2 2p 2p dðD/Þ A2p 4b 2p2 p  2 2p 1  A2 ¼  2 A BDT dZ p pþ2 pþ1 pþ1

ð168Þ



2b 2p2  2p pþ1

2p1



p

ð165Þ

ð167Þ

2

1218

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

For the fixed point of the dynamical system given by (164) and (165) with A = 1 with B = 0, one gets

 

 2 2 1 1 mþ1 C þ C C p p 2 p 1 4







 b¼ dðp þ 1Þ 2 1 1 1 mþ1 C p C p þ 12 2C p  C p þ 2

  3  ð3p1Þð2pþ1Þ Z s  2 1 1 Z 1 C þ C 2pðpþ1Þ p p 2 pþ1 1 ds

  ds5 ð169Þ þ rp2 2 2 1 1 2p2 p 1 1 cosh s coshp s C C p

2

Thus, from (167) and (168) one gets the coupled system of equations in DT and D/ for the fixed point with A = 1 and B = 0 as    3p2 2p d2 ðDT Þ 4b 2p2 p  2 dðDT Þ ð1Þ ð1Þ þ F2  F2 ¼ 0  ð170Þ p2 p þ 1 pþ2 dZ dZ 2   d2 ðD/Þ 2p 2p1 dA1 2p1 dA2 A  A2 ¼ ð171Þ pþ1 1 dZ dZ dZ 2 where in (170) b is given by (169). From (170), one can observe that there is a damping introduced in the soliton separation and the coefficient of the damping term is positive as 0 < p < 2. In the following figures, numerical simulations show that the suppression of the SSI is achieved, for power law, as proved in the QPT (Fig. 3(a) and (b)).

Fig. 3. (a) m = 1, p = 1/2; b ¼ 187 d; d = 0.001 b = 0.00257 and (b) m = 2, p = 1/2; b ¼ 480 d; d = 0.001, b = 0.002. 231

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

1219

3.2.2. Hamiltonian perturbations In presence of Hamiltonian perturbation terms, as given by (4), the dynamical system of the soliton parameters, by virtue of SPT, are dA ¼0 dZ

ð172Þ

dB ¼0 dZ

ð173Þ

   2  3 p1 1 1 2 1 1 3 C C þ C C þ 2 p p 2 p p dT 0  2 3cD 4 2    ðl þ 3cB Þ þ

  þ 15 ¼ B  A ð3k þ 2lÞ 2 dZ p2 C 12 þ 2p C 1p C 1p C 32 þ 2p

ð174Þ

so that by virtue of (102)–(105), dðDAÞ ð1Þ ð2Þ ¼ F 1 ðA; DT ; D/Þ  F 1 ðA; DT ; D/Þ dZ dðDBÞ ð1Þ ð2Þ ¼ F 2 ðA; DT ; D/Þ  F 2 ðA; DT ; D/Þ dZ

  1 1 C þ C 2p 2 p dðDT Þ  3c    2 DD½12D2 þ ðDDÞ2  ¼ DB  ADAð3k þ 2lÞ 1 2 1 dZ 4 4p C þ C 2

p

ð175Þ ð176Þ

p

  p1 1 1 C C þ 2 p p 3c 2  þ 2 DD½12D2 þ ðDDÞ   1 3 p C p C 2 þ 1p dðD/Þ A2p  A2p 2 ¼ 1 dZ pþ1

ð177Þ

ð178Þ

Eq. (178) can be rewritten, by virtue of (105), as     1 þ C 2p p dðDT Þ  dðD/Þ 3c dðD/Þ



 ¼ DB  ð3k þ 2lÞf1  2 g2 dZ 4 dZ 4p dZ C 12 þ 2p C 1p

    C p1 C 1 þ 1 p 2 p 3c dðD/Þ

  þ 2 f2 1 3 p dZ C p C 2 þ 1p 

C

1 2

ð179Þ

For in-phase injection of solitons with unequal amplitudes, 1 A ¼ ðA0 þ 1Þ 2 B¼0

ð180Þ ð181Þ

DA0 ¼ A0  1

ð182Þ

DB0 ¼ 0

ð183Þ

DT 0 ¼ T 0

ð184Þ

D/0 ¼ 0

ð185Þ

D/ ¼ Dd

ð186Þ

1220

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

Fig. 4.  = 0.1, p ¼ 12, c = 0.14, k = l = 0.

so that for DB = 0

    1 1   2   dðD/Þ C 2 þ p C p 3c dðD/Þ

   2 h2 DT ¼ T 0  ð3k þ 2lÞ  ð3k þ 2lÞh1 6 4 dZ 4p dZ C 12 þ 2p C 1p

    p1 1 1 3c dðD/Þ C p C 2 þ p

  þ 2 h2 p dZ C 1 C 31 p

where hj ðsÞ ¼

Z

2

ð187Þ

p

fj ðsÞ ds

ð188Þ

for j = 1, 2. Thus DT ¼ T 0 þ OðÞ

ð189Þ

Now, T0  O(1) so that DT 9 0 and thus the pulses do not collide during the transmission. This will be observed in the following numerical simulation (Fig. 4). 3.3. Parabolic law In this case, (72) yields ql ðZ; T Þ ¼

Al ½1 þ al coshfDl ðT  T l Þg

1 2

eiBl ðT T l Þþidl

ð190Þ

where

pffiffiffi Dl  vðAl Þ ¼ Al 2

ð191Þ

By (74), one gets i

oql 1 o2 ql 4 2 þ þ ðjq2l j þ mjql j Þql ¼ iR½ql ; ql   ðq2l ql þ 2jql j ql Þ oZ 2 oT 2 h i 2 2 4 2 2 2  m q3l ðql Þ þ 2jql j q2l ql þ 3jql j ql þ 3jql j ql q2l þ 6jql j jql j ql

ð192Þ

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

1221

where l = 1, 2 and l ¼ 3  l. Here, the separation 2

4

2

2

2

2

4

jqj q þ mjqj q ¼ ðjq1 j q1 þ q21 q2 þ 2jq1 j q2 Þ þ ðjq2 j q2 þ q22 q1 þ 2jq2 j q1 Þ þ m½jq1 j q1 þ q31 ðq2 Þ

2

þ 2jq1 j2 q21 q2 þ 3jq1 j4 q2 þ 3jq1 j2 q1 q22 þ 6jq1 j2 jq2 j2 q1  þ m½jq2 j4 q2 þ q32 ðq1 Þ2 2

4

2

2

2

þ 2jq2 j q22 q1 þ 3jq2 j q1 þ 3jq2 j q2 q21 þ 6jq1 j jq2 j q2 

ð193Þ

was used based on the degree of overlapping. By SPT, the evolution equations are dAl ðlÞ ¼ F 1 ðA; DT ; D/; mÞ þ M l dZ dBl ðlÞ ¼ F 2 ðA; DT ; D/; mÞ þ N l dZ dT l ¼ Bl  F 3 ðA; DT ; D/; mÞ þ Ql dZ

ð194Þ ð195Þ ð196Þ

ddl A2l B2l ¼ þ þ F 4 ðA; DT ; D/; mÞ þ P l dZ 4 2

ð197Þ

where, in this case Ml ¼

ð1Þ h1 ðAl Þ

Z

1

1 ð1Þ

N l ¼ h2 ðAl Þ

Z

1

ð1Þ

Pl ¼

1

ð1 þ al cosh sl Þ2 sinh sl

dsl

ð198Þ

dsl

ð199Þ

1

dsl

ð200Þ

  ^ ql ; q ei/l ð1  al sl sinh sl1Þ dsl I R l 1 ð1 þ al cosh sl Þ2

ð201Þ

Z

1

1 ð1Þ h4 ðAl Þ

  ^ ql ; q ei/l I R l

dsl

1

1

Ql ¼ h3 ðAl Þ

  ^ ql ; q ei/l R R l

Z

  ^ ql ; q ei/l R R l

ð1 þ al cosh sl Þ2 sl ð1 þ al cosh sl Þ2

1

Also, the following notations are used:      ^ ql ; q ¼ R ql ; q  q2 q þ 2jql j2 ql  m½q3 q 2 þ 2jql j2 q2 q þ 3jql j4 ql þ 3jql j2 q q2 þ 6jql j2 jql j2 ql  R l l l l l l l l l l ð202Þ For the parabolic law case, the study will be split into the following two subsections. 3.3.1. Non-Hamiltonian perturbations In presence of non-Hamiltonian perturbation terms, as given by, (4), the dynamical system of the soliton parameters, by SPT are     dA dA2mþ1 3 a1 mþ1 1 ¼ m m1 F m þ 1; m þ 1; m þ ; ; B dZ 2 2a p 2 2 a pffiffiffi Z 1 Z s  1 ds 2 þ ra2 ds 2 1 1 þ a cosh s 1 1 þ a cosh s " # rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi a a  1 1 a2 B 2 a1 2 3 1 1  tan  ba A  4 2b pffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 a þ 1 2ða2  1Þ aþ1 ða2  1Þ 32 a 1 # rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 " dB a2 a1 1 2A B 1 ¼ b  3 tan dZ E a þ 1 2ða2  1Þ ða2  1Þ2

ð203Þ

ð204Þ

1222

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

where E is the energy of the soliton given by rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 2A a1 1 E¼ jqj dT ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 aþ1 a 1 1 Z

1

2

ð205Þ

and F(a, b; c; z) is the Gauss’ hypergeometric function defined as F ða; b; c; zÞ ¼

1 CðcÞ X Cða þ nÞCðb þ nÞ zn CðaÞCðbÞ n¼0 Cðc þ nÞ n!

ð206Þ

while B(l, m) is the beta function defined as Z 1 Bðl; mÞ ¼ xl1 ð1  xÞml dx

ð207Þ

0

For the fixed point of the dynamical system, given by (203) and (204), with A = 1 and B = 0, one recovers

R    mþ1 1 s pffiffiffi R 1 1 ds F m þ 1; m þ 1; m þ 32 ; a1 B p ;2 2a r 2 1 1þa cosh s 1 1þa cosh s ds d qffiffiffiffiffiffi qffiffiffiffiffiffi þ m mþ1 ð208Þ b¼ 2 2 a a2 a2 tan1 a1  1 tan1 a1  1 3

ða2 1Þ2

aþ1

2ða2 1Þ

3

ða2 1Þ2

aþ1

2ða2 1Þ

Thus, one can obtain using, (102)–(105) d2 ðDT Þ dðDT Þ ð1Þ ð2Þ þ F2  F2 ¼ 0 þ bG dZ dZ 2

ð209Þ

where b is given by (208) and G > 0 represents the coefficient of ebDB in d(DB)/dZ = dB1/dZ  dB2/dZ. Now, Eq. (209) shows that there is a damping in the separation of solitons thus proving that there will be a suppression of the SSI in presence of the perturbation terms given by (4). The following numerical simulations show that the suppression of the SSI is achieved, for the parabolic law, as proved in the QPT (Fig. 5). 3.3.2. Hamiltonian perturbations In presence of Hamiltonian perturbation terms, as given by, (4), the dynamical system of the soliton parameters, by SPT are

Fig. 5. m = 0, r 5 0, d 5 0, b 5 0; d = 0.001, r = 0.001, b = 0.00895.

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

dA ¼0 dZ dB ¼0 dZ

1223

ð210Þ ð211Þ

rffiffiffiffiffiffiffiffiffiffiffi pffiffiffi dT 0 a 2 2A a1 1 ¼ B  pffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 E a 1 aþ1 dZ " rffiffiffiffiffiffiffiffiffiffiffi# pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffi  2 A a1 1 2  a  1  2 tan 3 ð3k þ 2lÞ 2E ða2  1Þ2 aþ1 " # r ffiffiffiffiffiffiffiffiffiffiffi  ffi pffiffiffi cA a4 A2 2j2 a  1 a2 A2 pffiffiffiffiffiffiffiffiffiffiffiffi 1 2  3 2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi tan a 1 E aþ1 2 4 a2  1

ð212Þ

From (102)–(105) one can now conclude that dðDAÞ ð1Þ ð2Þ ¼ F 1 ðA; DT ; D/Þ  F 1 ðA; DT ; D/Þ dZ dðDBÞ ð1Þ ð2Þ ¼ F 2 ðA; DT ; D/Þ  F 2 ðA; DT ; D/Þ dZ dðDT Þ dT 1 dT 2 ¼  dZ dZ dZ dðD/Þ 1 ¼ ADA dZ 2

ð213Þ ð214Þ ð215Þ ð216Þ

Eq. (215) can be rewritten as

  dðDT Þ dðD/Þ ¼ DB þ Gða; k; l; c; rÞf dZ dZ

ð217Þ

where G is the functional form that depends on the said parameters. For in-phase injection of solitons with unequal amplitudes, 1 A ¼ ðA0 þ 1Þ 2 B¼0

ð218Þ ð219Þ

DA0 ¼ A0  1 DB0 ¼ 0

ð220Þ ð221Þ

DT 0 ¼ T 0

ð222Þ

D/0 ¼ 0 D/ ¼ Dd

ð223Þ ð224Þ

so that, one can obtain from (217) and (222)–(224), for DB = 0,   dðD/Þ DT ¼ T 0 þ Gða; k; l; c; rÞh dZ where hj ðsÞ ¼

Z

fj ðsÞ ds

ð225Þ

ð226Þ

for j = 1, 2. Thus, DT ¼ T 0 þ OðÞ

ð227Þ

Now, T0  O(1) so that DT 9 0 and thus the pulses do not collide during the transmission. This is observed in the following numerical simulations (Fig. 6).

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Fig. 6. Parameters: m = 0.3, e = 0.1, a = 0.80, c = 0.35, r = 0.03.

3.4. Dual-power law In this case, (72) reduces to ql ðZ; T Þ ¼

Al ½1 þ al coshfDl ðT  T l Þg

1 2

eiBl ðT T l Þþidl

ð228Þ

where Dl  vðAl Þ ¼ Apl



2p2 1þp

12 ð229Þ

Here (74) modifies to

" #" # p   p   X  p pr r X p oql 1 o2 ql   pr  r i þ ¼ iR ql ; ql  q q ðq1 Þ ðq2 Þ ðq1 þ q2 Þ oZ 2 oT 2 r 1 2 r¼0 r r¼0 " #" #   2p  2p  X 2p 2pr r X 2p  2pr  r m q1 q2 ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r r r¼0 r¼0

ð230Þ

where, the separation

" #" # p   p   X p pr r X p  pr  r jqj q þ mjqj q ¼ q q ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r 1 2 r¼0 r r¼0 " #" #   2p  2p  X 2p 2pr r X 2p  2pr  r þm q1 q2 ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r r r¼0 r¼0 2p

4p

ð231Þ

was used based on the degree of overlapping. By SPT, the evolution equations are dAl ðlÞ ¼ F 1 ðA; DT ; D/; m; pÞ þ M l dZ dBl ðlÞ ¼ F 2 ðA; DT ; D/; m; pÞ þ N l dZ dT l ¼ Bl  F 3 ðA; DT ; D/; m; pÞ þ Ql dZ ddl A2p B2 l ¼ þ l þ F 4 ðA; DT ; D/; m; pÞ þ P l dZ 2p þ 2 2

ð232Þ ð233Þ ð234Þ ð235Þ

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

where, for the case of dual-power law, Z 1   ð2Þ ^ ql ; q ei/l R R M l ¼ h1 ðAl Þ l

dsl

1 dsl ð1 þ al cosh sl Þ2p Z 1   sinh sl ð2Þ ^ ql ; q ei/l N l ¼ h2 ðAl Þ I R 1 dsl l 1 ð1 þ al cosh sl Þ2p Z 1   sl ð2Þ ^ ql ; q ei/l R R Ql ¼ h3 ðAl Þ 1 dsl l 1 ð1 þ al cosh sl Þ2p Z 1   ð2Þ ^ ql ; q ei/l ð1  al s1 sinh sl1Þ dsl P l ¼ h4 ðAl Þ I R l 1 ð1 þ al cosh sl Þ2p

1225

ð236Þ

1

ð237Þ ð238Þ ð239Þ

In addition, the following notations are used:

" #" # p   p   X   p pr r X p 2p 4p    pr  r ^ R ql ; ql ¼ R ql ; ql þ ijql j ql þ imjql j ql  i q q ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r 1 2 r¼0 r r¼0 " #" #   2p  2p  X 2p 2pr r X 2p  2pr  r i q1 q2 ðq1 Þ ðq2 Þ ðq1 þ q2 Þ r r r¼0 r¼0

ð240Þ

For the case of dual-power law nonlinearity, the study is split into two subsections. 3.4.1. Non-Hamiltonian perturbations In presence of the perturbation terms, as given by, (4) the dynamical system of the soliton parameters, by SPT, are     dA 4dA2mþ2 mþ1 mþ1 mþ1 1 a1 mþ1 1 ¼ ; ; þ ; ; B F mþ1 dZ 2mþ1 p p p 2 2a p 2 p a p D ! Z s 4 Z 1 2rA 1 ds þ ds 1 1 2 p D 1 ð1 þ a cosh sÞ 1 ð1 þ a cosh sÞp          4bA2 2 1 1 3 1 a1 1 3 1 1 1 1 a1 1 1 2 ; ; þ ; D F 2þ ; ; þ ; B ; þB F B ; ð241Þ  1 1 p p 2 p 2a p 2 p p 2 p 2a p 2 Dap 2p   1 1 1 a1 1 dB bBD2 F 2 þ p ; 1 þ p ; 2 þ p ; 2a B 1 þ p ; 1



 ¼ 2 2 ð242Þ dZ 4p A B 1;1 F 1 ; 1 ; 1 þ 1 ; a1 p p 2

p

2a

p 2

For the fixed point of the dynamical system, given by (241) and (242), with A = 1 and B = 0, one recovers

   1p F mþ1 ; mþ1 ; mþ1 þ 1 ; a1 B mþ1 ; 3 p p p 2 2a p 2 d 1þp



 b¼ m m 2 p p 1 1 3 1 a1 1 3 2p 2 a F 2 þ p ; p ; 2 þ p ; 2a B p;2 ! 3 Z 1 Z s 1  rap 1 þ p 2p 1 1 ds

  þ p1 ds ð243Þ 1 1 2p2 1 ð1 þ a cosh sÞp 2p B 1 ; 3 F 2 þ 1 ; 1 ; 3 þ 1 ; a1 1 ð1 þ a cosh sÞp p 2

p p 2

p

2a

Thus, one can obtain using, (102)–(105) d2 ðDT Þ dðDT Þ ð1Þ ð2Þ þ F2  F2 ¼ 0 þ bG dZ dZ 2

ð244Þ

where b is given by (243) and G > 0 represents the coefficient of bDB in d(DB)/dZ = dB1/dZ  dB2/dZ. Now, Eq. (244) shows that there is a damping in the separation of solitons thus proving that there will be a suppression of the SSI in presence of the perturbation terms given by (4). Thus, in the following figures, the numerical simulations show that the suppression of the SSI is achieved, for the dual-power law, as proved in the QPT (Fig. 7).

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Fig. 7. m = 1, r 5 0, d 5 0, b 5 0; d = 0.001, r = 0.001, b = 0.00496.

3.4.2. Hamiltonian perturbations In presence of the perturbation terms, as give by, (4), the SPT, are dA ¼0 dZ dB ¼0 dZ

 2 F 2 þ 1 ; 1 ; 1 þ 3 ; a1 p p p 2 2a dT 0 cD

 ¼ B  ðl þ 3cB2 Þ  3 2 dZ 2p F 1p ; 1p ; 12 þ 1p ; a1 2a

  2 F 2 ; 2 ; 2 þ 1 ; a1 B 2 ; 1 p p p 2 2a p 2 A    ð3k þ 2lÞ 1 1 1 1 1 1 a1 p p 2 a F p ; p ; 2 þ p ; 2a B 1p ; 12 From (102)–(105), one can now conclude that dðDAÞ ð1Þ ð2Þ ¼ F 1 ðA; DT ; D/Þ  F 1 ðA; DT ; D/Þ dZ dðDBÞ ð1Þ ð2Þ ¼ F 2 ðA; DT ; D/Þ  F 2 ðA; DT ; D/Þ dZ dðDT Þ dT 1 dT 2 ¼  dZ dZ dZ dðD/Þ 1 ¼ ADA dZ 2 For in-phase injection of solitons with unequal amplitudes, 1 A ¼ ðA0 þ 1Þ 2 B¼0

dynamical system of the soliton parameters, by ð245Þ ð246Þ B B



1 3 ; p 2 1 1 ; p 2

 

ð247Þ

ð248Þ ð249Þ ð250Þ ð251Þ

ð252Þ ð253Þ

DA0 ¼ A0  1 DB0 ¼ 0

ð254Þ ð255Þ

DT 0 ¼ T 0

ð256Þ

D/0 ¼ 0 D/ ¼ Dd

ð257Þ ð258Þ

A. Biswas, S. Konar / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1202–1228

1227

Fig. 8. Parameters: m = 0.5, e = 0.1, p = 1/2, a = 0.8, c = 0.25, r = 0.05.

Eq. (250) can be rewritten as

  dðDT Þ dðD/Þ ¼ DB þ Gða; k; l; c; rÞf dZ dZ

ð259Þ

where G is the functional form that depends on the said parameters. For in-phase injection of solitons with unequal amplitudes, so that for DB = 0   dðD/Þ DT ¼ T 0 þ Gða; k; l; c; rÞh ð260Þ dZ where hj ðsÞ ¼

Z

fj ðsÞ ds

ð261Þ

for j = 1, 2. Thus, DT ¼ T 0 þ OðÞ

ð262Þ

Now, T0  O(1) so that DT 9 0 and thus the pulses do not collide during the transmission. This is observed in the following numerical simulations (Fig. 8). 4. Conclusions In this paper, the SSI of the NLSE in presence of Hamiltonian as well as non-Hamiltonian type perturbations was studied. The Hamiltonian perturbations included the third and fourth order dispersions, nonlinear dispersion term, self-steepening term and the frequency separation term. On the other hand, the non-Hamiltonian type perturbation terms included the nonlinear gain, saturable amplifiers and the bandpass filters. The four laws that are considered in this paper are the Kerr law, power law, parabolic law and the dual-power law. It was observed that the SSI can be suppressed in presence of these perturbation terms for all the four laws. The QPT, due to these perturbation terms, for all four laws, were developed and especially for the laws beyond the Kerr law is seen here for the first time. The analytical reasoning of the suppression of the SSI was established. Thus, in the applied soliton community, 2-solitons can be injected into a single channel, close to one another and also suppress their mutual interaction so that performance enhancement can be achieved. This conclusion is based on analytical results due to the quasi-particle theory of SSI that is supported by numerical simulation.

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In future, the SSI will be studied in the context of dispersion-managed solitons. Also, the study will be extended to the case of other laws of nonlinearities that are not considered in this paper namely saturable law, log law, exponential law and many others. Moreover, the case of SSI due to intra-pulse Raman scattering will be studied. Finally, this study of QPT can be extended to the case of 3-soliton, 4-soliton interaction and other higher numbers. All such studies are under way. Acknowledgements This research for the first author was fully supported by NSF Grant No. HRD-970668 and the support is thankfully appreciated. The work of the second author is partially supported by the Department of Science and Technology (DST), Government of India, through the R&D grant SP/S2/L-21/99. Therefore S.K. would like to acknowledge this support with thanks. References [1] Aceves AB, Angelis CD, Nalesso G, Santagiustina M. Higher-order effects in bandwidth-limited soliton propagation in optical fibers. Opt Lett 1994;19:2104–6. [2] Afanasjev VV. Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification. Opt Lett 1993;18:790–2. [3] Biswas A. Soliton–soliton interaction in optical fibers. J Nonlinear Opt Phys Mater 1999;8(4):483–95. [4] Biswas A. Quasi-stationary non-Kerr law optical solitons. Opt Fiber Technol 2003;9(4):224–59. [5] Chu PL, Desem C. Mutual interaction between solitons of unequal amplitudes in optical fibre. Electron Lett 1985;21:1133–4. [6] Chu PL, Desem C. Effect of third order dispersion of optical fibre on soliton interaction. Electron Lett 1985;21:228–9. [7] Desem C, Chu PL. Soliton interactions in presence of loss and periodic amplification in optical fibers. Opt Lett 1987;12:349–51. [8] Francois PL, Georges T. Reduction of averaged soliton interaction forces by amplitude modulation. Opt Lett 1993;18:583–5. [9] Georges T, Favre F. Influence of soliton interaction on amplifier noise-induced jitter: a first-order analytical solution. Opt Lett 1991;16:1656–8. [10] Georges T, Favre F. Modulation, filtering and initial phase control of interacting solitons. J Opt Soc Am B 1993;10:1880–9. [11] Gordon JP. Interaction forces among solitons in optical fibers. Opt Lett 1983;8:596–8. [12] Hasegawa A, Kodama Y. Solitons in optical communications. Oxford University Press; 1995. [13] Hermansson B, Yevick D. Numerical investigation of soliton interaction in optical fibers. Electron Lett 1983;19:570–1. [14] Karpman VI, Solov’ev VV. A perturbational approach to the two-soliton systems. Physica D 1981;3:487–502. [15] Kodama Y, Nozaki K. Soliton interaction in optical fibers. Opt Lett 1987;12(12):1038–40. [16] Kodama Y, Romagnoli M, Wabnitz S. Soliton stability and interaction in fiber lasers. Electron Lett 1992;28(21):1981–3. [17] Kodama Y, Wabnitz S. Reduction of soliton interaction forces by bandwidth limited amplification. Electron Lett 1991;27(21):1931–3. [18] Kodama Y, Wabnitz S. Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification. Electron Lett 1993;29(2):226–7. [19] Kodama Y, Wabnitz S. Reduction and suppression of soliton interaction by bandpass filters. Opt Lett 1993;18(16):1311–3. [20] Kodama Y, Wabnitz S. Analysis of soliton stability and interactions with sliding filters. Opt Lett 1994;19(3):162–4. [21] Nakazawa M, Kubota H. Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification. Electron Lett 1992;28:958–60. [22] Shiojiri E, Fuji Y. Transmission capability of an optical fibre communication system using index nonlinearity. Appl Opt 1985;24:358–60. [23] Uzunov IM, Stoev VD, Tsoleva TI. N-soliton interaction in trains of unequal soliton pulses in optical fibers. Opt Lett 1992;17:1417–9. [24] Uzunov IM, Stoev VD, Tsoleva TI. Influence of the initial phase difference between pulses on the N-soliton interaction in trains of unequal solitons in optical fibers. Opt Commun 1993;97(5–6):307–11. [25] Uzunov IM, Stoev VD, Tsoleva TI. N-soliton interaction in trains of unequal soliton pulses in optical fibers. Opt Lett 1992;17:1417–9. [26] Uzunov IM, Go¨lles M, Leine L, Lederer F. The effect of bandwidth limited amplification on soliton interaction. Opt Commun 1994;110(3–4):465–74. [27] Uzunov IM, Muschall R, Go¨lles M, Lederer F, Wabnitz S. Effect of nonlinear gain and filtering on soliton interaction. Opt Commun 1995;118(5–6):577–80. [28] Uzunov IM, Gerdjikov V, Go¨lles M, Lederer F. On the description of N-soliton interactions in optical fibers. Opt Commun 1996;125(4–6):237–42. [29] Wabnitz S. Control of soliton train transmission, storage, and clock recovery by cw light injection. J Opt Soc Am B 1996;13(12):2739–49. [30] Xu Z, Li L, Li Z, Zhou G. Soliton interaction under the influence of higher-order effects. Opt Commun 2002;210(3–6):375–84.