Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: Variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation

Physics Letters A 366 (2007) 223–229 www.elsevier.com/locate/pla

Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: Variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation Bo Tian a,b,e,∗ , Yi-Tian Gao c,d,b , Hong-Wu Zhu a a School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China b State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100083, China c CCAST (World Lab.), PO Box 8730, Beijing 100080, China d Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics,

Beijing University of Aeronautics and Astronautics, Beijing 100083, China e Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications,

Beijing 100876, China Received 12 December 2006; accepted 7 February 2007 Available online 21 March 2007 Communicated by A.R. Bishop

Abstract Symbolically investigated in this Letter is a variable-coefficient higher-order nonlinear Schrödinger (vcHNLS) model for ultrafast signalrouting, fiber laser systems and optical communication systems with distributed dispersion and nonlinearity management. Of physical and optical interests, with bilinear method extend, the vcHNLS model is transformed into a variable-coefficient bilinear form, and then an auto-Bäcklund transformation is constructed. Constraints on coefficient functions are analyzed. Potentially observable with future optical-fiber experiments, variable-coefficient brightons are illustrated. Relevant properties and features are discussed as well. Bäcklund transformation and other results of this Letter will be of certain value to the studies on inhomogeneous fiber media, core of dispersion-managed brightons, fiber amplifiers, laser systems and optical communication links with distributed dispersion and nonlinearity management. © 2007 Published by Elsevier B.V. PACS: 42.81.Dp; 42.65.Re; 05.45.Yv; 42.65.Tg; 42.65.Wi; 42.79.Sz Keywords: Variable-coefficient higher-order nonlinear Schrödinger model; Optical communication systems; Bäcklund transformation; Variable-coefficient bilinear form; Brightons; Symbolic computation

In various branches of physical and engineering sciences, nonlinear evolution equations (NLEEs), especially the variablecoefficient ones, have become more and more interesting [1–4] as computerized symbolic computation rapidly develops [1–4], among which the nonlinear-Schrödinger (NLS)-typed models have recently riveted much attention of the researchers in coastal engineering [5], blood mechanics [6], space plasmas [7], Rayleigh–Taylor dynamics [8] as well as long-distance optical-fiber communications and all-optical ultrafast switching [9–26]. Since the first soliton dispersion management experiment [27], investigations on the variable-coefficient NLS-typed models for optical fibers have become very fruitful [9–26]. On the other hand, for optical pulses in the femtosecond regime, with such higherorder effects as the third-order dispersion, self-steepening and decayed nonlinear response considered, the constant-coefficient higher-order NLS models have been extensively studied [4,9,25,26] (and references therein). However, an as-yet-not-widespread

* Corresponding author.

E-mail address: [email protected] (B. Tian). 0375-9601/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.physleta.2007.02.098

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subject is, for a realistic fiber with weakly dispersive and nonlinear dielectrics underlying variable coefficients, to take account of the aforementioned higher-order effects influenced by the spatial variations of the fiber parameters [9]. For this token, the variablecoefficient higher-order NLS (vcHNLS) model [4,9–11],     iψz + d(z)ψtt + h(z)ψ|ψ|2 + ik(z)ψ |ψ|2 t + il(z) ψ|ψ|2 t + ib(z)ψttt + iΓ (z)ψ = 0, (1) can be used to describe such femtosecond pulse propagation, applicable to, e.g., the design of ultrafast signal-routing, fiber laser systems and optical communication systems with distributed dispersion and nonlinearity management. Hereby ψ(z, t) is the complex envelope of electrical field in a co-moving frame with z and t representing the normalized propagation distance along the fiber and retarded time, the coefficient functions d(z), b(z), h(z), k(z), l(z) and Γ (z) are, respectively, related to the group velocity dispersion, third-order dispersion, Kerr nonlinearity, decayed nonlinear response, self-steepening and heat-insulating amplification/absorption effects. Ref. [4] has proposed the transformation from Eq. (1) to its known constant-coefficient counterpart without amplification/absorption with physical/optical examples as variable-coefficient burstons and brightons, where the term “brighton” is the shortened form of a bright soliton according to Ref. [28]. Refs. [9–11] have obtained some combined solitary wave solutions and multi-solitons of Eq. (1). In general, Eq. (1) is not integrable [9]. Also seen in the studies of optical fibers, Eq. (1) can be reduced to the (A) perturbed nonlinear Schrödinger model in Refs. [12, 13] for the long-distance propagation for very short optical solitons in a nonlinear optical fiber incorporating the effects of periodic phase conjugation and dispersion management with d(z) = d0 (z)/2, h(z) = h0 (z), k(z) = −ik0 (z), l(z) = −l0 (z), b(z) = −b0 (z)/6 and Γ (z) = 0, where the coefficients d0 (z) and b0 (z) represent the dispersion profiles between amplifiers, and accommodating the exponential factor due to both the linear loss and lumped amplification, h0 (z), k0 (z) and l0 (z) are respectively related to the Kerr coefficient, product of the Raman-time and Kerr coefficients, and ratio of the Kerr coefficient to carrier frequency; (B) generalized nonlinear Schrödinger model with periodically varying coefficients in Ref. [14] for the dispersion-managed fiber systems and soliton lasers with d(z) = ±D(z)/2, h(z) = N (z), k(z) = l(z) = b(z) = 0 and Γ (z) = γ0 − γ (z); (C) higher-order nonlinear Schrödinger model in Ref. [15] for the ultrashort light pulse propagation in certain optical communication systems with d(z) = α1 , h(z) = α2 , k(z) = −α5 , l(z) = −α4 , b(z) = −α3 and Γ (z) = 0, where αn ’s are all real; (D) extended third-order cubic nonlinear Schrödinger model in Ref. [16] for the slow evolution of the wave envelope in nonlinear highly-dispersive systems such as optical fibers, or perturbed/higher-order nonlinear Schrödinger model in Ref. [17] for the pulse propagation in optical fibers in the femtosecond regime, or generalized/extended nonlinear Schrödinger model in Refs. [13,18] for the femtosecond pulse propagation in nonlinear optical fibers with d(z) = 1/2, h(z) = 1, k(z) = β3 , l(z) = β2 , b(z) = β1 and Γ (z) = 0, where βn ’s are all real constants with (β2 + β3 ) and β3 corresponding to the Raman (related to the self-frequency shift) effect and retardation effect on the nonlinear part of the refractive index; (E) Hirota model with all the coefficients being constants and l + k = 0, for which dark soliton [26] and N -envelope-solitons [29] have been seen. In this Letter, we will extend the bilinear method, transform Eq. (1) into a variable-coefficient bilinear form via symbolic computation and then construct an auto-Bäcklund transformation. Variable-coefficient brightons come out as applications of the transformation. We will also discuss relevant properties and features. First, to achieve the Bäcklund transformation, we will assume that the coefficient functions d(z), b(z), h(z), k(z), l(z) and Γ (z) be all real, and extend the bilinear method, which has been shown powerful on the constant-coefficient NLEEs [30], by setting up the variable-coefficient dependent variable transformation, G(z, t) , (2) F(z, t) to transform Eq. (1) into a variable-coefficient bilinear form, where the differentiable functions A(z) and F (z, t) are both real, while G(z, t) is complex. Substituting Ansatz (2) back into Eq. (1), we perform symbolic computation to get ψ(z, t) = A(z)

d(z)A 2 d(z)A G 2 A G dA Dt G · F − Dt F · F + i 2 Dz G · F + 2 F dz F F F3 k(z)A3 |G|2 k(z)A3 G 2 h(z)A3 G|G|2 + i D G · F + i Dt G ∗ · F + t F3 F4 F4 l(z)A3 G 2 2 l(z)A3 |G|2 D G · F + i Dt G ∗ · F +i t F4 F4   b(z)A 3 3 b(z)A Γ (z)A G +i Dt G · F − i (Dt G · F) Dt2 F · F + i = 0, 2 4 F F F where G ∗ (z, t) is the complex conjugation of G(z, t), while binary operators, Dz and Dt , are defined by Ref. [30] as  Dzm Dtn a(z, t) · b(z, t) = (∂z − ∂z )m (∂t − ∂t  )n a(z, t)b(z , t  )z =z, t  =t (m, n, = 0, 1, 2, . . .). i

Equation splitting [31] of Eq. (3) then yields    A 1 dA 2 3 + Γ (z) G · F = 0, iDz + d(z)Dt + ib(z)Dt + i A dz F2

(3)

(4)

(5)

B. Tian et al. / Physics Letters A 366 (2007) 223–229

AG 2 2 2 h(z)A |G| − d(z)D F · F = 0, t F3

A3 G 2 k(z) + l(z) Dt G ∗ · F = 0, i 4 F

A  i 4 k(z) + 2l(z) A2 |G|2 − 3b(z)Dt2 F · F Dt G · F = 0. F From Eq. (7) we see that k(z) = −l(z),

225

(6) (7) (8)

(9)

which can be substituted back into Eq. (8) and compared with Eq. (6) so that another constraint on the coefficients appears as h(z) d(z) = . l(z) 3b(z)

(10)

In Eq. (5), we choose

1 dA + Γ (z) = 0 ⇒ A(z) = e− Γ (z) dz . (11) A dz To this stage, we in fact obtain, under Constraints (9) and (10), the variable-coefficient bilinear form of Eq. (1) via Transformation (2) with (11) as

iDz + d(z)Dt2 + ib(z)Dt3 G · F = 0, (12)

Dt2 F · F =

h(z)A2 2 |G| . d(z)

(13)

Next we begin to construct a Bäcklund transformation between ψ(z, t) = A(z)G(z, t)/F(z, t) and ψ  (z, t) = A(z)G  (z, t)/ both of which are supposed to be the solutions of Eq. (1), by considering



Ω ≡ F 2 iDz + d(z)Dt2 + ib(z)Dt3 G  · F  − F  2 iDz + d(z)Dt2 + ib(z)Dt3 G · F = 0, (14)

F  (z, t),

the prototype of which has been given in Ref. [32] for the typical second-order constant-coefficient NLS equation. To Eq. (14) we introduce the identities (Dx η · θ )χ 2 − θ 2 (Dx ν · χ) = θχDx (η · χ + θ · ν) − (ηχ + θ ν)(Dx θ · χ),  2        Dx η · θ χ 2 − θ 2 Dx2 ν · χ = ηθ Dx2 χ · χ − Dx2 θ · θ χν + θχDx2 (η · χ − θ · ν)

− (ηχ − θ ν)Dx2 θ · χ − 2Dx (ηχ + θ ν) · (Dx θ · χ) ,   

 3 Dx η · θ χ 2 − θ 2 Dx3 ν · χ = Dx3 (η · χ + θ · ν) θ χ − (ηχ + θ ν)Dx3 θ · χ

− 3Dx Dx (η · χ − θ · ν) · Dx (θ · χ) , perform symbolic computation and substitute Eq. (13), to get 

Ω = F F  i Dz + b(z)Dt3 (G  · F + F  · G) + d(z)Dt2 (G  · F − F  · G)

 − (G  F + F  G) i Dz + b(z)Dt3 F  · F − 2 ih(z)A2 Im(G  G ∗ )

− (G  F − F  G) d(z)Dt2 F  · F + 2 h(z)A2 Re(G  G ∗ ) 

− Dt 3 ib(z)Dt (G  · F − F  · G) + 2 d(z)(G  F + F  G) · (Dt F  · F) .

(15) (16) (17)

(18)

Again, equation splitting indicates that 3 ib(z)Dt (G  · F − F  · G) + 2 d(z)(G  F + F  G) = 0,

Dz + b(z)Dt3 F  · F − 2 h(z)A2 Im(G  G ∗ ) = 0,



i Dz + b(z)Dt3 (G  · F + F  · G) + d(z)Dt2 − λ(z) (G  · F − F  · G) = 0,

d(z)Dt2 − λ(z) F  · F + 2 h(z)A2 Re(G  G ∗ ) = 0,

(19) (20) (21) (22)

where we have used a decoupling function λ(z) instead of Ref. [32]’s constant. Into Eqs. (19)–(22), we next substitute F (z, t) = 1 and G(z, t) = 0, which represent a trivial solution of Eq. (1), ψ(z, t) = 0, aiming at obtaining an analytic expression for ψ  (z, t) = A(z)G  (z, t)/F  (z, t). Symbolic computation on Eqs. (20) and (22) together shows that

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λ(z) = μ2 d(z), 

F (z, t) = e

μt−μ3 b(z) dz

α+e

−μt+μ3 b(z) dz

(23) (24)

β,

where μ, α and β are all real constants with μ > 0 selected. Similarly, symbolic computation on Eqs. (19) and (21) together gives rise to d(z) 3γ = , (25) b(z) 2 i

G  (z, t) = eiγ t− 2 γ (γ

2 +3μ2 )

b(z) dz

(26)

,

with γ as another real constant. Constraint (30) follows up as below, and we end up with a family of the analytic variable-coefficient brightonic solutions of Eq. (1), i.e., i



G  (z, t) eiγ t− 2 γ (γ +3μ ) b(z) dz− Γ (z) dz

= ψ (z, t) = A(z)  F (z, t) eμt−μ3 b(z) dz α + e−μt+μ3 b(z) dz β    

i 1 α 1 2 2 eiγ t− 2 γ (γ +3μ ) b(z) dz− Γ (z) dz Sech μt − μ3 b(z) dz + ln , = √ 2 β 2 αβ 2

2



(27)

along with the constraints imposed on the variable coefficients as d(z) : b(z) =

3γ : 1, 2

γ , 2

24 e2 Γ (z) dz α βμ2 b(z) − l(z) = 0. k(z) : l(z) : h(z) = −1 : 1 :

(28) (29) (30)

The set of Equations (19)–(22) hereby constructed constitutes, under Constraints (28)–(30) and Expression (2) with (11), an autoBäcklund transformation to Eq. (1), the vcHNLS model for optical pulses in the femtosecond regime in a realistic fiber with weakly dispersive and nonlinear dielectrics underlying variable coefficients, with such effects as third-order dispersion, self-steepening and decayed nonlinear response considered. The last part of this Letter is the discussions and conclusions, as follows: 1. For such realistic fiber, the auto-Bäcklund transformation works as a system of equations relating a brighton of Eq. (1) to another (more complicated) brighton structure of Eq. (1) itself. In principle, therefore, we could progressively construct a multiple brighton “spectrum” for Eq. (1) beginning with a “seed brighton” [such as that from Solution (27)]. More information on the Bäcklund transformations, shown to be very powerful in constructing various solitonic solutions, conservation laws, etc., and their applications, can be referred to, e.g., Refs. [31,33]. The auto-Bäcklund transformation with Solutions (27), to nonlinear dispersive systems with spatial parameter variations, can be of use for the future work on transmission lines with brighton management and experiments of other femtosecond problems (e.g., lasers). 2. Under Constraints (28)–(30) and resulting from balance among the self-steepening, third-order dispersion and delayed nonlinear response, Solutions (27) indicates that the shape of the variable-coefficient brighton, brighton amplitude, inverse width of the brighton, frequency, center of the brighton and center of the phase of the brighton

are, respectively, γ 1 ln( βα ) and − γ2 (γ 2 + μ2 ) b(z) dz − 2μ ln( βα ). Sech[μt − μ3 b(z) dz + 12 ln( βα )], 2√1αβ e− Γ (z) dz , μ, −γ , μ2 b(z) dz − 2μ In line with the bird’s-eye view of Figs. 1 and 2, those analytic expressions say that an optimal control system could come out of suitable choice of the variable coefficients for each specific problem. 3. Solutions (27) and more complicated solutions of Eq. (1) via the auto-Bäcklund transformation come in fact from the balance between pulse asymmetries caused by less destructive effects of third-order dispersion and self-steepening through k(z), l(z) and b(z). Consequently, we expect that those brightons could be observed in properly designed optical-fiber experiments. 4. Figs. 1 and 2 provide us with variable-coefficient brighton intensity surfaces |ψ  (z, t)|2 , resulted from the combined contribution of the variable coefficients in Eq. (1). Those figures imply that even if the brightonic width and frequency remain unchanged, other quantities do vary in some complicated forms, such as the group velocity of brighton propagation, brighton peak intensity, shape of the brighton, center of the brighton and center of the phase of the brighton. Some of them, such as the brightonic amplitude, vary periodically for the function forms selected for variable coefficients in Figs. 1 and 2. Therefore, illustrated in Figs. 1 and 2 and similar to those claimed in Refs. [9,34], we are able to control, e.g., the brightonic velocity and shape by managing the variable coefficients in optical soliton communication systems, which are now dispersion managed and femtosecond scaled. In comparison, the typical NLS-typed models include only the pulse chirping and broadening from group velocity dispersion and the action of self-phase modulation due to the nonlinear refractive index [4,13].

B. Tian et al. / Physics Letters A 366 (2007) 223–229

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Fig. 1. Potentially-observable variable-coefficient brighton intensity profile |ψ  (z, t)|2 verses z and t via Solution (27) for the vcHNLS model, with the parameters and functions chosen as β = 3, α = 5, μ = 1/2, γ = −6, Γ (z) = Cos(2z + 6) and b(z) = 8 Sin(5z).

2 Fig. 2. The same as Fig. 1 except for different parameters and functions, as β = 4, α = 10, μ = 1, γ = −10, Γ (z) = e−(z−10) /10 and b(z) = 2 Sin(6z + 3).

5. The amplification/absorption Γ (z) does appear throughout the auto-Bäcklund transformation via constraints (28)–(30). Γ (z) obviously affects Figs. 1 and 2 from the viewpoint of an observer in the (z, t) space–time, who also discovers that among the ratios of the variable coefficients, k(z) : l(z) : h(z) are constants, and b(z) : d(z) is a constant too, while other ratios, such as k(z) : b(z), are all functions of Γ (z). In conclusion, we have, via symbolic computation, investigated Eq. (1), a vcHNLS model for ultrafast signal-routing, fiber laser systems and optical communication systems with distributed dispersion and nonlinearity management. Of physical and optical interests, we have extended the bilinear method and transformed the vcHNLS model into a variable-coefficient bilinear form. We have then constructed an auto-Bäcklund transformation and analyzed constraints on the coefficient functions. We have illustrated variable-coefficient brightons as well, which are potentially observable with future optical-fiber experiments. To end up the work, we have discussed relevant properties and features. The aforementioned auto-Bäcklund transformation and other results of this Letter will be of certain value to the studies on inhomogeneous fiber media, core of dispersion-managed brightons, fiber amplifiers, laser systems as well as optical communication links with distributed dispersion and nonlinearity management. Acknowledgements We express our sincere thanks to members of our discussion group for their timely and valuable comments. This work has been supported by the Key Project of Chinese Ministry of Education (No. 106033), by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060006024), Chinese Ministry of Education, and by the National Natural Science Foundation of China under Grant No. 60372095. Y.-T.G. would like to acknowledge the Cheung Kong Scholars Programme of the Ministry of Education of China and Li Ka Shing Foundation of Hong Kong.

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