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Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics
Optics Communications 262 (2006) 250–256 www.elsevier.com/locate/optcom
Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics T. Brugarino a
a,*
, M. Sciacca
b
Dip. di Metodi e Modelli Matematici, Facolta` d’Ingegneria, Universita’ di Palermo, Viale delle Scienze, 90128 Palermo, Italy b Dip. di Matematica ed Applicazioni, 90134 Palermo, Italy Received 13 May 2005; received in revised form 8 November 2005; accepted 28 December 2005
Abstract Taking into account many developments in fiber optics communications, we propose a higher nonlinear Schro¨dinger equation (HNLS) with variable coefficients, more general than that in [R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274], which governs the propagation of ultrashort pulses in a fiber optics with generic variable dispersion. The study of this equation is performed using the Painleve´ test and the zero-curvature method. Also, we prove the equivalence between this equation and its anomalous integrable counterpart (the so-called Sasa–Satsuma equation). Finally, in view of its physical relevance, we present a soliton solution which represents the propagation of ultrashort pulses in a dispersion decreasing fiber. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik Keywords: Optical solitons; Integrable systems; Zero curvature; Inhomogeneous nonlinear Schro¨dinger equation
1. Introduction One of the most important models of modern nonlinear science is the nonlinear Schro¨dinger equation (NLS), which appears in many branches of physics and applied mathematics, such as nonlinear optic. The best knows solutions of the NLS equation are solitons, which have been the objects of extensive theoretical and experimental studies during the last three decades because of their potential applications in long distance communication and all-optical ultrafast switching devices [1–8]. The optical solitons in fiber, which are localized in time optical pulses, evolve from a nonlinear change in the refractive index of the material (silica glass) known as Kerr effect, induced by the light intensity distribution. From this, when the combined effects of self-phase modulation (SPM) and
of group velocity dispersion (GVD) exactly compensate, the pulse propagates without any change in its shape. This was firstly suggested by Hasegawa and Tappert [1] theoretically, and then its experimental check has been successfully carried out by Mollenauer et al. [2]. The propagation of picosecond optical pulses (in particular t0 > 5 ps) in mono-mode optical fibers is governed by the well-known completely integrable NLS. Although the NLS equation includes GVD and SPM, it does not considers dissipative effects that in a real fiber are present at the order of 0.2 db/km for a wavelength of the pulse near to 1.55 lm. When we consider the propagation of pulse in a real optical fiber, the dissipative effects have to be included in the governing equation, which take the form of a more general equation than the NLS one: iuz þ a1 utt þ a2 ujuj2 þ ia3 u ¼ 0;
ð1Þ
*
Corresponding author. Tel.: +39 0916657320; fax: +39 091427258. E-mail addresses: [email protected] (T. Brugarino), micheles@ math.unipa.it (M. Sciacca). 0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.12.065
where a1, a2 and a3 are, respectively, the GVD, the SPM and the fiber loss coefficients.
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
Eq. (1) describes a variety of nonlinear optical phenomena, like bright solitons, modulation instability and dark solitons [3]. The concept of soliton control is an important development in the application of solitons in the fiber optics communication, and has been extensively studied in [8–14]. In fact, it is well-known that when the group velocity dispersion is slightly modified, the shape of the soliton pulse changes: for example, when the GVD decreases, the soliton pulse gets compressed as it propagates along the length of the fiber. The first dispersion managed soliton in a fiber with hyperbolic decreasing group velocity dispersion was realized experimentally in 1991 by the Dianov’s group [15]. After that, many other examples were considered, for instance in [8,12,13]. Here, the authors proposed NLS equation with variable coefficients, as equation which governs the propagation of soliton in a real fibers with variable GVD (along the axis of the fiber), and sometimes also SPM. In a dispersion decreasing single mode optical fiber (DDF), where the GVD monotonically and smoothly decreases from an initial value to a smaller value at the end of its length, pulse propagation is characterize by a generalized normalized NLS equation of the form [8]: iuz þ
a1 ðzÞ 2 utt þ ujuj þ ia3 u MðzÞt2 u ¼ 0; 2
ð2Þ
where u is the slowly varying complex pulse envelope, t is the normalized retarded time, z is the normalized propagation distance, the term M(z)t2u accounts for electro-optic phase modulation, a3 is the fiber loss coefficient and the GVD coefficient a1(z) has an hyperbolic dispersion decreasing profile. The property of DDF makes it a powerful tool in controlling optical solitons in soliton communication systems where high quality, stable soliton pulse compression and soliton train generation can be effectively realized. Fiber with a nearly exponential GVD profile have been built [16]. A practical technique for building up such DDFs consists in reducing the core diameter along fiber length in a controlled manner during the fiber-drawing process. Variations in the core diameter change the waveguide contribution to GVD coefficient and reduce its magnitude. In order to increase the bit rate in fiber optics communication systems for a single carrier frequency, it is necessary to decrease the pulse width t0. When ultrashort pulses (t0 < 1 ps) propagates in a fiber optics, in addition to the SPM it will be necessary to consider nonlinear effects of higher order like self-steepening (also called dispersion Kerr) and self-frequency shift due to stimulated Raman scattering (SRS); on the other hand, apart from GVD, the ultrashort pulse also suffers from third-order dispersion (TOD). The equation which governs the propagation of such ultrashort pulses inside a fiber optics, including in this way these effects of higher order, has been proposed by Hasegawa and Kodama in [17], and it has the following form:
2
251 2
2
iuz þ a1 utt þ a2 ujuj þ ia3 uþ ia4 uttt þ a5 uðjuj Þt þ ia6 ðujuj Þt ¼ 0; ð3Þ where a4 and a6 are the parameters related to TOD, and self-steepening, respectively, while a5, which in general is complex, is obtained by the retarded nonlinear response [3]. This last term corresponds to the SRS in which the higher frequency components of the soliton spectrum pump energy to the lower frequency components, producing a nonlinear dissipation and down-shift of the carrier frequency of the soliton. Eq. (3) refers to ultrashort pulse in a system of reference that moves at the group velocity (GV) vg given by the inverse of the coefficient of the first derivative in t. Regarding Eq. (3), it admits soliton-type solution in wide domains of its parameters [18], but this has no relation with its integrability. In [19], using the Wahlquist–Estabrook prolongation method, in [20], using the Painleve´ analysis, and in [21], using the Galilean transformation and generalized Galilean invariance, it is stated that Eq. (3) is completely integrable, besides the standard NLS, only in the following four cases: (a) the higher nonlinear Schro¨dinger equation (HNLS) [22] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 0 : 1 : 1Þ; (b) the HNLS [23] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 0 : 1 : 0Þ; (c) the Hirota equation [24,25] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 1 : 6 : 0Þ; (d) the Sasa–Satsuma equation [25–27] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 1 : 6 : 3Þ. From the above results, one can notice that the integrability is guaranteed if and only if the loss coefficient a3 and the real part of the nonlinear dispersion coefficient a5 are zero. A mathematical point of view on the well-posedness of the Cauchy problem for the HNLS equation (3) is contained in [28,29]. Many authors [30–32] have considered the generalized NLS with variable coefficients in order to determine the constraints under which the equation might be integrable. This constraints entail the transformation of the equation to its autonomous integrable counterpart. In accordance with the arguments of generalized secondorder NLS with variable coefficients and with the generalized third-order NLS equation with GVD variable proposed in [14], in this paper we generalize the third-order NLS (3) by supposing that the dispersion in the fiber is varied along the axis z of the fiber in a more general way with respect to [14]. This implies, first of all, that both group velocity dispersion and the third-order dispersion coefficients depend on z.
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Then, since we are considering a real fiber optics, this forces the presence in the equation of the linear term in u which represents the loss, and therefore we include a term that is a phase modulation, as in Eq. (2), which in general depends on the axis length z and the time t. Since we are supposing a fiber optics with variable dispersion, which can imply an asymmetry in the pulse shape during the propagation, particularly due to the third-order dispersion [3], we assume here that the group velocity vg of the pulse is not constant but it depends on z and t. This appears in our equation via the first derivative ut, whose coefficient is equal to the inverse coefficient of the group velocity vg. At the end, the HNLS with variable coefficient which we will discuss here has the form: 2
iuz þ AðzÞutt þ Bujuj þ ðCðz; tÞ þ iC 1 Þu þ iH 1 ðz; tÞut þ iM 1 ðzÞuttt þ ðP þ iP 1 Þuðjuj2 Þt þ iQ1 ðujuj2 Þt ¼ 0;
ð4Þ
where A(z), B, C(z, t), C1, M1(z) and Q1 are, respectively, the GVD, SPM, phase modulation, loss, TOD and selfsteepening coefficients, H1(z, t) is the inverse of the group velocity vg, P and P1, as presented by Kodama and Hasegawa in [17], represents the nonlinear dispersion or SRS. In [33], the Cauchy problem for the HNLS with variable coefficients (4), without the terms u and ut, is considered. The authors show the local well-posedness for this problem in some appropriate Sobolev space. In this work, we check under which condition solitontype pulse propagation is possible. In order to do that we apply the Painleve´ analysis [34,35] and check the zerocurvature property. After that, we will propose two sets of transformations that will map Eq. (4), under suitable constraints, into the Sasa–Satsuma equation. We conclude our paper with an application in a dispersion decreasing fiber, by choosing a particular expression for the GVD coefficient [14] and finding a soliton solution. 2. Painleve´ test Since Eq. (4) is not analytic in u, we set u ! u + iv and we perform the Painleve´ test in the new equations: vz AðzÞutt Buðu2 þ v2 Þ Cðz; tÞu þ C 1 v þ M 1 ðzÞvttt þ H 1 ðz; tÞvt 2Put u2 þ 2P 1 ut uv þ 2Q1 ut uv 2Pvt uv þ 2P 1 vt v2 þ Q1 vt u2 þ 3Q1 vt v2 ¼ 0; uz þ AðzÞvtt þ Bvðu2 þ v2 Þ þ Cðz; tÞv þ C 1 u þ M 1 ðzÞuttt
ð5Þ
2
þ 2Put uv þ 2P 1 ut u þ Q1 ut v þ 3Q1 ut u ¼ 0
b
1 X j¼0
j
vj ðz; tÞ/ðz; tÞ .
d¼
M1 2P 1 þ 3Q1
and the constraint P = 0. After that, inserting in Eq. (5) the two expansions (6) and (7), we obtain the following resonances: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 . r ¼ 1; 0; 3; 4; 3 2 ð2P 1 þ 3Q1 Þ Now, we have the following resonances r = 1, 0, 2, 3, 4, 4 for the following condition: 2P 1 þ Q1 ¼ 0.
ð9Þ
In order to obtain the compatibility condition, we simplify the calculation by the assumption: /(z, t) = t + g(z) [36]. From Eq. (8), we can deduce the relation between u0 and v0, and the possibility to consider four branches of which only two essentially distinguished: first branch pffiffiffiffiffiffiffiffiffi u0 ¼ v0 ¼ 3d; second branch pffiffiffiffiffiffiffiffiffi u0 ¼ v0 ¼ 3d. Concerning the first branch, for j = 1, we have: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3dð3dB AðzÞÞ 3dð3dB AðzÞÞ u1 ¼ and v1 ¼ . 3dQ1 3dQ1 As for the second branch, for j = 1, we have: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3dð3dB AðzÞÞ 3dð3dB AðzÞÞ u1 ¼ and v1 ¼ . 3dQ1 3dQ1 For j = 2, the constraint (9) gives the resonance for both branches. In the same way, for j = 3, we have a resonance under the following constraints on the coefficients: AðzÞz . 2AðzÞ
For j = 4, we have a double resonance under the following constraints on the coefficients for both branches: ð6Þ
H 1tt ¼ 0; AðzÞH 1t 6Q1 dC t ¼ 0.
ð7Þ
In conclusion, we can simplify all the constraints on the coefficients as follow:
j¼0
v ¼ /ðz; tÞ
ð8Þ
where
C 1 ¼ H 1t
2
with the following Laurent series for u and v: 1 X a j u ¼ /ðz; tÞ uj ðz; tÞ/ðz; tÞ ;
u20 þ v20 ¼ 6d/2t ;
AðzÞ 6dB ¼ 0;
þ H 1 ðz; tÞut þ 2Pvt v2 þ 2P 1 vt uv þ 2Q1 vt uv 2
Looking at the leading order, if we replace u = u0/(z, t)a and v = v0/(z, t)b in Eq. (5), to balance the dominant terms we must require a = b = 1; this analysis also gives the condition:
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
P ¼ 0; H 1 ¼ r1 KðzÞt þ K 1 ðzÞ; C ¼ KðzÞt þ K 2 ðzÞ; C 1 ¼ r1 KðzÞ
AðzÞz ; 2AðzÞ
where Q 3M 1 r1 ¼ 1 ¼ ; B AðzÞ C1 AðzÞz KðzÞ ¼ þ r1 2r1 AðzÞ
ð10Þ ð11Þ
and K1(z) and K2(z) are generic functions of z. Now, we can resume the above results as follow: Eq. (4) is integrable if and only if it assumes the following form: C1 AðzÞz 2 iuz þ AðzÞutt þ Bujuj þ þ t þ K 2 ðzÞ þ iC 1 u r1 2r1 AðzÞ C1 AðzÞz þ þ i r1 t þ K 1 ðzÞ ut r1 2r1 AðzÞ AðzÞ B 2 2 uttt i uðjuj Þt þ iBðujuj Þt ¼ 0. þ r1 i ð12Þ 3 2 Looking at the previous equation, we observe that the TOD coefficient depend, explicitly, on the GVD coefficient, whereas we find a more precise choice in the coefficients of group velocity and phase modulation, which we will simplify at the end of the paper. Another remark is that, as in the constant case, in order to have integrability we have to set P = 0 in the equation. This will produce a lack of asymmetry in the profile of the single pulse |u|. The study of Eq. (12) is of great interest since it could have a wide range of applications since it governs propagation of a solitary wave, whose temporal width is less than 1 ps, in a non-uniform dispersive media. In concrete applications, Eq. (12) can be interesting not only for the amplification and compression of ultrashort optical solitons, but also for the stable transmission of ultrashort managed soliton.
253
We shall choose the matrices A and B as: 2 3 ik gu g u1 6 7 ik 0 5; A ¼ 4 g u1 gu 0 ik 2 3 0 0 0 6 7 B ¼ ðk þ k2 þ k3 ÞI þ 4 0 m 0 5 0 0 m 2 3 2 3 0 gq g q 0 gu g u1 6 7 6 7 0 5 þ 4 g q 0 0 5 þ s4 g u1 0 gu 0 0 gq 0 0 2 3 2 2 2 2 gBr1 u u1 3 g Br1 u1 u 0 3 6 2 7 2 0 0 þ 4 3 g Br1 u1 u ð16Þ 5; 23 gBr1 u2 u1 0 0 where u1 = u* and: AðzÞz t 2AðzÞ þ þ C 1 t þ K 1 ðzÞ; 2AðzÞ 3r1 AðzÞr1 2 utt iAðzÞut ; q¼ 3 3 sffiffiffiffiffiffiffiffiffiffiffi
s¼
B rit e 1; 2AðzÞ 2 3 1 0 0 6 7 I ¼ 40 1 05 0 0 1
g¼
and 1 1 1 2iAðzÞ iK 1 ðzÞ m ¼ Br1 ut u1 Br1 uu1t iBuu1 þ þ iK 2 ðzÞ. 6 6 3 3r21 r1 Substituting A and B in the compatibility condition (15), we find that Eq. (15) is equivalent to the HNLSE (12) and its to conjugate. 4. Reduction to the GNLS equation with constant coefficient
3. Zero-curvature The zero-curvature and Lax pairs are two different commutator representation of nonlinear evolution equations. They play a relevant role in the study of integrable dynamical systems. The zero-curvature representation is a more general version of the Lax representation. In order to present the zero-curvature representation of Eq. (12), we consider the two linear equations: Wt ¼ AW;
ð13Þ
Wz ¼ BW;
ð14Þ
Starting from Eq. (12), which has the PP (Painleve´ property), in this section we propose two sets of transformations, of dependent and independent variables, which will transform Eq. (12) into the following expression for the Sasa–Satsuma equation: iU Z þ a1 U TT þ a2 U jU j2 a a2 1 þ r1 i U TTT i U ðjU j2 ÞT þ ia2 ðU jU j2 ÞT ¼ 0; ð17Þ 3 2 where r1 ¼
a6 3a4 ¼ . a2 a1
where Aðz; t; u; u1 ; kÞ and Bðz; t; u; u1 ; kÞ are 3 · 3 matrices, k is the isospectral parameter and W = [W1(z, t), W2(z, t), W3(z, t)]T. If require that Eqs. (13) and (14) are compatible, then A and B must satisfy the equation
To achieve this goal, we set the following expression:
Az Bt þ ½A; B ¼ 0.
into Eq. (12), and obtain:
ð15Þ
u ¼ U ðZ; T Þf ðzÞeigðz;tÞ
ð18Þ
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f ðzÞ ¼ eC1 z ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ffi Z 3C1 z B e pffiffiffiffiffiffiffiffiffi dz; ZðzÞ ¼ a1 a2 AðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi C z qffiffiffiffiffiffiffiffiffiffiffiffiffi3 3C z ffi C z Z a AðzÞpffiffiffiffiffiffiffi 1 þ a r K ðzÞ e Ba Ba1 e 1 ðBa1 Þ e 1 2 1 2 1 1 Ba1 C1 z pffiffiffiffiffiffiffiffiffiffiffiffiffi e dz; t T ðz; tÞ ¼ a2 AðzÞ r1 a32 AðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi ! Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 a1 B C1 z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 3r21 K 2 ðzÞ a32 AðzÞ 3a2 AðzÞ a1 BeC1 z 2 a32 AðzÞ3 e gðz; tÞ ¼ 1 tþ r1 a2 AðzÞ 3r21 a32 AðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi C z qffiffiffiffiffiffiffiffiffi 3 3 3 3C 1 z 1 a1 B e 3r1 K 1 ðzÞ a2 AðzÞ 3a2 r1 K 1 ðzÞ a1 Be dz.
In conclusion, Eq. (12), after a change of the dependent and independent variables, is equivalent to the Sasa–Satsuma equation (17). 5. A particular solution In this section, we consider the propagation of soliton pulse in a dispersion decreasing fiber (DDF) and, in particular, we set the group velocity dispersion [14]: 1 AðzÞ ¼ e2C1 z ; 2 which has the maximum value A(0) = 1/2 and the minimum one A(z) = exp(2C1L)/2, where L is the length of the fiber. When L ! 1 then the GVD coefficient go to zero. According to the numerical results, in the evolution of ultrashort pulses in optical fiber, the group velocity vg decreases due to the red shift of the soliton spectrum, to the dispersion and in particular to GVD [3]. In order to simplify the calculation from now on we take B = 1. Under these conditions Eq. (12) can be rewritten as: 1 2 iuz þ e2C1 z utt þ ujuj þ ðK 2 ðzÞ þ iC 1 Þu þ iK 1 ðzÞut 2 e2C1 z i 2 2 uttt uðjuj Þt þ iðujuj Þt ¼ 0; þ r1 i 2 6
ð19Þ
while the transformations becomes: f ðzÞ ¼ eC1 z ; 1 2C1 z e ; ZðzÞ ¼ 2C 1 Z T ðz; tÞ ¼ t K 1 ðzÞ dz; Z 1 gðz; tÞ ¼ ð1 1Þt þ K 2 ðzÞ dz r1 Z ð2 þ 2Þ 2C1 z ð1 1Þ þ K 1 ðzÞ dz; e þ r1 12r21 C 1 where we have set a1 = 1/2 and a2 = 1, which are the standard values for the GVD and SPM coefficients.
The Sasa–Satsuma equation: 1 2 iU Z þ U TT þ U jU j 2 i i 2 2 þ r1 U TTT U ðjU j ÞT þ iðU jU j ÞT ¼ 0 6 2
ð20Þ
is an expression of the HNLS, proposed by Kodama and Hasegawa, for a particular combination of the coefficients. This equation, as discussed in the Introduction, is completely integrable and therefore it admits soliton wave as solution. Now, if we know a solution of Eq. (20), it is possible to find a solution of (19) using one of the two sets of transformations (18). To achieve this goal, we put U ¼ yðT þ bZÞ expðiðHZ XT ÞÞ in Eq. (20) and, after equating the real and imaginary part of the resulting equation, we obtain the conditions: 1 ; 3r21 1 X¼ r1
H¼
and the ordinary differential equation for y(n) (where n = T + bZ): 1 þ 2r1 b 3 y nn þ 4y þ 3 y ¼ 0; ð21Þ r21 which has doubly periodic elliptic functions as solutions. By integrating Eq. (21) we have: 1 þ 2r1 b 2 y þ 2y 4 ¼ 0; y 2n þ 3 r21 which, for br1 < 1/2, has the following solution [37,38]: pffiffiffi y ¼ g sechð 2gðT þ bZÞÞ; where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6br1 g¼ . 2r21
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
Therefore, we have the following solitary wave solution for Eq. (20): pffiffiffi T Z ; U ¼ g sechf 2gðT þ bZÞg exp i r1 3r21 which is a soliton that moves at the group velocity group vg = 1/b. Now, we use the previous solution U and one of the two set of transformations in Eq. (18) and we obtain, choosing K2 ¼
e2C1 z 3r21
and K1 constant, the following soliton as solution of Eq. (19): pffiffiffi t K 1z C 1 z u ¼ ge sechð 2g/Þ exp i ; ð22Þ r1 r1 where b 2C1 z e . / ¼ t K 1z 2C 1 The above solution is the same for both transformations. The coefficient K1 is the inverse of the group velocity and the last term represent a frequency shift of the soliton from the carrier frequency x0, due to dispersion. The profile of the soliton solution (22) is expressed by: pffiffiffi ð23Þ juj ¼ geC1 z sechð 2g/Þ and it is shown in Fig. 1 for a particular choice of the coefficients b, r1, C1 and K1. The term exp(C1z) represents the decay behaviour of the crest of the soliton due to the loss of the fiber. Notice that a frequency shift also changes the soliton speed from its original values vg = 1/K1. In fact, by differentiating the soliton along the characteristic curves / = constant, we obtain the following relation between the group velocity vg and the effective group velocity v0g : 1 1 ¼ be2C1 z 0 vg vg and since b is negative we deduce that v0g 6 vg .
Fig. 1. The amplitude profile (23) with the following values of the parameters: b = 5; r1 = 0.2; C1 = 0.2; K1 = 4.8733 · 106.
255
If we observe the profile of the solution (23), and in particular the expression of /, unlike the case of a constant dispersion, the frequency shift decreases during the propagation and, for z sufficiently large, it can be neglected from the argument of the hyperbolic secant. Therefore, we have v0g vg . The frequency shift can also be removed from the soliton expression (22) choosing the carrier frequency appropriately or alternatively choosing a reference system which moves at the group velocity v0g , that is Te ¼ t K 1 z 2Cb 1 e2C1 z ; e ¼ z. Z
ð24Þ
6. Conclusions The great development of the fiber optics is due to the possibility to transmit solitary wave as pulses, which is rather natural if we launch pulses of given duration (t0 > 5 ps) in fiber; this is the case in which the equation governing the transmission of the pulse in the fiber optics is the NLS equation. On the contrary, in the case of ultrashort pulses, which are relevant to increase the transmission capacity in fibers, it becomes necessary to modify the equation so to include higher-order effects. In fact, in the particular case of constant dispersion fiber, it is unlikely to support stable propagation of ultrashort solitons (less than 10 ps) because of fiber loss, higher-order dispersive and nonlinear effects. It has been shown that dispersion decreasing fibers can support ultrashort soliton pulses. Because of the great interest for managed-soliton propagation, particularly in a DDF, we have proposed a higherorder NLS with some variable coefficients, which governs the propagation of a pulse in a fiber optics with variable dispersion, in a more general way with respect to [14]. Then, we have checked this HNLS equation from the integrable point of view using the Painleve´ analysis and by finding the Lax pair of the spectral problem associated. At the end, by means of a transformation which reduces the HNLS with variable coefficients to the HNLS with constant coefficients (Sasa–Satsuma equation), we propose a soliton solution in a DDF. The solution obtained is a soliton-pulse whose temporal width remains the same along the fiber, whose peak value decreases because of the exponential function in the amplitude, but whose effective group velocity increases during the propagation. Therefore, if it will be possible to build up fiber optics with variable local properties such that the nonlinear evolution equation is the completely integrable HNLS obtained in this paper, then the propagation of ultrashort stable optical soliton can be realized in a real optical fiber. This will improve not only the stability of the signal in a fiber optics communication but also the transmission speed. Since the integrability of Eq. (12) is analyzed without any hypothesis on the graphic of the dispersion, it could
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T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
be interesting to consider different analytic choices of it, for instance, to compress the pulse width. The complete integrability of Eq. (12) suggests that there should exist multi-soliton solutions of this model. We plan to discuss these points in a future publication. Acknowledgements The authors are indebted to the referees for helpful suggestions which improved this paper. Research supported by the MURST Italy through the project ‘‘Non Linear Mathematical Problems of Wave Propagation and Stability in Models of Continuous Media’’ and founds ‘‘60%’’. References [1] A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 142. [2] L.F. Mollenauer, R.H. Stolen, J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [3] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001. [4] A. Hasegawa, Optical Solitons in Fibers, Springer, Berlin, 1989. [5] R. Papannareddy, Introduction to Lightwave Communication Systems Artech House, Boston, 1997. [6] W. Hodel, H.P. Weber, Opt. Lett. 12 (11) (1987) 924. [7] M. Gedalin, T.C. Scott, Y.B. Band, Phys. Rev. Lett. 78 (1997) 448. [8] R. Ganapathy, V.C. Kuriakose, Chaos Solitons Fract. 15 (2003) 99. [9] M. Nakazawa, H. Kubota, K. Suzuki, E. Yamada, Chaos Solitons Fract. 10 (2002) 486. [10] A. Hasegawa, Physica D (Amsterdam) 123 (1998) 267.
[11] T.I. Lakoba, D.J. Kaup, Phys. Rev. E 58 (1998) 6728. [12] R. Hao, L. Li, Z. Li, W. Xue, G. Zhou, Opt. Commun. 236 (2004) 79. [13] L. Li, Z. Li, S. Li, G. Zhou, Opt. Commun. 234 (2004) 169. [14] R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274. [15] V.A. Bogatyrev, M.M. Bubnov, E.M. Dianov, et al., J. Lightwave Technol. 9 (1991) 561. [16] V. Bogatyrjov, M.M. Bubnov, E.M. Dianov, A.A. Sysoliatin, Pure Appl. Opt. 4 (1995) 345. [17] Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. 23 (1987) 510. [18] M.J. Potasek, M. Tabor, Phys. Lett. A 25 (1992) 2403. [19] J.H.B. Nijhof, G.H.M. Roelofs, J. Phys. A 25 (1992) 2403. [20] S.Y. Sakovich, J. Phys. Soc. Jpn. 66 (1997) 2527. [21] V.I. Karpman, Eur. Phys. J. B 39 (2004) 341. [22] D.J. Kaup, A.C. Newell, J. Math. Phys. 19 (4) (1978) 798. [23] H.H. Chen, Y.C. Lee, C.S. Liu, Phys. Scr. 20 (1979) 490. [24] R. Hirota, J. Math. Phys. 14 (7) (1973) 805. [25] A.A. Mohammad, M. Can, J. Nonlinear. Math. Phys. 3 (1996) 152. [26] N. Sasa, J. Satsuma, J. Phys. Soc. Jpn. 60 (1991) 409. [27] S. Ghosh, A. Kundu, S. Nandy, J. Math. Phys. 40 (1999) 1993. [28] C. Laurey, Nonlinear Anal. 29 (1997) 121. [29] H. Takaoka, Adv. Diff. Equ. 4 (1999) 561. [30] R. Grimshaw, Proc. Roy. Soc. Lond. A 368 (1979) 377. [31] P.A. Clarkson, Proc. Roy. Soc. Edin. A 109 (1988) 109. [32] L. Gagnon, P. Winternitz, J. Phys. A 26 (1993) 7061. [33] X. Carvajal, F. Linares, Diff. Int. Equ. 16 (2003) 1111. [34] J. Weiss, M. Tabor, G. Carnevale, J. Math. Phys. 24 (3) (1983) 522. [35] R. Conte, Phys. Lett. A 140 (1989) 383. [36] M. Jimbo, M.D. Kruskal, T. Miwa, Phys. Lett. 92 A (1982) 59. [37] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1973. [38] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics T. Brugarino a
a,*
, M. Sciacca
b
Dip. di Metodi e Modelli Matematici, Facolta` d’Ingegneria, Universita’ di Palermo, Viale delle Scienze, 90128 Palermo, Italy b Dip. di Matematica ed Applicazioni, 90134 Palermo, Italy Received 13 May 2005; received in revised form 8 November 2005; accepted 28 December 2005
Abstract Taking into account many developments in fiber optics communications, we propose a higher nonlinear Schro¨dinger equation (HNLS) with variable coefficients, more general than that in [R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274], which governs the propagation of ultrashort pulses in a fiber optics with generic variable dispersion. The study of this equation is performed using the Painleve´ test and the zero-curvature method. Also, we prove the equivalence between this equation and its anomalous integrable counterpart (the so-called Sasa–Satsuma equation). Finally, in view of its physical relevance, we present a soliton solution which represents the propagation of ultrashort pulses in a dispersion decreasing fiber. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik Keywords: Optical solitons; Integrable systems; Zero curvature; Inhomogeneous nonlinear Schro¨dinger equation
1. Introduction One of the most important models of modern nonlinear science is the nonlinear Schro¨dinger equation (NLS), which appears in many branches of physics and applied mathematics, such as nonlinear optic. The best knows solutions of the NLS equation are solitons, which have been the objects of extensive theoretical and experimental studies during the last three decades because of their potential applications in long distance communication and all-optical ultrafast switching devices [1–8]. The optical solitons in fiber, which are localized in time optical pulses, evolve from a nonlinear change in the refractive index of the material (silica glass) known as Kerr effect, induced by the light intensity distribution. From this, when the combined effects of self-phase modulation (SPM) and
of group velocity dispersion (GVD) exactly compensate, the pulse propagates without any change in its shape. This was firstly suggested by Hasegawa and Tappert [1] theoretically, and then its experimental check has been successfully carried out by Mollenauer et al. [2]. The propagation of picosecond optical pulses (in particular t0 > 5 ps) in mono-mode optical fibers is governed by the well-known completely integrable NLS. Although the NLS equation includes GVD and SPM, it does not considers dissipative effects that in a real fiber are present at the order of 0.2 db/km for a wavelength of the pulse near to 1.55 lm. When we consider the propagation of pulse in a real optical fiber, the dissipative effects have to be included in the governing equation, which take the form of a more general equation than the NLS one: iuz þ a1 utt þ a2 ujuj2 þ ia3 u ¼ 0;
ð1Þ
*
Corresponding author. Tel.: +39 0916657320; fax: +39 091427258. E-mail addresses: [email protected] (T. Brugarino), micheles@ math.unipa.it (M. Sciacca). 0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.12.065
where a1, a2 and a3 are, respectively, the GVD, the SPM and the fiber loss coefficients.
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
Eq. (1) describes a variety of nonlinear optical phenomena, like bright solitons, modulation instability and dark solitons [3]. The concept of soliton control is an important development in the application of solitons in the fiber optics communication, and has been extensively studied in [8–14]. In fact, it is well-known that when the group velocity dispersion is slightly modified, the shape of the soliton pulse changes: for example, when the GVD decreases, the soliton pulse gets compressed as it propagates along the length of the fiber. The first dispersion managed soliton in a fiber with hyperbolic decreasing group velocity dispersion was realized experimentally in 1991 by the Dianov’s group [15]. After that, many other examples were considered, for instance in [8,12,13]. Here, the authors proposed NLS equation with variable coefficients, as equation which governs the propagation of soliton in a real fibers with variable GVD (along the axis of the fiber), and sometimes also SPM. In a dispersion decreasing single mode optical fiber (DDF), where the GVD monotonically and smoothly decreases from an initial value to a smaller value at the end of its length, pulse propagation is characterize by a generalized normalized NLS equation of the form [8]: iuz þ
a1 ðzÞ 2 utt þ ujuj þ ia3 u MðzÞt2 u ¼ 0; 2
ð2Þ
where u is the slowly varying complex pulse envelope, t is the normalized retarded time, z is the normalized propagation distance, the term M(z)t2u accounts for electro-optic phase modulation, a3 is the fiber loss coefficient and the GVD coefficient a1(z) has an hyperbolic dispersion decreasing profile. The property of DDF makes it a powerful tool in controlling optical solitons in soliton communication systems where high quality, stable soliton pulse compression and soliton train generation can be effectively realized. Fiber with a nearly exponential GVD profile have been built [16]. A practical technique for building up such DDFs consists in reducing the core diameter along fiber length in a controlled manner during the fiber-drawing process. Variations in the core diameter change the waveguide contribution to GVD coefficient and reduce its magnitude. In order to increase the bit rate in fiber optics communication systems for a single carrier frequency, it is necessary to decrease the pulse width t0. When ultrashort pulses (t0 < 1 ps) propagates in a fiber optics, in addition to the SPM it will be necessary to consider nonlinear effects of higher order like self-steepening (also called dispersion Kerr) and self-frequency shift due to stimulated Raman scattering (SRS); on the other hand, apart from GVD, the ultrashort pulse also suffers from third-order dispersion (TOD). The equation which governs the propagation of such ultrashort pulses inside a fiber optics, including in this way these effects of higher order, has been proposed by Hasegawa and Kodama in [17], and it has the following form:
2
251 2
2
iuz þ a1 utt þ a2 ujuj þ ia3 uþ ia4 uttt þ a5 uðjuj Þt þ ia6 ðujuj Þt ¼ 0; ð3Þ where a4 and a6 are the parameters related to TOD, and self-steepening, respectively, while a5, which in general is complex, is obtained by the retarded nonlinear response [3]. This last term corresponds to the SRS in which the higher frequency components of the soliton spectrum pump energy to the lower frequency components, producing a nonlinear dissipation and down-shift of the carrier frequency of the soliton. Eq. (3) refers to ultrashort pulse in a system of reference that moves at the group velocity (GV) vg given by the inverse of the coefficient of the first derivative in t. Regarding Eq. (3), it admits soliton-type solution in wide domains of its parameters [18], but this has no relation with its integrability. In [19], using the Wahlquist–Estabrook prolongation method, in [20], using the Painleve´ analysis, and in [21], using the Galilean transformation and generalized Galilean invariance, it is stated that Eq. (3) is completely integrable, besides the standard NLS, only in the following four cases: (a) the higher nonlinear Schro¨dinger equation (HNLS) [22] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 0 : 1 : 1Þ; (b) the HNLS [23] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 0 : 1 : 0Þ; (c) the Hirota equation [24,25] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 1 : 6 : 0Þ; (d) the Sasa–Satsuma equation [25–27] ða1 : a2 : a3 : a4 : a6 : Imða5 Þ þ a6 ¼ 1=2 : 1 : 0 : 1 : 6 : 3Þ. From the above results, one can notice that the integrability is guaranteed if and only if the loss coefficient a3 and the real part of the nonlinear dispersion coefficient a5 are zero. A mathematical point of view on the well-posedness of the Cauchy problem for the HNLS equation (3) is contained in [28,29]. Many authors [30–32] have considered the generalized NLS with variable coefficients in order to determine the constraints under which the equation might be integrable. This constraints entail the transformation of the equation to its autonomous integrable counterpart. In accordance with the arguments of generalized secondorder NLS with variable coefficients and with the generalized third-order NLS equation with GVD variable proposed in [14], in this paper we generalize the third-order NLS (3) by supposing that the dispersion in the fiber is varied along the axis z of the fiber in a more general way with respect to [14]. This implies, first of all, that both group velocity dispersion and the third-order dispersion coefficients depend on z.
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Then, since we are considering a real fiber optics, this forces the presence in the equation of the linear term in u which represents the loss, and therefore we include a term that is a phase modulation, as in Eq. (2), which in general depends on the axis length z and the time t. Since we are supposing a fiber optics with variable dispersion, which can imply an asymmetry in the pulse shape during the propagation, particularly due to the third-order dispersion [3], we assume here that the group velocity vg of the pulse is not constant but it depends on z and t. This appears in our equation via the first derivative ut, whose coefficient is equal to the inverse coefficient of the group velocity vg. At the end, the HNLS with variable coefficient which we will discuss here has the form: 2
iuz þ AðzÞutt þ Bujuj þ ðCðz; tÞ þ iC 1 Þu þ iH 1 ðz; tÞut þ iM 1 ðzÞuttt þ ðP þ iP 1 Þuðjuj2 Þt þ iQ1 ðujuj2 Þt ¼ 0;
ð4Þ
where A(z), B, C(z, t), C1, M1(z) and Q1 are, respectively, the GVD, SPM, phase modulation, loss, TOD and selfsteepening coefficients, H1(z, t) is the inverse of the group velocity vg, P and P1, as presented by Kodama and Hasegawa in [17], represents the nonlinear dispersion or SRS. In [33], the Cauchy problem for the HNLS with variable coefficients (4), without the terms u and ut, is considered. The authors show the local well-posedness for this problem in some appropriate Sobolev space. In this work, we check under which condition solitontype pulse propagation is possible. In order to do that we apply the Painleve´ analysis [34,35] and check the zerocurvature property. After that, we will propose two sets of transformations that will map Eq. (4), under suitable constraints, into the Sasa–Satsuma equation. We conclude our paper with an application in a dispersion decreasing fiber, by choosing a particular expression for the GVD coefficient [14] and finding a soliton solution. 2. Painleve´ test Since Eq. (4) is not analytic in u, we set u ! u + iv and we perform the Painleve´ test in the new equations: vz AðzÞutt Buðu2 þ v2 Þ Cðz; tÞu þ C 1 v þ M 1 ðzÞvttt þ H 1 ðz; tÞvt 2Put u2 þ 2P 1 ut uv þ 2Q1 ut uv 2Pvt uv þ 2P 1 vt v2 þ Q1 vt u2 þ 3Q1 vt v2 ¼ 0; uz þ AðzÞvtt þ Bvðu2 þ v2 Þ þ Cðz; tÞv þ C 1 u þ M 1 ðzÞuttt
ð5Þ
2
þ 2Put uv þ 2P 1 ut u þ Q1 ut v þ 3Q1 ut u ¼ 0
b
1 X j¼0
j
vj ðz; tÞ/ðz; tÞ .
d¼
M1 2P 1 þ 3Q1
and the constraint P = 0. After that, inserting in Eq. (5) the two expansions (6) and (7), we obtain the following resonances: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 . r ¼ 1; 0; 3; 4; 3 2 ð2P 1 þ 3Q1 Þ Now, we have the following resonances r = 1, 0, 2, 3, 4, 4 for the following condition: 2P 1 þ Q1 ¼ 0.
ð9Þ
In order to obtain the compatibility condition, we simplify the calculation by the assumption: /(z, t) = t + g(z) [36]. From Eq. (8), we can deduce the relation between u0 and v0, and the possibility to consider four branches of which only two essentially distinguished: first branch pffiffiffiffiffiffiffiffiffi u0 ¼ v0 ¼ 3d; second branch pffiffiffiffiffiffiffiffiffi u0 ¼ v0 ¼ 3d. Concerning the first branch, for j = 1, we have: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3dð3dB AðzÞÞ 3dð3dB AðzÞÞ u1 ¼ and v1 ¼ . 3dQ1 3dQ1 As for the second branch, for j = 1, we have: pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 3dð3dB AðzÞÞ 3dð3dB AðzÞÞ u1 ¼ and v1 ¼ . 3dQ1 3dQ1 For j = 2, the constraint (9) gives the resonance for both branches. In the same way, for j = 3, we have a resonance under the following constraints on the coefficients: AðzÞz . 2AðzÞ
For j = 4, we have a double resonance under the following constraints on the coefficients for both branches: ð6Þ
H 1tt ¼ 0; AðzÞH 1t 6Q1 dC t ¼ 0.
ð7Þ
In conclusion, we can simplify all the constraints on the coefficients as follow:
j¼0
v ¼ /ðz; tÞ
ð8Þ
where
C 1 ¼ H 1t
2
with the following Laurent series for u and v: 1 X a j u ¼ /ðz; tÞ uj ðz; tÞ/ðz; tÞ ;
u20 þ v20 ¼ 6d/2t ;
AðzÞ 6dB ¼ 0;
þ H 1 ðz; tÞut þ 2Pvt v2 þ 2P 1 vt uv þ 2Q1 vt uv 2
Looking at the leading order, if we replace u = u0/(z, t)a and v = v0/(z, t)b in Eq. (5), to balance the dominant terms we must require a = b = 1; this analysis also gives the condition:
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
P ¼ 0; H 1 ¼ r1 KðzÞt þ K 1 ðzÞ; C ¼ KðzÞt þ K 2 ðzÞ; C 1 ¼ r1 KðzÞ
AðzÞz ; 2AðzÞ
where Q 3M 1 r1 ¼ 1 ¼ ; B AðzÞ C1 AðzÞz KðzÞ ¼ þ r1 2r1 AðzÞ
ð10Þ ð11Þ
and K1(z) and K2(z) are generic functions of z. Now, we can resume the above results as follow: Eq. (4) is integrable if and only if it assumes the following form: C1 AðzÞz 2 iuz þ AðzÞutt þ Bujuj þ þ t þ K 2 ðzÞ þ iC 1 u r1 2r1 AðzÞ C1 AðzÞz þ þ i r1 t þ K 1 ðzÞ ut r1 2r1 AðzÞ AðzÞ B 2 2 uttt i uðjuj Þt þ iBðujuj Þt ¼ 0. þ r1 i ð12Þ 3 2 Looking at the previous equation, we observe that the TOD coefficient depend, explicitly, on the GVD coefficient, whereas we find a more precise choice in the coefficients of group velocity and phase modulation, which we will simplify at the end of the paper. Another remark is that, as in the constant case, in order to have integrability we have to set P = 0 in the equation. This will produce a lack of asymmetry in the profile of the single pulse |u|. The study of Eq. (12) is of great interest since it could have a wide range of applications since it governs propagation of a solitary wave, whose temporal width is less than 1 ps, in a non-uniform dispersive media. In concrete applications, Eq. (12) can be interesting not only for the amplification and compression of ultrashort optical solitons, but also for the stable transmission of ultrashort managed soliton.
253
We shall choose the matrices A and B as: 2 3 ik gu g u1 6 7 ik 0 5; A ¼ 4 g u1 gu 0 ik 2 3 0 0 0 6 7 B ¼ ðk þ k2 þ k3 ÞI þ 4 0 m 0 5 0 0 m 2 3 2 3 0 gq g q 0 gu g u1 6 7 6 7 0 5 þ 4 g q 0 0 5 þ s4 g u1 0 gu 0 0 gq 0 0 2 3 2 2 2 2 gBr1 u u1 3 g Br1 u1 u 0 3 6 2 7 2 0 0 þ 4 3 g Br1 u1 u ð16Þ 5; 23 gBr1 u2 u1 0 0 where u1 = u* and: AðzÞz t 2AðzÞ þ þ C 1 t þ K 1 ðzÞ; 2AðzÞ 3r1 AðzÞr1 2 utt iAðzÞut ; q¼ 3 3 sffiffiffiffiffiffiffiffiffiffiffi
s¼
B rit e 1; 2AðzÞ 2 3 1 0 0 6 7 I ¼ 40 1 05 0 0 1
g¼
and 1 1 1 2iAðzÞ iK 1 ðzÞ m ¼ Br1 ut u1 Br1 uu1t iBuu1 þ þ iK 2 ðzÞ. 6 6 3 3r21 r1 Substituting A and B in the compatibility condition (15), we find that Eq. (15) is equivalent to the HNLSE (12) and its to conjugate. 4. Reduction to the GNLS equation with constant coefficient
3. Zero-curvature The zero-curvature and Lax pairs are two different commutator representation of nonlinear evolution equations. They play a relevant role in the study of integrable dynamical systems. The zero-curvature representation is a more general version of the Lax representation. In order to present the zero-curvature representation of Eq. (12), we consider the two linear equations: Wt ¼ AW;
ð13Þ
Wz ¼ BW;
ð14Þ
Starting from Eq. (12), which has the PP (Painleve´ property), in this section we propose two sets of transformations, of dependent and independent variables, which will transform Eq. (12) into the following expression for the Sasa–Satsuma equation: iU Z þ a1 U TT þ a2 U jU j2 a a2 1 þ r1 i U TTT i U ðjU j2 ÞT þ ia2 ðU jU j2 ÞT ¼ 0; ð17Þ 3 2 where r1 ¼
a6 3a4 ¼ . a2 a1
where Aðz; t; u; u1 ; kÞ and Bðz; t; u; u1 ; kÞ are 3 · 3 matrices, k is the isospectral parameter and W = [W1(z, t), W2(z, t), W3(z, t)]T. If require that Eqs. (13) and (14) are compatible, then A and B must satisfy the equation
To achieve this goal, we set the following expression:
Az Bt þ ½A; B ¼ 0.
into Eq. (12), and obtain:
ð15Þ
u ¼ U ðZ; T Þf ðzÞeigðz;tÞ
ð18Þ
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T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
f ðzÞ ¼ eC1 z ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ffi Z 3C1 z B e pffiffiffiffiffiffiffiffiffi dz; ZðzÞ ¼ a1 a2 AðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi C z qffiffiffiffiffiffiffiffiffiffiffiffiffi3 3C z ffi C z Z a AðzÞpffiffiffiffiffiffiffi 1 þ a r K ðzÞ e Ba Ba1 e 1 ðBa1 Þ e 1 2 1 2 1 1 Ba1 C1 z pffiffiffiffiffiffiffiffiffiffiffiffiffi e dz; t T ðz; tÞ ¼ a2 AðzÞ r1 a32 AðzÞ sffiffiffiffiffiffiffiffiffiffiffiffiffi ! Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 a1 B C1 z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 3r21 K 2 ðzÞ a32 AðzÞ 3a2 AðzÞ a1 BeC1 z 2 a32 AðzÞ3 e gðz; tÞ ¼ 1 tþ r1 a2 AðzÞ 3r21 a32 AðzÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi C z qffiffiffiffiffiffiffiffiffi 3 3 3 3C 1 z 1 a1 B e 3r1 K 1 ðzÞ a2 AðzÞ 3a2 r1 K 1 ðzÞ a1 Be dz.
In conclusion, Eq. (12), after a change of the dependent and independent variables, is equivalent to the Sasa–Satsuma equation (17). 5. A particular solution In this section, we consider the propagation of soliton pulse in a dispersion decreasing fiber (DDF) and, in particular, we set the group velocity dispersion [14]: 1 AðzÞ ¼ e2C1 z ; 2 which has the maximum value A(0) = 1/2 and the minimum one A(z) = exp(2C1L)/2, where L is the length of the fiber. When L ! 1 then the GVD coefficient go to zero. According to the numerical results, in the evolution of ultrashort pulses in optical fiber, the group velocity vg decreases due to the red shift of the soliton spectrum, to the dispersion and in particular to GVD [3]. In order to simplify the calculation from now on we take B = 1. Under these conditions Eq. (12) can be rewritten as: 1 2 iuz þ e2C1 z utt þ ujuj þ ðK 2 ðzÞ þ iC 1 Þu þ iK 1 ðzÞut 2 e2C1 z i 2 2 uttt uðjuj Þt þ iðujuj Þt ¼ 0; þ r1 i 2 6
ð19Þ
while the transformations becomes: f ðzÞ ¼ eC1 z ; 1 2C1 z e ; ZðzÞ ¼ 2C 1 Z T ðz; tÞ ¼ t K 1 ðzÞ dz; Z 1 gðz; tÞ ¼ ð1 1Þt þ K 2 ðzÞ dz r1 Z ð2 þ 2Þ 2C1 z ð1 1Þ þ K 1 ðzÞ dz; e þ r1 12r21 C 1 where we have set a1 = 1/2 and a2 = 1, which are the standard values for the GVD and SPM coefficients.
The Sasa–Satsuma equation: 1 2 iU Z þ U TT þ U jU j 2 i i 2 2 þ r1 U TTT U ðjU j ÞT þ iðU jU j ÞT ¼ 0 6 2
ð20Þ
is an expression of the HNLS, proposed by Kodama and Hasegawa, for a particular combination of the coefficients. This equation, as discussed in the Introduction, is completely integrable and therefore it admits soliton wave as solution. Now, if we know a solution of Eq. (20), it is possible to find a solution of (19) using one of the two sets of transformations (18). To achieve this goal, we put U ¼ yðT þ bZÞ expðiðHZ XT ÞÞ in Eq. (20) and, after equating the real and imaginary part of the resulting equation, we obtain the conditions: 1 ; 3r21 1 X¼ r1
H¼
and the ordinary differential equation for y(n) (where n = T + bZ): 1 þ 2r1 b 3 y nn þ 4y þ 3 y ¼ 0; ð21Þ r21 which has doubly periodic elliptic functions as solutions. By integrating Eq. (21) we have: 1 þ 2r1 b 2 y þ 2y 4 ¼ 0; y 2n þ 3 r21 which, for br1 < 1/2, has the following solution [37,38]: pffiffiffi y ¼ g sechð 2gðT þ bZÞÞ; where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6br1 g¼ . 2r21
T. Brugarino, M. Sciacca / Optics Communications 262 (2006) 250–256
Therefore, we have the following solitary wave solution for Eq. (20): pffiffiffi T Z ; U ¼ g sechf 2gðT þ bZÞg exp i r1 3r21 which is a soliton that moves at the group velocity group vg = 1/b. Now, we use the previous solution U and one of the two set of transformations in Eq. (18) and we obtain, choosing K2 ¼
e2C1 z 3r21
and K1 constant, the following soliton as solution of Eq. (19): pffiffiffi t K 1z C 1 z u ¼ ge sechð 2g/Þ exp i ; ð22Þ r1 r1 where b 2C1 z e . / ¼ t K 1z 2C 1 The above solution is the same for both transformations. The coefficient K1 is the inverse of the group velocity and the last term represent a frequency shift of the soliton from the carrier frequency x0, due to dispersion. The profile of the soliton solution (22) is expressed by: pffiffiffi ð23Þ juj ¼ geC1 z sechð 2g/Þ and it is shown in Fig. 1 for a particular choice of the coefficients b, r1, C1 and K1. The term exp(C1z) represents the decay behaviour of the crest of the soliton due to the loss of the fiber. Notice that a frequency shift also changes the soliton speed from its original values vg = 1/K1. In fact, by differentiating the soliton along the characteristic curves / = constant, we obtain the following relation between the group velocity vg and the effective group velocity v0g : 1 1 ¼ be2C1 z 0 vg vg and since b is negative we deduce that v0g 6 vg .
Fig. 1. The amplitude profile (23) with the following values of the parameters: b = 5; r1 = 0.2; C1 = 0.2; K1 = 4.8733 · 106.
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If we observe the profile of the solution (23), and in particular the expression of /, unlike the case of a constant dispersion, the frequency shift decreases during the propagation and, for z sufficiently large, it can be neglected from the argument of the hyperbolic secant. Therefore, we have v0g vg . The frequency shift can also be removed from the soliton expression (22) choosing the carrier frequency appropriately or alternatively choosing a reference system which moves at the group velocity v0g , that is Te ¼ t K 1 z 2Cb 1 e2C1 z ; e ¼ z. Z
ð24Þ
6. Conclusions The great development of the fiber optics is due to the possibility to transmit solitary wave as pulses, which is rather natural if we launch pulses of given duration (t0 > 5 ps) in fiber; this is the case in which the equation governing the transmission of the pulse in the fiber optics is the NLS equation. On the contrary, in the case of ultrashort pulses, which are relevant to increase the transmission capacity in fibers, it becomes necessary to modify the equation so to include higher-order effects. In fact, in the particular case of constant dispersion fiber, it is unlikely to support stable propagation of ultrashort solitons (less than 10 ps) because of fiber loss, higher-order dispersive and nonlinear effects. It has been shown that dispersion decreasing fibers can support ultrashort soliton pulses. Because of the great interest for managed-soliton propagation, particularly in a DDF, we have proposed a higherorder NLS with some variable coefficients, which governs the propagation of a pulse in a fiber optics with variable dispersion, in a more general way with respect to [14]. Then, we have checked this HNLS equation from the integrable point of view using the Painleve´ analysis and by finding the Lax pair of the spectral problem associated. At the end, by means of a transformation which reduces the HNLS with variable coefficients to the HNLS with constant coefficients (Sasa–Satsuma equation), we propose a soliton solution in a DDF. The solution obtained is a soliton-pulse whose temporal width remains the same along the fiber, whose peak value decreases because of the exponential function in the amplitude, but whose effective group velocity increases during the propagation. Therefore, if it will be possible to build up fiber optics with variable local properties such that the nonlinear evolution equation is the completely integrable HNLS obtained in this paper, then the propagation of ultrashort stable optical soliton can be realized in a real optical fiber. This will improve not only the stability of the signal in a fiber optics communication but also the transmission speed. Since the integrability of Eq. (12) is analyzed without any hypothesis on the graphic of the dispersion, it could
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