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Dark soliton solutions of the coupled Hirota equation in nonlinear fiber
6 August 2001
Physics Letters A 286 (2001) 321–331 www.elsevier.com/locate/pla
Dark soliton solutions of the coupled Hirota equation in nonlinear fiber S.G. Bindu, A. Mahalingam, K. Porsezian ∗ Department of Physics, Anna University, Chennai 600 025, India Received 20 November 2000; received in revised form 24 April 2001; accepted 7 May 2001 Communicated by A.R. Bishop
Abstract We consider the coupled Hirota equation which describes the pulse propagation in a coupled fiber with higher-order dispersion and self-steepening. Using the Painlevé analysis, we obtain the parametric conditions for the existence of bright and dark solitons. For the identified case(s), we also construct the common Lax pair and the soliton solutions are constructed for the dark soliton case. Also we extend our results to N-field propagation and the explicit Lax pair is constructed. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Recent experimental achievements have increased interest in the potential applications of optical dark solitons such as optical switching [1]. It is well known that optical bright solitons can be used for long distance communication to drastically increase the bit rate of fiber transmission systems. Dark solitons are reflectionless radiation modes of the wave guides, which also have a localised shape similar to bright solitons, but with complex envelope and nonvanishing asymptotics. In the case of temporal solitons, the group velocity dispersion is known to vanish at a wavelength of 1.3 µm and is positive at larger wavelengths and negative at shorter ones. Since silica optical fibers have always a +ve Kerr coefficient, the two different signs of group velocity dispersion support two different types of solitons, dark in the former case and bright in the latter case [2–4]. With the current interest of using solitons as pulse bits in long optical fibers for communication purposes, it is important for us to re-evaluate the practicality of using analytical techniques for predicting the behaviour of such bits. Since such pulses are near a pure soliton solution, it becomes feasible to use analytical soliton techniques and potentiality of obtaining useful analytical results becomes very high [5]. Slowly varying amplitude electromagnetic waves in a nonlinear medium are described by the nonlinear Schrödinger (NLS) equation [6]. In order to increase the bit rates, it is necessary to decrease the pulse width. As pulse lengths become comparable to the wavelength, however, the NLS equation becomes inadequate, additional terms have to be included and the resulting pulse * Corresponding author.
E-mail address: [email protected] (K. Porsezian). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 3 7 1 - 1
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propagation is called as higher-order nonlinear Schrödinger (HNLS) equation [7]. This equation includes effects like third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). The nonlinear response of a dielectric medium to electromagnetic radiation includes contribution from quadratic, cubic, and higher-order terms in the electric field. It is well known that TOD effects splitting-up of higherorder solitons. The inelastic Raman scattering is due to the delayed response of the medium which forces the pulse to undergo a frequency shift which is known as self-frequency shift. The effect of self-steepening is due to the intensity-dependent group velocity of the optical pulse, which gives the pulse a very narrow width in the course of propagation. Because of this, the peak of the pulse will travel slower than the wings. Among the effects associated with third-order nonlinearity, stimulated Raman scattering and stimulated Brillouin scattering limit the maximum input power available for transmission, whereas self-phase modulation directly influences the dispersion by modifying the pulse shape and thus can play a vital role, together with chromatic dispersion, in determining the transmission rate attainable in a given fiber. Wave propagation in optical fibers, with these higher-order effects, is governed by the HNLS equation. When solitons consist of several interacting modes, the stationary waves are described by a system of coupled nonlinear partial differential equations. In the case of wavelength division multiplexing (WDM) [8], one should consider at least two optical fields simultaneously. In 1974, Manakov proposed the coupled NLS equation [9]. In a similar way, we have proposed coupled HNLS equation [10]. The coupled HNLS equation proposed by Tasgal and Potasek [11] is the coupled version of the Hirota equation. Recently, the hierarchy of the integrable coupled NLS equation has been reported and it has been shown that the equation analysed by Tasgal and Potasek is the next hierarchy of the bright CNLS system. There has not been much work on the dark soliton solutions for the coupled Hirota equations so far. So the present analysis will uncover the integrability criterion for obtaining dark soliton solutions using various integrability techniques such as Painlevé analysis [12], Lax pair (AKNS) [13] and constructing the dark one- and two-soliton solutions through Hirota’s bilinearisation method [14].
2. Coupled Hirota equation As we have already discussed, the soliton aspects of Manakov model have been well studied by many authors [15–17]. If we are using high intensity ultra-short pulses through optical glass fiber, then the Manakov model is found to be inadequate and one has to incorporate the higher-order linear and nonlinear effects. These types of ultra-short pulses are widely used to increase the transmission capacity of information systems in the form of wavelength division multiplexing (WDM) network. In order to increase the transmission capacity of the network systems, one has to increase the number of channels with minimum frequency difference and the technique is called WDM method. In addition to the above, the coupled Hirota (CH) system explains pulse propagation in single mode fibers. But, in reality, even a single mode fiber admits birefringent effect and hence the two pulses are propagating in orthogonal directions. In this case, depending on the field strength, the field propagating along one direction may change the refractive index of the other one and vice versa. Excluding the effect of SRS, the co-propagation of two ultrashort pulses with the effects of TOD and SS, in this case, is governed by the generalised coupled equation of the form iq1t + c1 q1zz + 2 α|q1 |2 + β|q2|2 q1 − iε q1zzz + 2µ1 |q1 |2 + ν1 |q2 |2 q1z + ν1 q1 q2∗ q2z = 0, (1) 2 2 2 2 ∗ iq2t + c2 q2zz + 2 β|q1 | + γ |q2| q2 − iε q2zzz + ν2 |q1 | + 2µ2 |q2 | q2z + ν2 q1 q2 q1z = 0, (2) where qj is the complex amplitude of the pulse envelope and z and t represent the spatial and temporal coordinates. α and β refer to self-phase modulation (SPM) and cross-phase modulation (XPM), respectively. The above generalised version has been considered here for the purpose of analysing various possibilities of integrable cases from the point of view of Painlevé analysis. It has been already reported that Eqs. (1), (2) admits soliton solutions only for the conditions c1 = c2 , α = β = γ , µ1 = ν1 = µ2 = ν2 = 3 and c1 = −c2 ,
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α = −β = γ , µ1 = −ν1 = −µ2 = ν2 = 3. For the first condition, exact N -soliton solutions have been reported [11,18] which corresponds to bright solitons and latter condition is one of the new integrable systems, which is not well studied [19]. This case may deal with the bright–dark soliton pair. In this Letter, we have shown from the Painlevé analysis that there is one more integrability case corresponding to the dark–dark soliton pair, which has not been analysed for this system. In this context, it should be mentioned that for this system, Park and Shin have constructed the Bäcklund transformation and analysed the dark–dark, bright–dark and bright–bright pair of solutions [20]. In this Letter, we are mainly concerned with the dark soliton analysis due to its potential applications in communication, soliton switching and logic gate operations. This is because, in certain regards, such as inherent stability, reduction of jitter and so on, dark solitons are preferred instead of bright solitons [21].
3. Painlevé analysis The Painlevé analysis is one of the powerful methods for identifying the complete integrability properties of the nonlinear partial differential equations (NPDEs). Weiss et al. [12] introduced an algorithm for carrying out the Painlevé property of given NPDEs. Once the given equation passes the Painlevé property, we can look for the Lax pair, Hirota bilinear form and Bäcklund transformation for establishing the complete integrability properties of the system. To apply Painlevé analysis, we express q1 = p, q1∗ = q, q2 = r, q2∗ = s (where the asterisk represents the complex conjugate). By applying these new variables, Eqs. (1), (2) become ipt + c1 pzz + 2(αpq + βrs)p − iε pzzz + (2µ1 pq + ν1 rs)pz + ν1 psrz = 0, −iqt + c1 qzz + 2(αpq + βrs)q + iε qzzz + (2µ1 pq + ν1 rs)qz + ν1 qrsz = 0, irt + c2 rzz + 2(βpq + γ rs)r − iε rzzz + (ν2 pq + 2µ2 rs)rz + ν2 qrpz = 0, −ist + c2 szz + 2(βpq + γ rs)s + iε szzz + (ν2 pq + 2µ2 rs)sz + ν2 psqz = 0.
(3) (4) (5) (6)
The generalised Laurent series expansions of p, q, r and s are p = φk
∞
pj (z, t)φ j (z, t),
q = φl
j =0
r = φm
∞
∞
qj (z, t)φ j (z, t),
j =0
rj (z, t)φ j (z, t),
j =0
s = φn
∞
sj (z, t)φ j (z, t),
(7)
j =0
with p0 , q0 , . . . , s0 = 0; k, l, m and n are negative integers; pj , qj , . . . , sj are the set of expansion coefficients which are analytic in the neighbourhood of the noncharacteristic singular manifold φ(z, t) = z + ψ(t) = 0. Looking at the leading-order behaviour, we substitute in Eqs. (3)–(6) p ≈ p0 φ k , q ≈ q0 φ l , r ≈ r0 φ m , s ≈ s0 φ n and, upon balancing terms, we obtain k = l = m = n = −1, and the compatibility conditions are obtained as µ1 (p0 q0 ) + ν1 (r0 s0 ) = −3,
µ2 (r0 s0 ) + ν2 (p0 q0 ) = −3.
(8)
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Substituting the full Laurent series and keeping the leading-order terms alone, we obtain the following results: A −2µ1 p02 ν1 p0 s0 (j − 2) −2ν1 p0 r0 −2µ1q02 A −2ν1 q0 s0 ν1 r0 q0 (j − 2) , (9) ν2 r0 q0 (j − 2) −2ν2 p0 r0 B −2µ2 r02 −2ν2 s0 q0 ν2 p0 s0 (j − 2) −2µ2 s02 B where A and B are given by A = (j − 1)(j − 2)(j − 3) + µ1 (j − 2)p0 q0 − 3(j − 2), B = (j − 1)(j − 2)(j − 3) + µ2 (j − 2)r0 s0 − 3(j − 2). From the detailed analysis, we find that Eqs. (1) and (2), in addition to bright soliton case [22], also admit a sufficient number of positive resonances and arbitrary functions when µ1 = µ2 = ν1 = ν2 = −3,
c1 = c2 = −1,
α = −β = γ = 1.
(10)
Thus the resonances are found to be j = −1, 0, 0, 0, 1, 2, 2, 3, 4, 4, 4, 5.
(11)
As usual, the resonance at j = −1 corresponds to the arbitrariness of the singularity manifold φ. Upon substituting the full Laurent series into Eqs. (3)–(6) and collecting the coefficients of different powers of φ, we find that Eqs. (1), (2) admit sufficient number of arbitrary functions for the above conditions. Having identified the integrability nature of Eqs. (1), (2), our next aim is to construct the Lax pair and dark soliton solutions for the obtained choices of parameters.
4. Lax pair for the coupled Hirota equation In this section, we generalise the 2 ×2 AKNS method to a 3 ×3 eigenvalue problem. To establish the integrability properties, we construct the linear eigenvalue problem in the following form: Ψz = U Ψ,
Ψt = V Ψ,
where Ψ = (Ψ1 Ψ2 Ψ3 )T .
Here, the Lax operators U and V are given in the form −iλ/2 −k1 q1 −k1 q2 U = k1 q1∗ iλ/2 0 , k1 q2∗ 0 iλ/2
(12)
(13)
λ being the isospectral parameter and k1 is an arbitrary constant. In general, the matrix V is written as Vij =
3 n=0
(n)
Vij λn .
(14)
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The integrability condition for U and V is Ut − Vz + [U, V ] = 0. Using this condition and choosing the appropriate constants of integration, the V matrix is obtained as iε/2 0 0 A2 εk1 q1 εk1 q2 2 V = λ3 0 −iε/2 0 + λ −εk1 q1∗ −A2 0 0 0 −iε/2 0 −A2 −εk1 q2∗ G H J −iεk12 (|q1 |2 + |q2 |2 ) iεk1q1z − 2iA2 k1 q1 iεk1 q2z − 2iA2 k1 q2 ∗ + λ iεk1 q1z +K L M , + 2iA2k1 q1∗ iεk12 |q1 |2 iεk12 q1∗ q2 ∗ N P Q iεk1 q2z + 2iA2k1 q2∗ iεk12 q2∗ q1 iεk12 |q2 |2 (15) where ∗ ∗ G = −2A2k12 |q1 |2 + |q2|2 − εk12 q1 q1z − q1∗ q1z + q2 q2z − q2∗ q2z , H = −εk1 q1zz + 2A2k1 q1z − 2εk13 q1 |q1 |2 + |q2 |2 , J = −εk1 q2zz + 2A2 k1 q2z − 2εk13 q2 |q1 |2 + |q2 |2 , ∗ ∗ ∗ L = −εk12 q1∗ q1z − q1z + 2A2 k1 q1z + 2εk13 q1∗ |q1 |2 + |q2 |2 , q1 + 2A2 k12 |q1 |2 , K = εk1 q1zz ∗ ∗ M = −εk12 q1∗ q2z − q2 q1z + 2A2 k12 q1∗ q2 , N = εk1 q2zz + 2A2 k1 q2z + 2εk13 q2∗ |q1 |2 + |q2 |2 , ∗ ∗ + 2A2 k12 q2∗ q1 , Q = −εk12 q2∗ q2z − q2z q2 + 2A2k12 |q2 |2 . P = −εk12 q2∗ q1z − q1 q2z Compatibility condition for the above Lax pair gives the following form of equations: k1 q1t + 2A2 k1 q1zz + 4k13 A2 |q1 |2 + |q2 |2 q1 − iε −ik1q1zzz − 3ik13 |q1 |2 + |q2 |2 q1z − 3ik13 q1 q1∗ q1z + q2∗ q2z = 0, k1 q2t + 2A2 k1 q2zz + 4k13 A2 |q1 |2 + |q2 |2 q2 − iε −ik1q2zzz − 3ik13 |q1 |2 + |q2 |2 q2z − 3ik13 q2 q1∗ q1z + q2∗ q2z = 0.
(16)
(17)
We have found that the above equations give the bright soliton version of the coupled Hirota (CH) equations for the choice [11]: k1 = 1,
A2 = −i/2.
(18)
For the above choice of parameters, the Lax pair and the inverse scattering method have been analysed [11] and N -soliton solutions were reported through Hirota bilinear form [23]. For the choice k1 = i,
A2 = i/2,
(19)
one can obtain dark soliton version of the coupled Hirota equation.
5. Lax pair for the N -coupled Hirota equation If the WDM using solitons is to be compatible with present linear systems, it is necessary that one should be able to propagate not only two fields but more than two. Hence, it is highly significant that N -field propagation should
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be properly analysed. Keeping this in mind, in this section, we provide the Lax pair for N -fields and derive the system of dark N -coupled Hirota equation. This is achieved by generalising the above results to (N + 1) × (N + 1) eigenvalue problem, so that one can get dark soliton solutions for N -field propagation. The linear eigenvalue problem is written as ψz = U ψ,
ψt = V ψ,
ψ = (ψ1 ψ2 ψ3 . . . ψN ψN+1 )T ,
(20)
where
−iλ/2
−k1 q1
−k1 q2
...
−k1 qN
iλ/2 ... 0 k1 q1∗ 0 iλ/2 ... 0 , U = k1 q2∗ (21) .. .. .. .. .. . . . . . ∗ 0 0 ... iλ/2 k1 qN εk1 q1 εk1 q2 . . . εk1 qN iε/2 0 0 ... 0 A2 −A2 0 ... 0 −iε/2 0 ... 0 0 −εk1 q1∗ 0 −A2 ... 0 0 −iε/2 . . . 0 + λ2 −εk1 q2∗ V = λ3 0 .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . ∗ −εk1 qN 0 0 ... −A2 0 0 0 . . . −iε/2 −iεk12 N |qj |2 iεk1q1z − 2iA2 k1 q1 iεk1q2z − 2iA2 k1 q2 . . . iεk1 qNz − 2iA2 k1 qN j =1 ∗ + 2iA2k1 q1∗ iεk12 |q1 |2 iεk12 q1∗ q2 ... iεk12 q1∗ qN iεk1 q1z ∗ ∗ 2 ∗ 2 2 2 ∗ iεk1 q2 q1 iεk1 |q2 | ... iεk1 q2 qN + λ iεk1 q2z + 2iA2k1 q2 .. .. .. .. .. . . . . . ∗ + 2iA2k1 qN∗ iεk12 qN∗ q1 iεk12 qN∗ q2 ... iεk12 |qN |2 iεk1 qNz B1 B2 B3 . . . BN C2 C3 . . . CN C1 + D1 D2 D3 . . . DN , (22) .. .. .. .. .. . . . . . P1 P2 P3 . . . PN where B1 = −2A2k12
N j =1
|qj |
2
− εk12
N qj qj∗z − qj z qj∗ ,
B2 = −εk1 q1zz + 2A2 k1 q1z − 2εk13 q1
j =1
B3 = −εk1 q2zz + 2A2 k1 q2z − 2εk13 q2
N
|qj |2 ,
j =1 N j =1
|qj |2 ,
BN = −εk1 qNzz + 2A2k1 qNz − 2εk13 qN
N j =1
|qj |2 ,
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∗ ∗ C1 = −εk1 q1zz + 2A2 k1 q1z + 2εk13 q1∗
N
|qj |2 ,
j =1
∗ C3 = −εk12 q1∗ q2z − q1z q2 + 2A2 k12 q1∗ q2 , ∗ ∗ + 2A2 k1 q2z + 2εk12 q2∗ D1 = −εk1 q2zz
N
|qj |2 ,
∗ ∗ q2 q2z − q2z q2 + 2A2 k12 |q2 |2 ,
∗ ∗ + 2A2k1 qNz + 2εk13 q1∗ P1 = −εk1 qNzz
N j =1
P3 = −εk12
∗ C2 = −εk12 q1∗ q1z − q1z q1 + 2A2 k12 |q1 |2 ,
∗ CN = −εk12 q1∗ qNz − q1z qN + 2A2 k12 q1∗ qN ,
j =1
D3 = −εk12
327
|qj |2 ,
∗ ∗ qN q2z − qNz q2 + 2A2k12 q2 qN∗ ,
∗ D2 = −εk12 q2∗ q1z − q2z q1 + 2A2 k12 q2∗ q1 , ∗ DN = −εk12 q2∗ qNz − q2z qN + 2A2 k12 q2∗ qN , ∗ P2 = −εk12 qN∗ q1z − qNz q1 + 2A2 k12 q1 qN∗ , ∗ PN = −εk12 qN∗ qNz − qNz qN + 2A2 k12 |qN |2 .
(23)
The compatibility condition gives rise to the following system of equations: k1 q1t + 2A2 k1 q1zz + 4k13 A2 |q1 |2 + |q2 |2 + · · · + |qN |2 q1 − iε −ik1q1zzz − 3ik13 |q1 |2 + |q2 |2 + · · · + |qN |2 q1z − 3ik13 q1 q1∗ q1z + q2∗ q2z + · · · + qN∗ qNz = 0, .. .
k1 qNt + 2A2 k1 qNzz + 4k13 A2 |q1 |2 + |q2 |2 + · · · + |qN |2 qN − iε −ik1qNzzz − 3ik13 |q1 |2 + |q2 |2 + · · · + |qN |2 qNz − 3ik13 qN q1∗ q1z + q2∗ q2z + · · · + qN∗ qNz = 0. (24)
6. Dark one-soliton solution Having identified the condition for dark soliton case using Painlevé singularity structure analysis and constructed the Lax pair, our next aim is to generate the soliton solutions. From the Lax pair and using the Bäcklund transformations (BT), one can generate soliton solutions. As we are not able to construct BT for dark soliton system, in the following, we generate the dark soliton solutions through Hirota’s bilinear method. Before constructing the bilinear form for the coupled dark Hirota system, it would be rather convenient to transform the dark-CH equations into a set of complex modified KdV equations with the help of the following transformations: T t Z − , where j = 1, 2 and T = t, Z = z + . qj (z, t) = Qj (Z, T ) exp i (25) 2 3ε 27ε 3ε Then, Eqs. (17), (18) become Q1T − ε Q1ZZZ − 6|Q1 |2 + 3|Q2 |2 Q1Z − 3Q1 Q∗2 Q2Z = 0, Q2T − ε Q2ZZZ − 6|Q2 |2 + 3|Q1 |2 Q2Z − 3Q2 Q∗1 Q1Z = 0.
(26)
In this context, it should be mentioned that even though the coupled mKdV equation analysed here is not directly relevant in fiber optics, it has many applications in other fields. The twin hole dark solitary waves in nonintegrable systems were found in various physical settings such as the propagation of terahertz electromagnetic pulses in media characterized by the simultaneous presence of second- and third-order nonlinearities and the parametric interaction in diffractive quadratic nonlinear media. However, the cmKdV equations can be transformed to the CH equation, which describes ultrashort pulse propagation in birefringent fibers or coupled wave propagation in single
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mode fibers, through transformations (25). Once the soliton solutions are obtained for Eqs. (26), then it is easy to get the soliton solutions for the CH system using the same transformations. The Hirota bilinear for the dark-CH equations can be constructed by applying the transformation for the field variables as Q1 (Z, T ) =
G(Z, T ) , F (Z, T )
H (Z, T ) , F (Z, T )
Q2 (Z, T ) =
(27)
where G(Z, T ) and H (Z, T ) are complex functions and F (Z, T ) is a real function. Using Eqs. (27), the bilinear forms of Eqs. (26), are obtained as β1 (G · F ) = 0,
β2 (H · F ) = 0,
β2 (F · F ) = −2 GG∗ + H H ∗ ,
DZ (H · F ) = 0,
(28)
where β1 = DT − εDZ3 − 3ελDZ and β2 = DZ2 + λ, with λ a constant to be determined and the Hirota bilinear operators DZ and DT are defined as ∂ m ∂ ∂ n ∂
− − G(Z, T )F (Z , T ) . DZm DTn G(Z, T ) · F (Z, T ) = (29)
∂Z ∂Z ∂T ∂T Z=Z , T =T In order to obtain one-soliton solutions, we assume G = τ1 (1 + χg1 ),
H = τ2 (1 + χh1 )
and F = 1 + χf1 ,
(30)
where τ1 and τ2 are complex constants. Equating different powers of χ , we obtain χ 0:
λ = −2 |τ1 |2 + |τ2 |2 ,
χ:
β1 (1 · f1 + g1 · 1) = 0,
χ 2:
(31)
β1 (1 · f1 + h1 · 1) = 0, 2 β2 (1 · f1 + f1 · 1) = −4 |τ1 | + |τ2 |2 , β1 (h1 · f1 ) = 0, 2 2 β2 (f1 · f1 ) = −2 |τ1 | g1 + |τ2 |2 h21 .
(32)
β1 (g1 · f1 ) = 0,
(33)
The above equation admits the following solutions: (0) g1 = h1 − f1 = − exp ω1 T + c1 Z + ξ1 .
(34)
From Eqs. (31)–(33), we can obtain ω1 = εc1 c12 + 3λ and c12 = −2λ.
(35)
From the above results, the dark one-soliton solution of cmKdV equation is obtained as c1 Z c13 εT Q1 = τ1 tanh − , 2 4
c1 Z c13 εT Q2 = τ2 tanh − . 2 4
(36)
Using transformations (25), we can easily obtain the corresponding dark one-soliton solution of the CH equation. Thus, the possibility of a new type of dark–dark soliton propagation in CH system is established clearly. This dark soliton solution is plotted as shown in Fig. 1.
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329
Fig. 1.
7. Dark two-soliton solution For constructing the dark two-soliton solutions we assume H = τ2 1 + χh1 + χ 2 h2 G = τ1 1 + χg1 + χ 2 g2 ,
and F = 1 + χf1 + χ 2 f2 ,
(37)
where, τ1 , τ2 are complex constants and g1 , g2 , g3 , h1 , h2 , h3 , f1 , f2 are real functions. Using the usual Hirota identities, we obtain the following set of equations for different orders of χ : χ:
χ 2:
χ 3:
χ 4:
β1 (1 · f1 + g1 · 1) = 0,
β1 (1 · f1 + h1 · 1) = 0, 2 β2 (1 · f1 + f1 · 1) = −4 |τ1 | g1 + |τ2 |2 h1 , β1 (1 · f2 + h1 · f1 + h2 · 1) = 0, 2 2 β2 (1 · f2 + f1 · f1 + f2 · 1) = −2 |τ1 | g1 + 2g2 + |τ2 |2 h21 + 2h2 ,
(38)
β1 (1 · f2 + g1 · f1 + g2 · 1) = 0,
β1 (h1 · f2 + h2 · f1 ) = 0, 2 β2 (f1 · f2 + f2 · f1 ) = −4 |τ1 | g1 g2 + |τ2 |2 h1 h2 ,
(39)
β1 (g1 · f2 + g2 · f1 ) = 0,
β1 (h2 · f2 ) = 0, β2 (f1 · f2 + f2 · f1 ) = −2 |τ1 |2 g22 + |τ2 |2 h22 .
(40)
β1 (g2 · f2 ) = 0,
(41)
We take solutions of these equations to be of the form g1 = h1 = Z1 exp[ξ1 ] + Z2 exp[ξ2 ], g2 = h2 = A12 Z1 Z2 exp[ξ1 + ξ2 ],
f1 = exp[ξ1 ] + exp[ξ2 ], f2 = A12 exp[ξ1 + ξ2 ],
(42)
where ξj = Pj T − Ωj Z,
Pj = εΩj3 + 3ελΩj ,
j = 1, 2.
(43)
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Fig. 2.
The values of Zj are found to be Zj =
2|τ1 |2 + |τ2 |2 − Ωj2
(44)
2(|τ1 |2 + |τ2 |2 )
with λ = −2(|τ1 |2 + |τ2 |2 ). The value of A12 is found to be A12 =
(Z2 − Z1 ){−(P2 − P1 ) + ε(Ω23 − Ω13 ) − 3ελ(Ω2 − Ω1 )} (1 − Z1 Z2 ){−(P2 + P1 ) + ε(Ω23 + Ω13 ) − 3ελ(Ω2 + Ω1 )}
.
(45)
By using Eqs. (42)–(45), we can find the dark two-soliton solution explicitly. This dark two-soliton solution is plotted as shown in Fig. 2, which clearly shows that after collision, the pulses retain their shape with a slight change in their phase. Thus, in this Letter, we have reported the dark soliton version of CHNLS equations and also obtained the dark one-soliton and two-soliton solutions for both equations using the Hirota bilinear method.
8. Conclusion In this Letter, we have analysed the dark soliton version of coupled Hirota equations. Through Painlevé analysis we demonstrated that the results are in agreement with bright soliton case reported in the literature [11] and a new choice of parameters were given for the propagation of dark solitons. We also constructed the explicit Lax pair for both bright and dark soliton conditions. We also generalised our results to N -field propagation and constructed the one and two solitons through Hirota bilinear method. It was found that the system admits dark soliton propagation when the coefficient of self-steepening is negative, −6 to be precise. The construction of specific Lax pair confirms the integrability of the dark soliton version of this equation. The one- and two-soliton solutions were obtained by means of Hirota’s bilinear technique. Here, it should be mentioned that we have analysed only the dark–dark soliton pair whereas there are other possibilities like bright–bright, bright–dark soliton propagations in the coupled Hirota system. Work in the coupled state of bright–dark soliton pair is in progress and the results will be published elsewhere. From the plots of the dark one- and two-soliton solutions, it can be clearly seen that dark solitons exist for the CH system as the femtosecond optical pulses retain their dark solitary wave nature even in the presence of higher-order effects like TOD, SS. The presence of higher-order terms are felt by their influence on the velocity of dark solitons.
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But otherwise they leave the solitons’ shape intact. From the plot for two-soliton solution, it can be concluded that the presence of higher-order terms certainly influences the phase and velocity of dark solitons. Yet, they maintain their inelastic behaviour since after collision, they retain their shape and intensity only with a slight change in their phase. As the dark solitons are preferred to the bright solitons because of their inherent stability, resistance to the influence of noise and fiber loss, we believe that the study of dark solitons in higher-order systems will be useful for future applications.
Acknowledgement K.P. wishes to thank AICTE, DST and NBHM, Government of India, for their financial support through major projects.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
A. Hasegawa, Y. Kodama, Solitons in Optical Communication, Oxford University Press, 1995. Y.S. Kivshar, Opt. Quantum Electron. 30 (1998) 571. Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. 23 (1987) 510. G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed., Academic, New York, 1995. Y.S. Kivshar, B. Luther-Davies, Phys. Rep. 298 (1998) 81. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 142. L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 171. S.V. Manakov, Sov. Phys. JETP 38 (1974) 248. K. Porsezian, P. Shanmugha Sundaram, A. Mahalingam, Phys. Rev. E 50 (1994) 1543. R.S. Tasgal, M.J. Potasek, J. Math. Phys. 33 (1992) 1208. J. Weiss, M.T. Tabor, G. Carnevale, J. Math. Phys. 24 (1983) 552. P.D. Lax, Comm. Pure Appl. Math. 21 (1968) 467. R. Hirota, in: Lecture Notes in Mathematics, Vol. 515, Springer, Berlin, 1976, p. 40. K. Porsezian, K. Nakkeeran, Phys. Rev. Lett. 76 (1996) 3955. K. Porsezian, P. Shanmugha Sundaram, A. Mahalingam, J. Phys. A: Math. Gen. 32 (1999) 8731. R. Radhakrishnan, M. Lakshmanan, J. Phys. A 28 (1995) 2683. R. Radhakrishnan, M. Lakshmanan, M. Daniel, J. Phys. A 28 (1995) 7299. K. Porsezian, J. Nonlinear Math. Phys. 5 (1998) 126. Q.-H. Park, H.J. Shin, Opt. Comm. 178 (2000) 233. Y.S. Kivshar, B. Luther-Davies, Phys. Rep. 298 (1998) 81. R.M. Miura, in: Lecture Notes in Mathematics, Vol. 515, Springer, Berlin, 1976. R. Radhakrishnan, M. Lakshmanan, Phys. Rev. E 28 (1995) 2683.
Physics Letters A 286 (2001) 321–331 www.elsevier.com/locate/pla
Dark soliton solutions of the coupled Hirota equation in nonlinear fiber S.G. Bindu, A. Mahalingam, K. Porsezian ∗ Department of Physics, Anna University, Chennai 600 025, India Received 20 November 2000; received in revised form 24 April 2001; accepted 7 May 2001 Communicated by A.R. Bishop
Abstract We consider the coupled Hirota equation which describes the pulse propagation in a coupled fiber with higher-order dispersion and self-steepening. Using the Painlevé analysis, we obtain the parametric conditions for the existence of bright and dark solitons. For the identified case(s), we also construct the common Lax pair and the soliton solutions are constructed for the dark soliton case. Also we extend our results to N-field propagation and the explicit Lax pair is constructed. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Recent experimental achievements have increased interest in the potential applications of optical dark solitons such as optical switching [1]. It is well known that optical bright solitons can be used for long distance communication to drastically increase the bit rate of fiber transmission systems. Dark solitons are reflectionless radiation modes of the wave guides, which also have a localised shape similar to bright solitons, but with complex envelope and nonvanishing asymptotics. In the case of temporal solitons, the group velocity dispersion is known to vanish at a wavelength of 1.3 µm and is positive at larger wavelengths and negative at shorter ones. Since silica optical fibers have always a +ve Kerr coefficient, the two different signs of group velocity dispersion support two different types of solitons, dark in the former case and bright in the latter case [2–4]. With the current interest of using solitons as pulse bits in long optical fibers for communication purposes, it is important for us to re-evaluate the practicality of using analytical techniques for predicting the behaviour of such bits. Since such pulses are near a pure soliton solution, it becomes feasible to use analytical soliton techniques and potentiality of obtaining useful analytical results becomes very high [5]. Slowly varying amplitude electromagnetic waves in a nonlinear medium are described by the nonlinear Schrödinger (NLS) equation [6]. In order to increase the bit rates, it is necessary to decrease the pulse width. As pulse lengths become comparable to the wavelength, however, the NLS equation becomes inadequate, additional terms have to be included and the resulting pulse * Corresponding author.
E-mail address: [email protected] (K. Porsezian). 0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 3 7 1 - 1
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propagation is called as higher-order nonlinear Schrödinger (HNLS) equation [7]. This equation includes effects like third-order dispersion (TOD), self-steepening (SS) and stimulated Raman scattering (SRS). The nonlinear response of a dielectric medium to electromagnetic radiation includes contribution from quadratic, cubic, and higher-order terms in the electric field. It is well known that TOD effects splitting-up of higherorder solitons. The inelastic Raman scattering is due to the delayed response of the medium which forces the pulse to undergo a frequency shift which is known as self-frequency shift. The effect of self-steepening is due to the intensity-dependent group velocity of the optical pulse, which gives the pulse a very narrow width in the course of propagation. Because of this, the peak of the pulse will travel slower than the wings. Among the effects associated with third-order nonlinearity, stimulated Raman scattering and stimulated Brillouin scattering limit the maximum input power available for transmission, whereas self-phase modulation directly influences the dispersion by modifying the pulse shape and thus can play a vital role, together with chromatic dispersion, in determining the transmission rate attainable in a given fiber. Wave propagation in optical fibers, with these higher-order effects, is governed by the HNLS equation. When solitons consist of several interacting modes, the stationary waves are described by a system of coupled nonlinear partial differential equations. In the case of wavelength division multiplexing (WDM) [8], one should consider at least two optical fields simultaneously. In 1974, Manakov proposed the coupled NLS equation [9]. In a similar way, we have proposed coupled HNLS equation [10]. The coupled HNLS equation proposed by Tasgal and Potasek [11] is the coupled version of the Hirota equation. Recently, the hierarchy of the integrable coupled NLS equation has been reported and it has been shown that the equation analysed by Tasgal and Potasek is the next hierarchy of the bright CNLS system. There has not been much work on the dark soliton solutions for the coupled Hirota equations so far. So the present analysis will uncover the integrability criterion for obtaining dark soliton solutions using various integrability techniques such as Painlevé analysis [12], Lax pair (AKNS) [13] and constructing the dark one- and two-soliton solutions through Hirota’s bilinearisation method [14].
2. Coupled Hirota equation As we have already discussed, the soliton aspects of Manakov model have been well studied by many authors [15–17]. If we are using high intensity ultra-short pulses through optical glass fiber, then the Manakov model is found to be inadequate and one has to incorporate the higher-order linear and nonlinear effects. These types of ultra-short pulses are widely used to increase the transmission capacity of information systems in the form of wavelength division multiplexing (WDM) network. In order to increase the transmission capacity of the network systems, one has to increase the number of channels with minimum frequency difference and the technique is called WDM method. In addition to the above, the coupled Hirota (CH) system explains pulse propagation in single mode fibers. But, in reality, even a single mode fiber admits birefringent effect and hence the two pulses are propagating in orthogonal directions. In this case, depending on the field strength, the field propagating along one direction may change the refractive index of the other one and vice versa. Excluding the effect of SRS, the co-propagation of two ultrashort pulses with the effects of TOD and SS, in this case, is governed by the generalised coupled equation of the form iq1t + c1 q1zz + 2 α|q1 |2 + β|q2|2 q1 − iε q1zzz + 2µ1 |q1 |2 + ν1 |q2 |2 q1z + ν1 q1 q2∗ q2z = 0, (1) 2 2 2 2 ∗ iq2t + c2 q2zz + 2 β|q1 | + γ |q2| q2 − iε q2zzz + ν2 |q1 | + 2µ2 |q2 | q2z + ν2 q1 q2 q1z = 0, (2) where qj is the complex amplitude of the pulse envelope and z and t represent the spatial and temporal coordinates. α and β refer to self-phase modulation (SPM) and cross-phase modulation (XPM), respectively. The above generalised version has been considered here for the purpose of analysing various possibilities of integrable cases from the point of view of Painlevé analysis. It has been already reported that Eqs. (1), (2) admits soliton solutions only for the conditions c1 = c2 , α = β = γ , µ1 = ν1 = µ2 = ν2 = 3 and c1 = −c2 ,
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α = −β = γ , µ1 = −ν1 = −µ2 = ν2 = 3. For the first condition, exact N -soliton solutions have been reported [11,18] which corresponds to bright solitons and latter condition is one of the new integrable systems, which is not well studied [19]. This case may deal with the bright–dark soliton pair. In this Letter, we have shown from the Painlevé analysis that there is one more integrability case corresponding to the dark–dark soliton pair, which has not been analysed for this system. In this context, it should be mentioned that for this system, Park and Shin have constructed the Bäcklund transformation and analysed the dark–dark, bright–dark and bright–bright pair of solutions [20]. In this Letter, we are mainly concerned with the dark soliton analysis due to its potential applications in communication, soliton switching and logic gate operations. This is because, in certain regards, such as inherent stability, reduction of jitter and so on, dark solitons are preferred instead of bright solitons [21].
3. Painlevé analysis The Painlevé analysis is one of the powerful methods for identifying the complete integrability properties of the nonlinear partial differential equations (NPDEs). Weiss et al. [12] introduced an algorithm for carrying out the Painlevé property of given NPDEs. Once the given equation passes the Painlevé property, we can look for the Lax pair, Hirota bilinear form and Bäcklund transformation for establishing the complete integrability properties of the system. To apply Painlevé analysis, we express q1 = p, q1∗ = q, q2 = r, q2∗ = s (where the asterisk represents the complex conjugate). By applying these new variables, Eqs. (1), (2) become ipt + c1 pzz + 2(αpq + βrs)p − iε pzzz + (2µ1 pq + ν1 rs)pz + ν1 psrz = 0, −iqt + c1 qzz + 2(αpq + βrs)q + iε qzzz + (2µ1 pq + ν1 rs)qz + ν1 qrsz = 0, irt + c2 rzz + 2(βpq + γ rs)r − iε rzzz + (ν2 pq + 2µ2 rs)rz + ν2 qrpz = 0, −ist + c2 szz + 2(βpq + γ rs)s + iε szzz + (ν2 pq + 2µ2 rs)sz + ν2 psqz = 0.
(3) (4) (5) (6)
The generalised Laurent series expansions of p, q, r and s are p = φk
∞
pj (z, t)φ j (z, t),
q = φl
j =0
r = φm
∞
∞
qj (z, t)φ j (z, t),
j =0
rj (z, t)φ j (z, t),
j =0
s = φn
∞
sj (z, t)φ j (z, t),
(7)
j =0
with p0 , q0 , . . . , s0 = 0; k, l, m and n are negative integers; pj , qj , . . . , sj are the set of expansion coefficients which are analytic in the neighbourhood of the noncharacteristic singular manifold φ(z, t) = z + ψ(t) = 0. Looking at the leading-order behaviour, we substitute in Eqs. (3)–(6) p ≈ p0 φ k , q ≈ q0 φ l , r ≈ r0 φ m , s ≈ s0 φ n and, upon balancing terms, we obtain k = l = m = n = −1, and the compatibility conditions are obtained as µ1 (p0 q0 ) + ν1 (r0 s0 ) = −3,
µ2 (r0 s0 ) + ν2 (p0 q0 ) = −3.
(8)
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Substituting the full Laurent series and keeping the leading-order terms alone, we obtain the following results: A −2µ1 p02 ν1 p0 s0 (j − 2) −2ν1 p0 r0 −2µ1q02 A −2ν1 q0 s0 ν1 r0 q0 (j − 2) , (9) ν2 r0 q0 (j − 2) −2ν2 p0 r0 B −2µ2 r02 −2ν2 s0 q0 ν2 p0 s0 (j − 2) −2µ2 s02 B where A and B are given by A = (j − 1)(j − 2)(j − 3) + µ1 (j − 2)p0 q0 − 3(j − 2), B = (j − 1)(j − 2)(j − 3) + µ2 (j − 2)r0 s0 − 3(j − 2). From the detailed analysis, we find that Eqs. (1) and (2), in addition to bright soliton case [22], also admit a sufficient number of positive resonances and arbitrary functions when µ1 = µ2 = ν1 = ν2 = −3,
c1 = c2 = −1,
α = −β = γ = 1.
(10)
Thus the resonances are found to be j = −1, 0, 0, 0, 1, 2, 2, 3, 4, 4, 4, 5.
(11)
As usual, the resonance at j = −1 corresponds to the arbitrariness of the singularity manifold φ. Upon substituting the full Laurent series into Eqs. (3)–(6) and collecting the coefficients of different powers of φ, we find that Eqs. (1), (2) admit sufficient number of arbitrary functions for the above conditions. Having identified the integrability nature of Eqs. (1), (2), our next aim is to construct the Lax pair and dark soliton solutions for the obtained choices of parameters.
4. Lax pair for the coupled Hirota equation In this section, we generalise the 2 ×2 AKNS method to a 3 ×3 eigenvalue problem. To establish the integrability properties, we construct the linear eigenvalue problem in the following form: Ψz = U Ψ,
Ψt = V Ψ,
where Ψ = (Ψ1 Ψ2 Ψ3 )T .
Here, the Lax operators U and V are given in the form −iλ/2 −k1 q1 −k1 q2 U = k1 q1∗ iλ/2 0 , k1 q2∗ 0 iλ/2
(12)
(13)
λ being the isospectral parameter and k1 is an arbitrary constant. In general, the matrix V is written as Vij =
3 n=0
(n)
Vij λn .
(14)
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The integrability condition for U and V is Ut − Vz + [U, V ] = 0. Using this condition and choosing the appropriate constants of integration, the V matrix is obtained as iε/2 0 0 A2 εk1 q1 εk1 q2 2 V = λ3 0 −iε/2 0 + λ −εk1 q1∗ −A2 0 0 0 −iε/2 0 −A2 −εk1 q2∗ G H J −iεk12 (|q1 |2 + |q2 |2 ) iεk1q1z − 2iA2 k1 q1 iεk1 q2z − 2iA2 k1 q2 ∗ + λ iεk1 q1z +K L M , + 2iA2k1 q1∗ iεk12 |q1 |2 iεk12 q1∗ q2 ∗ N P Q iεk1 q2z + 2iA2k1 q2∗ iεk12 q2∗ q1 iεk12 |q2 |2 (15) where ∗ ∗ G = −2A2k12 |q1 |2 + |q2|2 − εk12 q1 q1z − q1∗ q1z + q2 q2z − q2∗ q2z , H = −εk1 q1zz + 2A2k1 q1z − 2εk13 q1 |q1 |2 + |q2 |2 , J = −εk1 q2zz + 2A2 k1 q2z − 2εk13 q2 |q1 |2 + |q2 |2 , ∗ ∗ ∗ L = −εk12 q1∗ q1z − q1z + 2A2 k1 q1z + 2εk13 q1∗ |q1 |2 + |q2 |2 , q1 + 2A2 k12 |q1 |2 , K = εk1 q1zz ∗ ∗ M = −εk12 q1∗ q2z − q2 q1z + 2A2 k12 q1∗ q2 , N = εk1 q2zz + 2A2 k1 q2z + 2εk13 q2∗ |q1 |2 + |q2 |2 , ∗ ∗ + 2A2 k12 q2∗ q1 , Q = −εk12 q2∗ q2z − q2z q2 + 2A2k12 |q2 |2 . P = −εk12 q2∗ q1z − q1 q2z Compatibility condition for the above Lax pair gives the following form of equations: k1 q1t + 2A2 k1 q1zz + 4k13 A2 |q1 |2 + |q2 |2 q1 − iε −ik1q1zzz − 3ik13 |q1 |2 + |q2 |2 q1z − 3ik13 q1 q1∗ q1z + q2∗ q2z = 0, k1 q2t + 2A2 k1 q2zz + 4k13 A2 |q1 |2 + |q2 |2 q2 − iε −ik1q2zzz − 3ik13 |q1 |2 + |q2 |2 q2z − 3ik13 q2 q1∗ q1z + q2∗ q2z = 0.
(16)
(17)
We have found that the above equations give the bright soliton version of the coupled Hirota (CH) equations for the choice [11]: k1 = 1,
A2 = −i/2.
(18)
For the above choice of parameters, the Lax pair and the inverse scattering method have been analysed [11] and N -soliton solutions were reported through Hirota bilinear form [23]. For the choice k1 = i,
A2 = i/2,
(19)
one can obtain dark soliton version of the coupled Hirota equation.
5. Lax pair for the N -coupled Hirota equation If the WDM using solitons is to be compatible with present linear systems, it is necessary that one should be able to propagate not only two fields but more than two. Hence, it is highly significant that N -field propagation should
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be properly analysed. Keeping this in mind, in this section, we provide the Lax pair for N -fields and derive the system of dark N -coupled Hirota equation. This is achieved by generalising the above results to (N + 1) × (N + 1) eigenvalue problem, so that one can get dark soliton solutions for N -field propagation. The linear eigenvalue problem is written as ψz = U ψ,
ψt = V ψ,
ψ = (ψ1 ψ2 ψ3 . . . ψN ψN+1 )T ,
(20)
where
−iλ/2
−k1 q1
−k1 q2
...
−k1 qN
iλ/2 ... 0 k1 q1∗ 0 iλ/2 ... 0 , U = k1 q2∗ (21) .. .. .. .. .. . . . . . ∗ 0 0 ... iλ/2 k1 qN εk1 q1 εk1 q2 . . . εk1 qN iε/2 0 0 ... 0 A2 −A2 0 ... 0 −iε/2 0 ... 0 0 −εk1 q1∗ 0 −A2 ... 0 0 −iε/2 . . . 0 + λ2 −εk1 q2∗ V = λ3 0 .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . ∗ −εk1 qN 0 0 ... −A2 0 0 0 . . . −iε/2 −iεk12 N |qj |2 iεk1q1z − 2iA2 k1 q1 iεk1q2z − 2iA2 k1 q2 . . . iεk1 qNz − 2iA2 k1 qN j =1 ∗ + 2iA2k1 q1∗ iεk12 |q1 |2 iεk12 q1∗ q2 ... iεk12 q1∗ qN iεk1 q1z ∗ ∗ 2 ∗ 2 2 2 ∗ iεk1 q2 q1 iεk1 |q2 | ... iεk1 q2 qN + λ iεk1 q2z + 2iA2k1 q2 .. .. .. .. .. . . . . . ∗ + 2iA2k1 qN∗ iεk12 qN∗ q1 iεk12 qN∗ q2 ... iεk12 |qN |2 iεk1 qNz B1 B2 B3 . . . BN C2 C3 . . . CN C1 + D1 D2 D3 . . . DN , (22) .. .. .. .. .. . . . . . P1 P2 P3 . . . PN where B1 = −2A2k12
N j =1
|qj |
2
− εk12
N qj qj∗z − qj z qj∗ ,
B2 = −εk1 q1zz + 2A2 k1 q1z − 2εk13 q1
j =1
B3 = −εk1 q2zz + 2A2 k1 q2z − 2εk13 q2
N
|qj |2 ,
j =1 N j =1
|qj |2 ,
BN = −εk1 qNzz + 2A2k1 qNz − 2εk13 qN
N j =1
|qj |2 ,
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∗ ∗ C1 = −εk1 q1zz + 2A2 k1 q1z + 2εk13 q1∗
N
|qj |2 ,
j =1
∗ C3 = −εk12 q1∗ q2z − q1z q2 + 2A2 k12 q1∗ q2 , ∗ ∗ + 2A2 k1 q2z + 2εk12 q2∗ D1 = −εk1 q2zz
N
|qj |2 ,
∗ ∗ q2 q2z − q2z q2 + 2A2 k12 |q2 |2 ,
∗ ∗ + 2A2k1 qNz + 2εk13 q1∗ P1 = −εk1 qNzz
N j =1
P3 = −εk12
∗ C2 = −εk12 q1∗ q1z − q1z q1 + 2A2 k12 |q1 |2 ,
∗ CN = −εk12 q1∗ qNz − q1z qN + 2A2 k12 q1∗ qN ,
j =1
D3 = −εk12
327
|qj |2 ,
∗ ∗ qN q2z − qNz q2 + 2A2k12 q2 qN∗ ,
∗ D2 = −εk12 q2∗ q1z − q2z q1 + 2A2 k12 q2∗ q1 , ∗ DN = −εk12 q2∗ qNz − q2z qN + 2A2 k12 q2∗ qN , ∗ P2 = −εk12 qN∗ q1z − qNz q1 + 2A2 k12 q1 qN∗ , ∗ PN = −εk12 qN∗ qNz − qNz qN + 2A2 k12 |qN |2 .
(23)
The compatibility condition gives rise to the following system of equations: k1 q1t + 2A2 k1 q1zz + 4k13 A2 |q1 |2 + |q2 |2 + · · · + |qN |2 q1 − iε −ik1q1zzz − 3ik13 |q1 |2 + |q2 |2 + · · · + |qN |2 q1z − 3ik13 q1 q1∗ q1z + q2∗ q2z + · · · + qN∗ qNz = 0, .. .
k1 qNt + 2A2 k1 qNzz + 4k13 A2 |q1 |2 + |q2 |2 + · · · + |qN |2 qN − iε −ik1qNzzz − 3ik13 |q1 |2 + |q2 |2 + · · · + |qN |2 qNz − 3ik13 qN q1∗ q1z + q2∗ q2z + · · · + qN∗ qNz = 0. (24)
6. Dark one-soliton solution Having identified the condition for dark soliton case using Painlevé singularity structure analysis and constructed the Lax pair, our next aim is to generate the soliton solutions. From the Lax pair and using the Bäcklund transformations (BT), one can generate soliton solutions. As we are not able to construct BT for dark soliton system, in the following, we generate the dark soliton solutions through Hirota’s bilinear method. Before constructing the bilinear form for the coupled dark Hirota system, it would be rather convenient to transform the dark-CH equations into a set of complex modified KdV equations with the help of the following transformations: T t Z − , where j = 1, 2 and T = t, Z = z + . qj (z, t) = Qj (Z, T ) exp i (25) 2 3ε 27ε 3ε Then, Eqs. (17), (18) become Q1T − ε Q1ZZZ − 6|Q1 |2 + 3|Q2 |2 Q1Z − 3Q1 Q∗2 Q2Z = 0, Q2T − ε Q2ZZZ − 6|Q2 |2 + 3|Q1 |2 Q2Z − 3Q2 Q∗1 Q1Z = 0.
(26)
In this context, it should be mentioned that even though the coupled mKdV equation analysed here is not directly relevant in fiber optics, it has many applications in other fields. The twin hole dark solitary waves in nonintegrable systems were found in various physical settings such as the propagation of terahertz electromagnetic pulses in media characterized by the simultaneous presence of second- and third-order nonlinearities and the parametric interaction in diffractive quadratic nonlinear media. However, the cmKdV equations can be transformed to the CH equation, which describes ultrashort pulse propagation in birefringent fibers or coupled wave propagation in single
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mode fibers, through transformations (25). Once the soliton solutions are obtained for Eqs. (26), then it is easy to get the soliton solutions for the CH system using the same transformations. The Hirota bilinear for the dark-CH equations can be constructed by applying the transformation for the field variables as Q1 (Z, T ) =
G(Z, T ) , F (Z, T )
H (Z, T ) , F (Z, T )
Q2 (Z, T ) =
(27)
where G(Z, T ) and H (Z, T ) are complex functions and F (Z, T ) is a real function. Using Eqs. (27), the bilinear forms of Eqs. (26), are obtained as β1 (G · F ) = 0,
β2 (H · F ) = 0,
β2 (F · F ) = −2 GG∗ + H H ∗ ,
DZ (H · F ) = 0,
(28)
where β1 = DT − εDZ3 − 3ελDZ and β2 = DZ2 + λ, with λ a constant to be determined and the Hirota bilinear operators DZ and DT are defined as ∂ m ∂ ∂ n ∂
− − G(Z, T )F (Z , T ) . DZm DTn G(Z, T ) · F (Z, T ) = (29)
∂Z ∂Z ∂T ∂T Z=Z , T =T In order to obtain one-soliton solutions, we assume G = τ1 (1 + χg1 ),
H = τ2 (1 + χh1 )
and F = 1 + χf1 ,
(30)
where τ1 and τ2 are complex constants. Equating different powers of χ , we obtain χ 0:
λ = −2 |τ1 |2 + |τ2 |2 ,
χ:
β1 (1 · f1 + g1 · 1) = 0,
χ 2:
(31)
β1 (1 · f1 + h1 · 1) = 0, 2 β2 (1 · f1 + f1 · 1) = −4 |τ1 | + |τ2 |2 , β1 (h1 · f1 ) = 0, 2 2 β2 (f1 · f1 ) = −2 |τ1 | g1 + |τ2 |2 h21 .
(32)
β1 (g1 · f1 ) = 0,
(33)
The above equation admits the following solutions: (0) g1 = h1 − f1 = − exp ω1 T + c1 Z + ξ1 .
(34)
From Eqs. (31)–(33), we can obtain ω1 = εc1 c12 + 3λ and c12 = −2λ.
(35)
From the above results, the dark one-soliton solution of cmKdV equation is obtained as c1 Z c13 εT Q1 = τ1 tanh − , 2 4
c1 Z c13 εT Q2 = τ2 tanh − . 2 4
(36)
Using transformations (25), we can easily obtain the corresponding dark one-soliton solution of the CH equation. Thus, the possibility of a new type of dark–dark soliton propagation in CH system is established clearly. This dark soliton solution is plotted as shown in Fig. 1.
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Fig. 1.
7. Dark two-soliton solution For constructing the dark two-soliton solutions we assume H = τ2 1 + χh1 + χ 2 h2 G = τ1 1 + χg1 + χ 2 g2 ,
and F = 1 + χf1 + χ 2 f2 ,
(37)
where, τ1 , τ2 are complex constants and g1 , g2 , g3 , h1 , h2 , h3 , f1 , f2 are real functions. Using the usual Hirota identities, we obtain the following set of equations for different orders of χ : χ:
χ 2:
χ 3:
χ 4:
β1 (1 · f1 + g1 · 1) = 0,
β1 (1 · f1 + h1 · 1) = 0, 2 β2 (1 · f1 + f1 · 1) = −4 |τ1 | g1 + |τ2 |2 h1 , β1 (1 · f2 + h1 · f1 + h2 · 1) = 0, 2 2 β2 (1 · f2 + f1 · f1 + f2 · 1) = −2 |τ1 | g1 + 2g2 + |τ2 |2 h21 + 2h2 ,
(38)
β1 (1 · f2 + g1 · f1 + g2 · 1) = 0,
β1 (h1 · f2 + h2 · f1 ) = 0, 2 β2 (f1 · f2 + f2 · f1 ) = −4 |τ1 | g1 g2 + |τ2 |2 h1 h2 ,
(39)
β1 (g1 · f2 + g2 · f1 ) = 0,
β1 (h2 · f2 ) = 0, β2 (f1 · f2 + f2 · f1 ) = −2 |τ1 |2 g22 + |τ2 |2 h22 .
(40)
β1 (g2 · f2 ) = 0,
(41)
We take solutions of these equations to be of the form g1 = h1 = Z1 exp[ξ1 ] + Z2 exp[ξ2 ], g2 = h2 = A12 Z1 Z2 exp[ξ1 + ξ2 ],
f1 = exp[ξ1 ] + exp[ξ2 ], f2 = A12 exp[ξ1 + ξ2 ],
(42)
where ξj = Pj T − Ωj Z,
Pj = εΩj3 + 3ελΩj ,
j = 1, 2.
(43)
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Fig. 2.
The values of Zj are found to be Zj =
2|τ1 |2 + |τ2 |2 − Ωj2
(44)
2(|τ1 |2 + |τ2 |2 )
with λ = −2(|τ1 |2 + |τ2 |2 ). The value of A12 is found to be A12 =
(Z2 − Z1 ){−(P2 − P1 ) + ε(Ω23 − Ω13 ) − 3ελ(Ω2 − Ω1 )} (1 − Z1 Z2 ){−(P2 + P1 ) + ε(Ω23 + Ω13 ) − 3ελ(Ω2 + Ω1 )}
.
(45)
By using Eqs. (42)–(45), we can find the dark two-soliton solution explicitly. This dark two-soliton solution is plotted as shown in Fig. 2, which clearly shows that after collision, the pulses retain their shape with a slight change in their phase. Thus, in this Letter, we have reported the dark soliton version of CHNLS equations and also obtained the dark one-soliton and two-soliton solutions for both equations using the Hirota bilinear method.
8. Conclusion In this Letter, we have analysed the dark soliton version of coupled Hirota equations. Through Painlevé analysis we demonstrated that the results are in agreement with bright soliton case reported in the literature [11] and a new choice of parameters were given for the propagation of dark solitons. We also constructed the explicit Lax pair for both bright and dark soliton conditions. We also generalised our results to N -field propagation and constructed the one and two solitons through Hirota bilinear method. It was found that the system admits dark soliton propagation when the coefficient of self-steepening is negative, −6 to be precise. The construction of specific Lax pair confirms the integrability of the dark soliton version of this equation. The one- and two-soliton solutions were obtained by means of Hirota’s bilinear technique. Here, it should be mentioned that we have analysed only the dark–dark soliton pair whereas there are other possibilities like bright–bright, bright–dark soliton propagations in the coupled Hirota system. Work in the coupled state of bright–dark soliton pair is in progress and the results will be published elsewhere. From the plots of the dark one- and two-soliton solutions, it can be clearly seen that dark solitons exist for the CH system as the femtosecond optical pulses retain their dark solitary wave nature even in the presence of higher-order effects like TOD, SS. The presence of higher-order terms are felt by their influence on the velocity of dark solitons.
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But otherwise they leave the solitons’ shape intact. From the plot for two-soliton solution, it can be concluded that the presence of higher-order terms certainly influences the phase and velocity of dark solitons. Yet, they maintain their inelastic behaviour since after collision, they retain their shape and intensity only with a slight change in their phase. As the dark solitons are preferred to the bright solitons because of their inherent stability, resistance to the influence of noise and fiber loss, we believe that the study of dark solitons in higher-order systems will be useful for future applications.
Acknowledgement K.P. wishes to thank AICTE, DST and NBHM, Government of India, for their financial support through major projects.
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