Dark solitons in N-coupled higher order nonlinear Schrödinger equations

Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328 www.elsevier.com/locate/cnsns

Dark solitons in N-coupled higher order nonlinear Schro¨dinger equations P. Seenuvasakumaran a, A. Mahalingam

b,*

, K. Porsezian

c

a

b

Department of Physics, Muthurangam Government Arts College, Vellore 632 002, India Department of Physics, Centre for Laser Technology, Anna University, Chennai 600 025, India c Department of Physics, Pondicherry University, Pondicherry 605 014, India Received 15 July 2005; received in revised form 20 April 2006; accepted 15 July 2006 Available online 12 January 2007

Abstract By considering the third order dispersion, self-steepening and stimulated Raman scattering effects, we analyse the dark soliton propagation in N-coupled higher order nonlinear Schro¨dinger equations. Using Painleve´ analysis, we prove that this system is completely integrable. The result is confirmed further by the presentation of Lax pair. Using the Hirota method, the construction of soliton solution is discussed. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.50.Rh; 42.65.Tg; 42.79.Sz; 42.81.Dp; 02.30.Ik; 02.30.Jr; 05.45.Yv Keywords: Complete integrable system; Third order dispersion; Stimulated Raman scattering; Lax pair; Painleve´ analysis

1. Introduction There is a growing interest in studying the propagation of optical soliton pulses in fibers. This is because of their potential applications in fiber-optic-based communication system, soliton laser and switching devices. The idea of soliton based all-optical communication systems with loss compensated by optical amplifications has provided hints of potential advantage for solitons over conventional system. The major attraction for the soliton communication system arises from the fact that repeater spacings for this kind of system could be much larger than that required by the conventional systems. It has been envisaged that solitons can replace the conventional systems not only in long-distance communication systems, but also in short-distance systems like LAN etc., [1–8].

*

Corresponding author. E-mail address: [email protected] (A. Mahalingam).

1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.07.014

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

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Hasegawa and Tappert [3,4] predicted both bright and dark solitons. They showed, in anomalous (negative) Group Velocity Dispersion (GVD) regime of optical fibers, the possible propagation of bright solitons and dark solitons in the normal (positive) GVD regime. After the invention of high-intensity lasers, Mollenauer et al. [5] confirmed their results experimentally. Since then the soliton propagation has been verified in a number of elegant experiments [8]. Emplit et al. [9] and Kro¨kel et al. [10] observed dark solitons in fibers independently. Blow and Doran [11] verified the same result numerically. These dark pulses undergo soliton like pulse propagation and emerges from the fiber almost unchanged even in the presence of significant broadening and chirping. The properties of dark solitons in nonlinear optical fibers have been discussed in a number of theoretical and experimental papers [12–20]. These dark pulses consist of rapid dips in the intensity of a continuous-wave (CW) background. Dark solitons can be created without a threshold value in the input pulse power, which is not possible in the bright soliton case. The force of attraction between the dark solitons is always repulsive whereas it is either attractive or repulsive in the case of bright solitons depending upon their relative phase. Theoretically, the soliton-type pulse propagation is governed by the well known nonlinear Schro¨dinger (NLS) equation derived from the Maxwell’s equations. However, this ordinary NLS equation cannot explain the propagation of soliton-type pulses in the femtosecond range. For transmitting the ultrashort-pulses, which are in the femtosecond range, one has to consider the higher order effects such as the third order dispersion (TOD), Self-steepening (SS) and stimulated Raman scattering (SRS). This is because the ultrashort-pulses (USP) suffer from these effects, as experimentally reported by Mitschke and Mollenauer [17]. Kodama and Hasegawa [2] have proposed that dynamics of femtosecond pulse propagation is governed by higher order NLS (HNLS) equation. The integrability properties of this system have been well documented [19–29]. In recent years, CNLS and Hirota coupled HNLS type equations have been analysed for soliton or solitary wave type solutions with the introduction of additional parameters such as, inelastic collisions of solitons have been observed. Recently, Sakovich and Tsuchida [28] have done singularity structure analysis and reported many new integrable type coupled NLS type equations. In order to increase the transmission capacity of a communication system, more channels are to be simultaneously propagated. For this, Wavelength Division Multiplexing (WDM) technique is preferred. In such systems, at least two optical fields are to be transmitted and the pulse is governed by a coupled NLS equation. Manakov [19] proposed a coupled NLS equation taking into account the fact that the optical field consists of left and right polarizations. Here, we consider the N-coupled HNLS equation for dark-soliton propagation taking into account the higher order effects. This article is arranged in the following manner. Section 2 discusses about the Lax pair for the N-coupled bright and dark soliton system. Section 3, deals with the Painleve´ analysis of the N-coupled HNLS system. In Section 4, we discuss the formation of Lax Pair for N-coupled HNLS system. Finally, in Section 5, the existence of dark solitons in the N-coupled HNLS system is confirmed by obtaining the one-soliton solution by means of Hirota’s bilinear technique. As the bilinearization will not give two-soliton solution, we discuss trilinear form briefly. 2. Lax pair for N-coupled NLS system The Lax pair assures the complete integrability of any soliton possessing nonlinear system. Once the Lax pair is known, we can obtain N-soliton solutions by means of inverse scattering transform method. Here, we follow the AKNS formalism to obtain the Lax pair. The linear eigenvalue problem for optical solitons in N-CNLS system can be constructed as follows: wx ¼ U w; wt ¼ V w;

T

where; w ¼ ðw1 w2 . . . wN Þ :

The Lax operators U and V are given in the following form:

ð1Þ

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0

ik=2

m1 q1

m1 q2

ik=2

0



0

0

ik=2



0

B B m2 q1 B B m q 2 2 U ¼B B B . B .. @

.. .

m2 qN 0 1 0 B B0 1 B B 2B 0 0 V ¼ a2 k B B. B .. @ 0 0 B B B B B B  iB B B B B B @

   m1 qN

0 0 0 1

N P

ik=2 0 0  0 C B  B q1  0C C B C B     0 C þ 2ia2 kB q2 C B B . .. C C B .. .A @ 1

qN þ1

! jqj j

2

q1x

q2x

j¼1 2

q1x

jq1 j

q2x .. . qðN þ1Þx

C C C C C C C C A



0

 1

0

1

q1 qðN þ1Þ

0

q1

q2



0

0



0 1



qN þ1

1

C 0 C C C C C .. C . C A

ð2Þ

0



qðN þ2Þx C C C C     q1 qðN þ1Þ C C C C C C C .. C . A 2    jqðN þ1Þ j

where N = n + 1. Here, k is the eigenvalue parameter and m1 and m2 are constants whose choices make the resultant equation to be either for bright solitons or dark solitons as shown below: Case (i): a2 ¼  2i ; m1 ¼ m2 ¼ 1; m1 ¼ i; m2 ¼ i; m1 ¼ i; m2 ¼ i; m1 ¼ m2 ¼ 1 The compatibility condition U t  V x þ ½U ; V  ¼ 0, gives the N-coupled nonlinear Schro¨dinger equation for bright solitons of the form: ! N X 2 iq1t þ q1xx þ 2 jqj j q1 ¼ 0; j¼1

.. . iqNt þ qNxx þ 2

ð3Þ N X

! jqj j

2

qN ¼ 0:

j¼1

Case (ii): a2 ¼ 2i ; m1 ¼ 1; m2 ¼ 1; m1 ¼ 1; m2 ¼ 1; m1 ¼ m2 ¼ i The compatibility condition gives the N-coupled nonlinear Schro¨dinger equation for dark solitons: ! N X 2 iq1t  q1xx þ 2 jqj j q1 ¼ 0; j¼1

.. . iqNt  qNxx þ 2

ð4Þ N X

! jqj j2 qN ¼ 0:

j¼1

The dark soliton property in NLS system has been completely analysed. Hence, in this article we have considered only the higher order NLS systems.

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

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3. N-coupled higher order NLS system As we have discussed in the introduction, the NLS equation explains the pulse propagation in a nonlinear optical fiber, but it has its own limitations. For example, when the optical pulse is of the order of femtoseconds, the NLS equation becomes inadequate, as higher order effects like TOD, SS and SRS should be included. In such a case, the governing equation is the one, widely known as higher order NLS (HNLS) equation, first derived by Kodama and Hasegawa [2]. The third order dispersion makes higher order solitons to split into fundamental solitons. The effect of SS is due to intensity dependent group velocity of the optical pulse, which gives the pulse a very narrow width in the course of propagation. The peak of the pulse will travel slower than the wings resulting the steepening of the pulse. The SRS is due to the delayed response of the medium, which forces the pulse to undergo a frequency shift, known as self-frequency shift [8]. When the TOD and SS are taken into account together with group velocity and self-phase modulation terms of the NLS system, the governing equation is known as Hirota equation, whose bright solitons properties were analyzed by many authors [22,25]. For the first time, considering TOD, SS and SRS we have proposed couple HNLS and studied the integrability through singularity structure analysis. Recently, we extended the above results to N-coupled HNLS system [27]. On the other hand if the SRS is included together with TOD and SS, one would get the HNLS equation. Radhakrishnan et al. [26] have considered both bright and dark soliton propagation in HNLS systems, which are not integrable from the point of view of Painleve´ analysis. The effect of TOD on dark solitons have been discussed by Kivshar and Afanasjev [16], who showed that near the zero point of group velocity dispersion, dark solitons exist as humps, instead of dips. It was proved that the solitary wave acts as a source, generating trailing oscillations, which with the leading front propagates with group velocity vg. In this section, Painleve´ analysis is carried out to find out new integrability conditions for the case of dark solitons. The N-coupled higher order nonlinear Schro¨dinger equation in the normal dispersion region is given in the following form: ! ( ! ! ) N N N X X X 2 2 2 q1t ¼ iq1xx þ 2i jqj j q1 þ e q1xxx þ a1 jqj j q1x þ a2 jqj j q1 ; j¼1

q2t ¼ iq2xx þ 2i

N X

! jqj j

2

(

q2 þ e q2xxx þ a1

j¼1

qNt ¼ iqNxx þ 2i

N X

j¼1

! jqj j

2

q2x þ a2

j¼1

.. . N X

j¼1

! jqj j

2

( qN þ e qNxxx þ a1

j¼1

N X j¼1

N X

!x ) jqj j

2

q2 ;

j¼1

! jqj j

2

qNx þ a2

N X j¼1

ð5Þ

x

! jqj j

2

) qN :

x

Here, q is the slowly varying amplitude of the pulse envelope and a1 and a2 are arbitrary constants. The parameter e represents the relative width of the spectrum that arises due to quasi-monochromocity and it is assumed that 0 < e < 1. The above equation has been analysed by many researchers in detail for bright solitons [19–29]. But here we discuss the dark-soliton case only. In order to understand the nature of the solutions of the above equation, first we apply the Painleve´ analysis to prove the integrability conditions. This method is one of the systematic and powerful methods to identify the integrability cases of the nonlinear partial differential equations, and leads to the confirmation of the existence of solitons. To identify the new integrable systems, a novel method for applying Painleve´ test was introduced by Weiss, Tabor and Carnevale with simplifications due to Kruskal involves seeking a solution of a given partial differential equation in the form [21]: qðx; tÞ ¼ /a

1 X

qj ðtÞ/j ðx; tÞ;

q0 6¼ 0;

ð6Þ

j¼0

with /ðx; tÞ ¼ x þ wðtÞ ¼ 0, where w(t) is an arbitrary analytic function of t, and qj ðtÞ; j ¼ 0; 1; 2; . . . ; is an analytic function of t, in the neighborhood of a noncharacteristic movable singularity manifold / = 0. The Painleve´ analysis not only provides the first valuable test for identifying whether a given partial differential

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equation is completely integrable, but also other important information relating to completely integrable equations, which include the construction of Ba¨cklund transformation, Lax pair, Hirota’s bilinear representation, special and rational solutions, etc. Many of these results are obtained by truncating the Laurent series at a constant level term. Let us perform this analysis for the following N-coupled mode system. ! ( ! ! ) N N N X X X 2 2 2 q1t ¼ iq1xx þ 2i jqj j q1 þ e q1xxx þ a1 jqj j q1x þ a2 jqj j q1 ; j¼1

q1t

¼

iq1xx

 2i

N X

j¼1

! jqj j

2

(

q1

þ e q1xxx þ a1

j¼1

! 2

jqj j

(

¼

iq2xx

 2i

N X

2

q1x

jqj j

þ a2

q2 þ e q1xxx þ a1

N X

! jqj j

2

jqj j

2

(

q2

þ e q1xxx þ a1

j¼1

jqj j

q2x þ a2

N X

)

2

q1 !

2

q2x

jqj j

þ a2

j¼1

N X

)

2

jqj j

q2 ;

j¼1

!

;

x

N X

j¼1

!

x

!

j¼1

N X

j¼1

q2t

!

j¼1

N X

q2t ¼ iq2xx þ 2i

N X

j¼1

x

! jqj j

)

2

q2

j¼1

ð7Þ

;

x

.. . qNt ¼ iqNxx þ 2i

N X

! jqj j

2

(

qN þ e q1xxx þ a1

j¼1

qNt

¼ iqNxx  2i

N X

N X

! jqj j

2

qNx þ a2

j¼1

! jqj j

2

qN

( þ e q1xxx þ a1

j¼1

N X

N X

! jqj j

qN ;

j¼1

! 2

jqj j

qNX

j¼1

þ a2

N X j¼1

)

2 x

! jqj j

)

2

qN

:

x

For the leading order analysis, we substitute qN  q0N ul in Eq. (7). On balancing the dominant terms we obtain l = 1. In order to find the powers at which the arbitrary functions can enter into the Laurent series, otherwise known as resonances, we substitute the following expression: qN ðx; tÞ ¼

1 X

qNj ðtÞ/j1 ;

ð8Þ

j¼0

in Eq. (7) and keeping the leading order terms alone we obtain the resonances as, j ¼ 1; ð2N  1Þ zeros; ð2N  1Þ

twos; 3; 2N

fours:

ð9Þ

From the resonance analysis, it can be clearly seen that the resonance would be integers only when, a1 = 2a2 a new integrability condition is obtained. Here, it is interesting to note that when a1 and a2 are positive integers it corresponds to the bright soliton case, whereas when they are negative integers then it corresponds to dark soliton case. For the uncoupled HNLS system this has been proved by Mihalache et al. [12] and Mahalingam and Porsezian [20]. For each resonance value, there is a compatibility condition, which must be identically satisfied, so that Eq. (7) has a general solution of the form (6). 4. Lax pair for N-coupled HNLS system If the solitons are to be preferred in the WDM type of transmission, we should be able to propagate more than two fields. Here we provide the Lax pair for N-fields and derive the system of N-coupled HNLS equation. From which we can get dark soliton solutions for N-field propagation. The liner eigenvalue problem for the N-coupled HNLS can be presented in the following form: wx ¼ U w; wt ¼ V w;

where w ¼ ðw1 w2 w3    wN Þ

T

ð10Þ

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

where,

0

ik=2 B k q B 1 1 B B k 1 r1 B B k 1 q B 2 U ¼B B k 1 r2 B B . B .. B B @ k 1 qN k 1 rN

k 1 q1 ik=2 0 0 0

k 1 r1 0 ik=2 0 0

k 1 q2 0 0 ik=2 0

    ik=2

0 0

0 0

0 0

 

    

    

   

ik=2 0

0

0 B B B 3B V ¼k B B B @

ie=2 0 0 0  0 ie=2 0 0  0 0 ie=2 0  .. . 0 0 0  

A2 B ek 1 q 1 1 B  0 B B ek 1 r1 C B  0 C B ek 1 q C 2 B 2B  0 C C þ k B ek 1 r C 2 B C B .. A B . B B 0 ie=2 @ ek 1 qN ek 1 rN

  



0

N    2  2 P  2 qj þ rj  B iek 1 B j¼1 B B iek q  2iA k q 1 1x 2 1 1 B B B iek 1 r  2iA2 k 1 r þ kB 1x 1 B .. B B . B B iek q  2iA k q 1 Nx 2 1 N @

0

iek 1 rNx  2iA2 k 1 rN M 11 M 12 M 13  

1 k 1 rN 0 C C C 0 C C 0 C C C 0 C C C C C C 0 A

k 1 qN 0 0 0 

ð11Þ

ik=2 ek 1 q1 A2 0 0 0

0 0

iek 1 q1x  2iA2 k 1 q1

iek 1 r1x  2iA2 k 1 r1



iek 21 jq1 j2

iek 21 q1 r1



iek 21 r1 q1

iek 21 jr1 j2



iek 21 rN q1 1

1 ek 1 rN 0 C C C 0 C C 0 C C C 0 C C C C C C 0 0 0    A2 0 A 0    0 A2 1 iek 1 qNx  2iA2 k 1 qN iek 1 rNx  2iA2 k 1 rN C C C 2  C iek 1 q1 rN C C 2  C iek 1 r1 rN C C .. C C . C C iek 21 jqN j2 A 2 2 iek 1 jrN j

ek 1 r1 0 A2 0 0

.. . iek 21 qN q1 

1323



ek 1 q2 0 0 A2 0

ek 1 r2 0 0 0 A2

    

    

    

ek 1 qN 0 0 0 0

M 1n

BM B 21 þB B .. @ .

M 22



 

M 2n C C .. C C . A

M n1

M n2



 

M nn

ð12Þ

Here, k1 and A2 are constants whose choice of values gives either bright or dark soliton version of the N-coupled HNLS system. The expressions of Mij are listed in Appendix. The compatibility condition U t  V x þ ½U ; V  ¼ 0 gives rise to the following N-coupled HNLS system, ! ( ! ! ) N N N X X X 2 2 2 3 3 3 k 1 q1t þ 2A2 k 1 q1xx þ 4k 1 A2 jqj j q1  ie ik 1 q1xxx  ia1 k 1 jqj j q1x  ia2 k 1 jqj j q1 ¼ 0; j¼1

k 1 q2t þ 2A2 k 1 q2xx þ 4k 31 A2

N X j¼1

.. .

k 1 qNt þ 2A2 k 1 qNxx þ 4k 31 A2

! jqj j

2

(

q2  ie ik 1 q2xxx  ia1 k 31

j¼1 N X

j¼1

! 2

jqj j

q2x  ia2 k 31

j¼1

N X j¼1

!x ) jqj j

2

q2

¼ 0;

x

8 ! 9 N > > > > 2 3 P > > > ik 1 qNxxx  ia1 k 1 ! jqj j qNx > > > N = < X j¼1 2 ! ¼0 jqj j qN  ie > > N P > > j¼1 2 > > 3 > > > > ; : ia2 k 1 j¼1 jqj j qN x

ð13Þ

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It has been found that the above equations give the bright-soliton CHNLS system for the choice k1 = 1, A2 = i/2 and for the choice k1 = i, A2 = i/2, one can obtain the dark-soliton solution. Thus, for this N-coupled HNLS system, complete integrability can be proved. From this Lax pair, we can obtain Ba¨cklund transformation and generate multi soliton solutions. 5. Hirota’s bilinear method Even though, Lax pair has been obtained for the N-coupled HNLS system, it is very difficult to generate soliton solutions using this approach, as the resultant Ricatti equations are complicate to solve. Hence, we proceed further to obtain the soliton solutions by using Hirota’s bilinear technique, which is simple and straightforward. This technique [22] is a novel method to generate soliton solutions and construct soliton solutions for nonlinear partial differential equations. Here, we transform the N-coupled HNLS equations into a set of N-coupled complex modified KdV equations using the following transformations:   Z T q1 ðx; tÞ ¼ Q1 ðZ; T Þ exp i  ; 3e 27e2   Z T  ; q2 ðx; tÞ ¼ Q2 ðZ; T Þ exp i 3e 27e2 .. ð14Þ .   Z T qN ðx; tÞ ¼ QN ðZ; T Þ exp i  ; 3e 27e2 t T ¼ t; Z ¼ x þ : 3e Using this transformation, the complex modified KdV equations are obtained as, ( ) ! ! N N X X 2 2 Q1t  e Q1ZZZ þ a1 jQj j Q1Z þ a2 jQj j Q1 ¼ 0 ; j¼1

( Q2t  e Q2ZZZ þ a1

N X

j¼1

! 2

jQj j

Q2Z þ a2

j¼1

.. .

(

QNt  e QNZZZ þ a1

N X

N X

!Z jQj j

2

j¼1

! jQj j

2

QNZ þ a2

j¼1

N X j¼1

) Q2 ¼ 0 ; ð15Þ

Z

! jQj j

2

) QN ¼ 0 :

Z

The Hirota bilinear form for theN-coupled HNLS equations can be constructed by applying the transformation for the field variables as Qj ðZ; T Þ ¼

Gj ðZ; T Þ ; F ðZ; T Þ

ð16Þ

where Gj ðZ; T Þ are complex functions and F ðZ; T Þ is a real function with respect to Z and T. Here j ¼ 1; . . . ; N  1. Using Eq. (16), the decoupled bilinear forms of Eq. (15) are given as: ðDT  eD3Z þ 3ekDZ ÞðGj  F Þ ¼ 0; 2

ðD2Z  kÞðF  F Þ ¼ 4ðjGj j Þ; DZ ðGj  Gj Þ ¼ 0; where k is a constant to be determined and the Hirota bilinear operators Dx and Dt are defined as   m  n  o o o o m n 0 0    Dx Dt Gðx; tÞ  F ðx; tÞ ¼ Gðx; tÞF ðx ; t Þ : ox ox0 ot ot0 x¼x0 ;t¼t0

ð17Þ

ð18Þ

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

1325

To obtain dark soliton solution we assume Gj ¼

N X

g0j ð1 þ vgj1 Þ;

and F ¼

j¼0

N X

1 þ vfj1 ;

ð19Þ

j¼1

where g0j’s are complex constants and gj1 and fj1’s are real functions. Here j ¼ 1; . . . ; N  1. Collecting the coefficients of v0, we have ! N 1 X 2 k¼4 jg0j j : ð20Þ j¼1

The coefficients of v leads to the following equations: ðDT  eD3Z þ 3ekDZ Þð1  fj1 þ gj1  1Þ ¼ 0; !! N 1 X 2 ðD2Z  kÞð1  fj1 þ fj1  1Þ þ 8 jg0j j ¼ 0;

ð21Þ

j¼1

2

The coefficients of v leads to the following equations: ðDT  eD3Z þ 3ekDZ Þðgj1  fj1 Þ ¼ 0; " ! # N 1  X 2 2 2 g0j  g ¼ 0: ðD  kÞðfj1  fj1 Þ þ 4 Z

ð22Þ

j1

j¼1

These equations suggest that they can be solved if we assume, h i ð0Þ g1j ¼ f1j ¼  exp xj1 T þ c1j Z þ nj1 where,

"

x1j ¼

ec1j ðc21j

 3kÞ and

c21j

# N  X 2 g0j  : ¼ 2k ¼ 8

ð23Þ

j¼1

Using, Eqs. (23), and (20), the dark one-soliton solution of cmKdV equation is obtained as ! " ( )# c21j eT 1 ð0Þ c1j Z  Q1 ¼ g0j tanh þ nj1 ; 2 2 .. .

ð24Þ

! " ( )# c21j eT 1 ð0Þ QN ¼ g0j tanh c1j Z  þ nj1 : 2 2 Using the transformations (14), we can easily obtain the corresponding dark one-soliton solution of the N-coupled HNLS Eq. (13). The dark one-soliton solution is shown in Fig. 1. It is interesting to note that the shape of soliton solution obtained in this work matches with the one obtained by Park and Shin, who used the Ba¨cklund transformation method [24]. Though we are able to construct one-soliton solution for Eq. (13), from the two-soliton solution, we found that the third condition in Eq. (17) is not satisfied. Taking this fact into consideration, very recently Gilson et al. [23] gave an alternative and effective method to overcome this type of problem. Using this idea, we obtain the trilinear form for dark soliton case in the form,

2 2 2 F 2 DX  eD3t ðG  F Þ þ 3eDt ðG  F ÞD2t ðF  F Þ  6eDt ðG  F ÞðjGj þ jH j Þ  3ejGj Dt ðG  F Þ  3eG2 Dt ðG  F Þ  3eGH  Dt ðH  F Þ  3eGHDt ðH   F Þ ¼ 0 Using the ideas suggested in Ref. [23], one can generate multi dark soliton solutions.

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Fig. 1. The dark one-soliton solution.

6. Conclusion In this article, we have analysed the dark soliton propagation in N-coupled HNLS system. We have extended our earlier works to the N-coupled systems because of the potential applications of systems, in which more than one wave will be propagating, like WDM, optical switching etc. Dark solitons are considered to be more exciting than bright solitons due to their repulsive nature, which helps in increasing the bit rate and noise-withstanding capabilities. Using the Painleve´ analysis, a new class of integrable systems has been identified for which the Lax pair has been also obtained, which proves beyond doubt that the system under investigation is indeed integrable. This system admits dark soliton propagation even when the higher order Selfsteepening and Raman scattering terms are negative in the normal dispersion regime. The one-soliton solution is obtained by applying the simple but elegant Hirota’s bilinear method and is plotted. From the plot, it is proved that the N-coupled HNLS system is suitable for the propagation of dark solitons. Analyses of the switching characteristics and other properties of these dark solitons are under progress and will be published elsewhere. Acknowledgements KP wishes to thank the DST, UGC (Research Award) and CSIR, Government of India, for the financial support through projects. PSK wishes to acknowledge UGC, India for availing him Faculty Improvement Programme (FIP). Appendix The elements of the last matrix in (12) are as follows: N  N    X X 2 2 M 11 ¼ 2A2 k 21 jqj j þ jrj j  ek 21 qj qjx  qjx qj þ rj rjx  rjx rj ; j¼1

j¼1

M 12 ¼ ek 1 q1xx þ 2A2 k 1 q1x  2ek 31 q1

N  X

 jqj j2 þ jrj j2 ;

j¼1

M 13 ¼ ek 1 r1xx þ 2A2 k 1 r1x  2ek 31 r1

N  X

 2 2 jqj j þ jrj j ;

j¼1

M 14 ¼ ek 1 q2xx þ 2A2 k 1 q2x  2ek 31 q2

N  X j¼1

 jqj j2 þ jrj j2 ;

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

M 15 ¼ ek 1 r2xx þ 2A2 k 1 r2x  2ek 31 r2

N   X jqj j2 þ jrj j2 ; j¼1

.. . M 1ðn1Þ ¼ ek 1 qNxx þ 2A2 k 1 qNx  2ek 31 qN

N  X

 2 2 jqj j þ jrj j ;

j¼1

M 1n ¼ ek 1 rNxx þ 2A2 k 1 rNx  2ek 31 rN

N   X 2 2 jqj j þ jrj j ; j¼1

M 21 ¼ ek 1 q1xx þ 2A2 q1x þ 2ek 31 q1 M 22 ¼

ek 21



2

q1 q1x



q1x q1





N  X

 2 2 jqj j þ jrj j ;

j¼1

þ 2A2 k 21 q1 q1 ;

M 23 ¼ ek 1 q1 r1x  q1x r1 þ 2A2 k 21 q1 r1 ; .. . M 2ðn1Þ ¼ ek 21 ðq1 qNx  qNx q1 Þ þ 2A2 k 21 q1 qN ; M 2n ¼ ek 21 ðq1 rNx  q1x r1 Þ þ 2A2 k 21 q1 rN ; N   X 2 2 jqj j þ jrj j ; M 31 ¼ ek 1 r1xx þ 2A2 r1x þ 2ek 31 r1 j¼1

M 32 ¼

ek 21 ðr1 q1x



þ 2A2 k 21 r1 q1 ;

r1x q1 Þ

M 33 ¼ ek 21 ðr1 r1x  r1x r1 Þ þ 2A2 k 21 r1 r1 ; .. . M 3ðn1Þ ¼ ek 21 ðr1 qNx  r1x qN Þ þ 2A2 k 21 r1 qN ; M 3n ¼ ek 21 ðr1 rNx  r1x rN Þ þ 2A2 k 21 r1 rN ; N   X 2 2 jqj j þ jrj j ; M 41 ¼ ek 1 q2xx þ 2A2 k 1 q2x þ 2ek 31 q2 j¼1

M 42 ¼

ek 21 ðq2 q1x

M 43 ¼

ek 21 ðq2 r1x

2A2 k 21 q2 q1 ;



q2x q1 Þ

þ



q2x r1 Þ

þ 2A2 k 21 q2 r1 ;

.. . M 4ðn1Þ ¼ ek 21 ðq2 qNx  q2x qN Þ þ 2A2 k 1 q2 qN ; M 4N ¼ ek 21 ðq2 rNx  q2x rN Þ þ 2A2 k 21 q2 rN ; N   X 2 2 jqj j þ jrj j ; M 51 ¼ ek 1 r2xx þ 2A2 k 1 r2x þ 2ek 31 r2 j¼1

M 52 ¼

ek 21 ðr2 q1x

M 53 ¼

ek 21 ðr2 r1x

2A2 k 21 r2 q1 ;



r2x q1 Þ

þ



r2x r1 Þ

þ 2A2 k 21 r2 r1 ;

M 5ðn1Þ ¼ ek 21 ðr2 qNx  r2x qN Þ þ 2A2 k 21 r2 qN ; M 5n ¼ ek 21 ðr2 rNx  r2x rN Þ þ 2A2 k 21 r2 rN ;

1327

1328

P. Seenuvasakumaran et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1318–1328

M ðn1Þ1 ¼ ek 1 qNxx þ 2A2 k 1 qNx þ 2ek 31 qN

N  X

 jqj j2 þ jrj j2 ;

j¼1

M n1 ¼ ek 1 rNxx þ 2A2 k 1 rNx þ 2ek 31 rN

N  X

 2 2 jqj j þ jrj j ;

j¼1

M ðn1Þn ¼ ek 21 ðrN qNx  rNx qN Þ þ 2A2 k 21 rN qN ; M nn ¼ ek 21 ðqN rNx  qNx rN Þ þ 2A2 k 21 qN rN ; where rN ¼

eiH qN ;

  2 2 xþ t : Hðx; tÞ ¼ 3 9

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