Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects

Nowadays we can see many interesting applications of solitons in different areas of physical sciences such as plasma physics (1), nonlinear optics (2; 3), Bose-Einstein condensate (4; 5), fluid mechanics (6), and so on. Solitons are so robust particles that they are unlikely to breakdown under small perturbations. The most interesting factor about the soliton, however, is that their interactions with the medium through which it propagates is elastic. Recent researches on nonlinear optics have shown that dispersion-managed pulse can be more useful if the pulse is in the form of a power series of a stable localized pulse which is called soliton. Optical solitons have been the objects of extensive theoretical and experimental studies during the last four decades, because of their potential applications in long distance communication. In 1973, the pioneering results of Hasegawa and Tappert (7) proved that the major constraint in the optical fiber, namely, the group velocity dispersion (GVD) could be exactly counterbalanced by the self-phase modulation (SPM). SPM is the dominant nonlinear effect in silica fibers due to the Kerr effect. The theoretical results of Hasegawa and Tappert were greatly supported by the experimental demonstration of optical solitons by Mollenauer et al. (8) in 1980. Since then many theoretical and experimental works have been done to achieve a communication system based on optical solitons. The solitons, localized-in-time optical pulses, evolve from a nonlinear change in the refractive index of the material, known as Kerr effect, induced by the light intensity distribution. When the combined effects of the intensity-dependent refractive index nonlinearity and the frequency-dependent pulse dispersion exactly compensate for one another, the pulse propagates without any change in its shape, being self-trapped by the waveguide nonlinearity. The propagation of optical solitons in a nonlinear dispersive optical fiber is governed by the well-known completely integrable nonlinear Schrodinger (NLS) equation


Introduction
Nowadays we can see many interesting applications of solitons in different areas of physical sciences such as plasma physics (1), nonlinear optics (2; 3), Bose-Einstein condensate (4; 5), fluid mechanics (6), and so on.Solitons are so robust particles that they are unlikely to breakdown under small perturbations.The most interesting factor about the soliton, however, is that their interactions with the medium through which it propagates is elastic.Recent researches on nonlinear optics have shown that dispersion-managed pulse can be more useful if the pulse is in the form of a power series of a stable localized pulse which is called soliton.Optical solitons have been the objects of extensive theoretical and experimental studies during the last four decades, because of their potential applications in long distance communication.In 1973, the pioneering results of Hasegawa and Tappert (7) proved that the major constraint in the optical fiber, namely, the group velocity dispersion (GVD) could be exactly counterbalanced by the self-phase modulation (SPM).SPM is the dominant nonlinear effect in silica fibers due to the Kerr effect.The theoretical results of Hasegawa and Tappert were greatly supported by the experimental demonstration of optical solitons by Mollenauer et al. (8) in 1980.Since then many theoretical and experimental works have been done to achieve a communication system based on optical solitons.The solitons, localized-in-time optical pulses, evolve from a nonlinear change in the refractive index of the material, known as Kerr effect, induced by the light intensity distribution.When the combined effects of the intensity-dependent refractive index nonlinearity and the frequency-dependent pulse dispersion exactly compensate for one another, the pulse propagates without any change in its shape, being self-trapped by the waveguide nonlinearity.The propagation of optical solitons in a nonlinear dispersive optical fiber is governed by the well-known completely integrable nonlinear Schr ödinger (NLS) equation where q is the complex amplitude of the pulse envelope, τ and z represent the spatial and temporal coordinates, and the + and − sign of ǫ before the dispersive term denote the anomalous and normal dispersive regimes, respectively.In the anomalous dispersive regime, this equation possesses a bright soliton solution, and in the normal dispersive regime it possesses dark solitons.The bright soliton and dark soliton solutions can be derived by the inverse-scattering transform method with vanishing (9; 11) and nonvanishing boundary conditions (10).However, if optical pulses are shorter, the standard NLS equation becomes inadequate.Therefore, some higher-order effects such as third-order dispersion, self-steepening, and stimulated Raman scattering, will play important roles in the propagation of optical pulses.
In such a case, the governing equation is the one known widely as the higher-order NLS equation, first derived by Kodama and Hasegawa (12).The effect of these effects in uncoupled and coupled systems for bright solitons is well explained (13; 14).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 a 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 more slowly than the wings.
In practice, the refractive index or the core diameter of the optical fiber are fucntions of the axial coordinate, which means that the fiber is actually axially inhomogeneous.In this case, the parameters which characterize the dispersive and nonlinear properties of the fiber exhibit variations and the corresponding nonlinear wave equations are NLS equations with variable coefficients.Moreover, the problem of ultrashort pulse propagation in nonlinear and axially inhomogeneous optical fibers near the zero dispersion point is more complicated because the high order effects have to be taken into account as well.In order to understand such phenomena, we consider the higher-order NLS (HNLS) equation with variable coefficients where u is the slowly varying envelope of the pulse, d 1 , d 2 , d 3 , d 4 , d 5 and d 6 are the z-dependent real parameters related to GVD, SPM, third-order dispersion (TOD), self-steepening, and stimulated Raman scattering (SRS), and the heat-insulating amplification or loss, respectively.Though Eq. ( 2) was first derived in the year 1980s, only for the past few years, it has attracted much attention among the researchers from both theoretical and experimental points of view.For example, Porsezian and Nakkeeran (13) derive all parametric conditions for soliton-type pulse propagation in HNLS equation using the Painlevé analysis, and generalize the Ablowitz-Kaup-Newell-Segur method to the 3 × 3 eigenvalue problem to construct the Lax pair for the integrable case.Papaioannou et al. (15) give an analytical treatment of the effect of axial inhomogeneity on femtosecond solitary waves near the zero dispersion point which governed by the variable-coefficient HNLS equation.The exact bright and dark soliton wave solutions of this variable-coefficient equation are derived and their behaviors in the presence of the inhomogeneity are analyzed.Mahalingam and Porsezian (16) analyze the propagation of dark solitons with higher-order effects in optical fibers by Painlevéa n a l y s i s and Hirota bilinear method.Xu et al. (17) investigate the modulation instability and solitons on a cw background in an optical fiber with higher-order effects.In addition, there have recently been several papers giving W-shaped solitary wave solution in the HNLS equation.However, in recent years the studies of Eq. ( 2) have not been widespread.In this chapter, we consider equation (2) again and derive some exact soliton solutions in explicit form for specified soliton management conditions.We first change the variable-coefficient HNLS equation into the well-known constant-coefficient HNLS equation through similarity transformation.Then the Lax pairs for two integrable cases of the constant-coefficient HNLS equation are constructed explicitly by prolongation technique, and the novel exact bright N-soliton solutions for the bright soliton version of HNLS equation are obtained by Riemann-Hilbert formulation.Finally, we examine the dynamics and present the features of the optical solitons.It is seen that the bright two-soliton solution of the HNLS equation behaves in an elastic manner characteristic of all soliton solutions.These results are useful in the design of transmission lines with spatial parameter variations and soliton management to future research.

Similarity transformation
A direct and efficient method for investigating the variable-coefficient nonlinear wave equation is to transform them into their constant-coefficient counterparts by similarity transformation.To do so, we firstly take the similarity transformation (18; 19) u = ρ q (T, X) e i(a 1 τ+a 2 ) , ( 3 )   to reduce Eq. ( 2) to the constant-coefficient HNLS equation where q = q(T, X) is the complex amplitude of the pulse envelope, the parameter ǫ (0 < ǫ < 1) denotes the relative width of the spectrum that arises due to the quasi-monochromocity, α 1 , α 2 , α 3 , α 4 and α 5 are the real constant parameters.In Eq. ( 3), ρ, T, a 1 and a 2 are functions of z,andX is a function of τ and z.Substituting Eq. (3) into Eq.( 2) and asking q (T, X) to satisfy the constant-coefficient HNLS equation ( 4), we have a set of partial differential equations (PDEs) where the subscript denotes the derivative with respect to z and τ.Solving this set of PDEs, we have X = kτ + f and where ρ = ρ 0 e d 6 z , k, ρ 0 and c are constants, and T and d 6 are arbitrary functions of z.Sothe similarity transformation (3) becomes Therefore, if we can get the exact soliton solutions of the constant-coefficient HNLS equation ( 4) we can obtain the exact soliton solutions for HNLS equation (2) through Eq. ( 5).In the next section, we will investigate the integrable condition of equation ( 4) by prolongation technique.

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Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects

Prolongation structures of the constant-coefficient HNLS equation
In this section, we investigate the prolongation structures of the constant-coefficient HNLS equation ( 4) by means of the prolongation technique (20)(21)(22).Firstly, the complex conjugate of the dependent variable q in Eq. ( 4) is denoted as q * = u.Then, Eq. ( 4) and its conjugate become Next we introduce four new variables p, r, v and w by and define a set of differential 2-form I = {θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 } on solution manifold M = {T, X, u, v, w, p, q, r} ,where When these differential 2-forms restricted on the solution manifold M become zero, we recover the original constant-coefficient HNLS equation ( 4).It is easy to verify that I is a differential closed idea, i.e. dI ⊂ I.
We further introduce n differential 1-forms ( 8 ) where i = 1, 2, ••• , n, Fi and Gi are functions of u, v, w, p, q, r, ζ i a n da r ea ssumedt ob eb ot h linearly dependent on For the sake of simplification, we drop the indices by rewriting ζ i as ζ, F i as F and G i as G.When restricting on solution manifold, the differential 1-forms Ω i are null, i.e.Ω i = 0 which is just the linear spectral problem ζ X = Fζ and ζ T = Gζ.Following the well-known prolongation technique, the extended set of differential form Ĩ = I Ω i must be a closed ideal under exterior differentiation, i.e. d Ĩ ⊂ Ĩ.BecausedI ⊂ I ⊂ Ĩ, we only need to let d Ω i ⊂ Ĩ, which denotes that where f i j (j = 1, 2, 3, 4, 5, 6) are functions of (T, X),a n dη i = g i (T, X)dX + h i (T, X)dT are differential 1-forms.

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Recent Progress in Optical Fiber Research www.intechopen.comOptical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects 5 When Eq. ( 9) is written out in detail, after dropping the indices we have the following PDEs about F and G as Solving Eq. ( 10), we have the expressions of F and G as where L = {x 0 , x 1 , x 2 , ••• , x 15 } is an incomplete Lie algebra which is called prolongation algebra and it satisfies the following commutation relations It is known that nontrivial matrix representations of prolongation algebra L correspond to nontrivial prolongation structures.To find the matrix representation of L, following the procedure of Fordy (23), we try to embed it into Lie algebra sl(n, C).Starting from the case of n = 2, we found that sl(2, C) is the whole algebra for some special coefficients α j (j = 1, 2, 3, 4, 5).For the case of n = 3, we can also find that sl(3, C) will be the whole algebra for some other special coefficients α j (j = 1, 2, 3, 4, 5).In this paper, we only examine the case of sl(2, C) algebra.
From the above commutation relations, we have the special relations among elements x 1 , x 2 and x 5 as from which we know that x 1 and x 2 are nilpotent elements and x 5 is a neutral element.So we have α 5 = ±6 δ 2 α 3 and
In what folows, we consider the scattering problem of the Eq.(22a).By doing so, the variable T is fixed and is a dummy variable.We first introduce the matrix Jost solutions J ± (X, λ) of (22a) with the asymptotic condition where I is a 2 × 2 unit matrix.Here the subscripts in J ± refer to which end of the X-axis the boundary conditions are set.Then due to tr(Q)=0andAbel'sformulawehavedet(J ± )=1 for all X.Next we denote E = e −iλΛX .S i n c eΨ ≡ J + E and Φ ≡ J − E are both solutions of the first equation in (15), they must be linearly related, i.e. where is the scattering matrix, and R is the set of real numbers.Notice that det(S(λ)) = 1s i n c e det(J ± )=1.If we denote (Φ, Ψ) as a collection of columns,

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Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects www.intechopen.comby using the same formulation as (24; 25; 27), we have the Jost solution is analytic in λ ∈ C + , and Jost solution is analytic in λ ∈ C − ,with and In addition, it is easy to see that and In addition, if we express S −1 as Hence we have constructed two matrix functions P + and P − which are analytic in C + and C − , respectively.On the real line, using Eqs.( 25), ( 27) and (28), it is easily to see that with This determines a matrix Riemann-Hilbert problem with asymptotics which provide the canonical normalization condition for this Riemann-Hilbert problem.If this problem can be solved, one can readily reconstruct the potential q(X, T) as follows.Notice that P + is the solution of the spectral problem (22a).Thus if we expand P + at large λ as and inserting this expansion into (22a), then comparing O(1) terms in (34), we find that

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Recent Progress in Optical Fiber Research www.intechopen.com Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects 9 Thus, recalling the definition of Q the potentials q is reconstructed immediately as where P + 1 =(P ij ).In addition, from the definitions of P + , P − and Eq. ( 25) we have The symmetry properties of the potential Q and V in (23) give rise to symmetry properties in the scattering matrix as well as in the Jost functions.In fact, after some computation we have J ± satisfies the involution property analytic solutions P ± satisfy the involution property and S satisfies the involution property Let λ k and λk are zero points of detP + and detP − , respectively.We see from (37) that (λ k , λk ) are zeros of the scattering coefficients ŝ22 (λ) and s 22 (λ).Due to the above involution property, we have the symmetry relation λk For simplicity, we assume that all zeros (λ k , λk ), k = 1, 2, •, N are simple zeros of ŝ22 (λ) and s 22 (λ),t h e ne a c hk e r n a lo fP + (λ k ) and P − ( λk ) contains only a single column vector v k and row vector vk , Taking the Hermitian of the above equations and using the involution properties, we have To obtain the soliton solutions, we set G = I in (32).In this case, the solutions to this special Riemann-Hilbert problem have been derived in (25; 26) as where The zeros λ k and λk are T-independent.To find the spatial and temporal evolutions for vectors v k (T, X), we take the X-derivative to equation P + v k = 0.By using (22a), one gets Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects www.intechopen.comthus we have Similarly, taking T-derivative to equation P + v k = 0 and using (22b), one has thus we have Solving ( 46) and (48) we get where (v k0 , vk0 ) are constant vectors.
In summary, the N-bright soliton solutions to Eq. ( 14) are obtained from the analytical functions P + 1 in (43) together with the potential reconstruction formula (36) as where the vectors v j are given by (49).Without loss of generality, we take ′ with b k constants.And if we denote the general N-soliton solution to Eq. ( 14) can be written out explicitly as with In what follows, we investigate the dynamics of the one-soliton and two-soliton solutions in Eqs.(14) in detail.

Examples of single and two bright solitons in Eq. (14)
To get the single bright soliton solution for Eq. ( 14), we set N = 1 in (52) to have X 0 +iω 0 , the single soliton solution (54) can be rewritten as constants.This solution is similar to the solitary wave solution in the standard NLS equation (1).Its amplitude function has the shape of a hyperbolic secant with peak amplitude 2η 1 /δ, and its velocity depends on several parameters, which is 12 T h e phase θ of this solution depends linearly both on space X and time T. We show this single soliton solution in Fig. 1 with parameters The two-soliton solution in Eq. ( 14) corresponds to N = 2 in the general N-soliton solution (52) with (53).This solution can also be written out explicitly, however, we prefer to showing 59 Optical Solitons in a Nonlinear Fiber Medium with Higher-Order Effects www.intechopen.com52) with (53) are Under these parameters, the velocity of the two solitons are different.It is observed that interactions between two soliton don't change the shape and velocity of the solitons, and there is no energy radiation emitted to the far field.Thus the interaction of these solitons is elastic, which is a remarkable property which signals that the HNLS equation ( 14) is integrable.Fig. 2(b) displays a bound state in Eq. ( 14), and the soliton parameters here are Under these parameters, the two constituent solitons have equal velocities, thus they will stay together to form a bound state which moves at the common speed.It can be seen that the width of this solution changes periodically with time, thus this solution is called breather soliton.

Single soliton solutions
We choose two cases of soliton management parameters d j (j = 1, 2, 3, 4, 5, 6) to study the dynamics of the single solitons in HNLS equation (2).Firstly, if we take the soliton management parameters to satisfy the variables ρ, T and X in similarity transformation (3) are So the single soliton solution in HNLS equation ( 2) with coefficients (59) is where q (T, X) satisfies Eq. (55) and T, X satisfy Eq. (60).
Secondly, if we take the soliton management parameters to satisfy the variables ρ, T and X in similarity transformation (3) are In this case the single soliton solution in HNLS equation ( 2) with coefficients (62) is where q (T, X) satisfies Eq. (55) and T, X satisfy Eq. ( 63).In Fig. 3, we show the single soliton solutions (61) and ( 64) in HNLS equation ( 2) with coefficients (59) and (62), respectively.Here the solution parameters are given in (56) and ρ 0 = 0.5, c = 1, k = 2.It is observered that when the soliton management parameters d j (j = 1, 2, 3, 4, 5) are linearly dependent on variable z and d 6 = 0 (see Eq. ( 59)), the trajectory of the optical soliton is a localized parabolic curve, as shown in Fig. 3(a).When the soliton management parameters d j (j = 1, 2, 3, 4, 5) are periodically dependent on variable z and d 6 = 0 (see Eq. ( 62)), the trajectory of the optical soliton is a periodical localized nonlinear wave, as shown in Fig. 3(b).

Conclusions
In summary, we have studied the variable-coefficient higher order nonlinear Schr ödinger equation which describes the wave propagation in a nonlinear fiber medium with higher-order effects such as third order dispersion, self-steepening and stimulated Raman scattering.By means of similarity transformation, we first change this variable-coefficient equation into the constant-coefficient HNLS equation.Then we investigate the integrability of the constant-coefficient HNLS equation by prolongation technique and find two Lax integrable HNLS equations.The exact bright N-soliton solutions for the bright soliton version of HNLS equation are obtained using Riemann-Hilbert formulation.Finally, the dynamics of the optical solitons in both constant-coefficient and variable-coefficient HNLS equations is examined and the effects of higher-order effects on the velocity and shape of the optical soliton

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