Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schrödinger Equation with Self-steepening and Self-frequency Shift

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1 Commun. Theor. Phys. (Beijing, China) 45 (2006) pp c International Academic Publishers Vol. 45, No. 4, April 15, 2006 Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schrödinger Equation with Self-steepening and Self-frequency Shift ZONG Feng-De, DAI Chao-Qing, and ZHANG Jie-Fang Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua , China (Received August 16, 2005) Abstract By making use of the generalized sine-gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive and the quintic nonlinear Schrödinger equation with self-steepening, and self-frequency shift. Moreover, we discuss the formation conditions of the bright and dark solitary waves. PACS numbers: Dp, Yv, Tg Key words: fourth-order dispersive nonlinear Schrödinger equation, bright optical solitary wave, dark optical solitary wave In the past decades, terrestrial and submarine communication systems have scored an incredible growth of their transmission capacity. Optical solitons have been the objects of extensive theoretical and experimental studies due to their potential applications in long distance communication and all-optical ultrafast switching devices. It is well known that picosecond pulses are well described by the classic nonlinear Schrödinger equation (NLSE), which accounts for the group velocity dispersion (GVD) and selfphase modulation (SPM). To enlarge the information capacity, it is necessary to transmit ultrashort (subpicosecond and femtosecond) optical pulse at a high bit rate. However, when the stronger and stronger intensity of the incident light field and shorter and shorter pulses are considered, the classic NLSE fails in the physical description of the propagations of light pulses in fibers because higherorder dispersion terms and the non-kerr nonlinearity effects cannot be neglected. For example, when short pulses are considered (to nearly 50 fs), the third-order dispersion (TOD) is not negligible. The TOD will produce asymmetrical broadening in the time domain for the ultrashort soliton pulses. Moreover, as the pulse width becomes even narrower (below 10 fs), the fourth order dispersion (FOD) must also be taken into account. [1] When the optical field frequency approaches a resonant frequency of the optical fibers material, accounting for higher-order nonlinearities in optical fibers is related to stronger optical fields and the corresponding lager powers launched into them. Therefore, the cubic-quintic nonlinearities should be introduced into the NLSE. [2] For the up-to-4th-order dispersion and the quintic nonlinearity to be considered, the pulse must be extremely narrow and the optical intensity must be very high. Under such conditions, the self-steepening (SS) and self-frequency shift (SFS) must be included in the NLSE. The SS, otherwise called the Kerr dispersion, is due to the intensity dependence of group velocity. This forces the peak of the pulse to travel slower than the wings, which causes an asymmetrical spectral broadening of the pulse. The SFS due to stimulated Raman scattering results in an increasing redshift in the pulse spectrum, in which the long wavelength components experience Reman gain at the expense of the short wavelength components. [3] It has been recognized that the SFS is a potentially detrimental effect in soliton communication systems. [4] Hasegawa and Tappert [5] theoretically predicted bright soliton solutions in the anomalous dispersion region of the fiber and dark solitons (which in intensity profile contains a dip in a uniform background) in the normal dispersion regime. Technically, this means that any optical pulse injected in a lossless fiber evolves into a soliton as it propagates and thus solitons are natural forms of signal carriers in a similar way as are the Fourier modes in a linear transmission system. Bright soliton propagation was first verified by Mollenauer et al. [6] in single-mode low loss (0.2 db/km) optical fiber over a short (700 m, effectively loss) distance. The generation of dark pulses was first demonstrated by Emplit et al. [7] using amplitude and phase filtering. There exist several essential differences between bright and dark solitons. One of them consists of the existence of multiple bound states that can form bright solitons in clear contrast with dark solitons. In addition, given a fixed optical frequency and background intensity, there is a continuous range of dark solitons with different blackness parameters (the so-called gray, black, and darker than black solitons). On the other hand, for a fixed frequency and intensity, just one bright soliton solution is possible. Finally, dark solitons have a phase profile, which is an anti-symmetric function of time, whereas The project supported by National Natural Science Foundation of Zhejiang Province of China under Grant No. Y Corresponding author, jf zhang@zjnu.cn

2 722 ZONG Feng-De, DAI Chao-Qing, and ZHANG Jie-Fang Vol. 45 bright solitons have a constant phase. It is this phase function that is a major reason for the difficulty in generating and studying dark solitons experimentally. Generally, in experiments and numerical simulations, dark pulses were created on a background carrier pulse of a finite width, and the latter had a shape of a long bright pulse. [8] Zhao and Bourkoff [9] compared bright and dark pulse properties, showing that the dark pulse spreads more slowly with loss and is less sensitive to noise than a bright one. Extending the NLS equation to the generalized nonlinear Schrödinger (GNS) equations received much attention due to these models being more realistic and many important applications of different types of GNS equations. Generally, corrections to the linear dispersion should be taken into account together with nonlinear corrections to the cubic nonlinear term being the same order. In this paper, we consider the fourth-order dispersive NLSE with the quintic nonlinearity, self-steepening and self-frequency shift, iu z β 2 2 u tt + γ 1 u 2 u = i β 3 6 u ttt + β 4 24 u tttt γ 2 u 4 u + (α 1 + iα 2 )( u 2 u) t + (α 3 + iα 4 )u( u 2 ) t, (1) where u(z, t) is the slowly varying envelope of the electromagnetic field, t represents the time (in the groupvelocity frame), z represents the distance along the direction of propagation (the longitudinal coordinate), β 2 = 2 k/ ω 2 ω=ω0, β 3 = 3 k/ ω 3 ω=ω0, β 4 = 4 k/ ω 4 ω=ω0 represent GVD (second order dispersion), TOD, FOD, respectively. k is the axial wave number, ω 0 is the carrier wave frequency. γ 1 and γ 2 are the cubic and quintic nonlinearities coefficients, respectively. α 1 describes the combined effect (CE) between nonlinear gain and/or absorption processes and nonresonant. [10] The term proportional to α 2 results from the first derivative of the slowly varying part of the nonlinear polarization. It is responsible for self-steepening (SS) and shock formation at a pulse edge. The last term is a nonlinear gradient term which results from the time-retarded induced Raman process. In fact, α 3 is usually responsible for the soliton self-frequency shift (SFS), and α 4 is here called nonlinear dispersion (ND). In our analysis, the parameters do not need to be small, so we are not limited to the perturbation regime. For picosecond light pulse, these terms on the righthand side of Eq. (1) can be omitted, and equation (1) is reduced to the classic NLSE. When the terms on the right-hand side except γ 2 are negligible, equation (1) is the cubic-quintic NLSE, whose exact travelling wave solutions were discussed in Ref. [11]. Recently, some authors [12] have found bright optical soliton for Eq. (1) with the TOD and all the α terms being nil (β 3 = α 1,...,4 = 0). Moreover, Karpman et al. [13] have investigated the resonant radiation and evolution of a soliton described by the nonlinear Schrödinger equation with the third and fourth derivatives, i.e. the last five terms on the right-hand side of Eq. (1) disappear. When β 4 = γ 2 = α 1 = α 3 = 0, equation (1) is shown that dark and bright soliton solutions exist. [14,15] More recently, we derive the combined solitary wave, bright soliton, dark soliton, and W-shape solitary wave [16] when α 1,...,4 = 0 in Eq. (1). When α 1 = α 3 = 0, we obtain the bright and dark solitary wave solutions. [17] However, the exact analytic solutions of Eq. (1) have not been previously obtained. It is always useful to construct exact analytical solutions and it is worth while to investigate the exact solutions (in particular solitary wave solutions). A method based on the inverse scattering transformation (IST) is very powerful when the NLSE is integrable. However, it is rather sophisticated and inconvenient for one who is not familiar with IST. The general HONLS equations are not completely integrable and cannot be exactly solved by IST. The nonintegrability usually originates not only from the higher-order nonlinear terms but also the higher-dispersion terms. And therefore, they do not have soliton solutions, but they do have solitary wave solutions. Soliton should be called under two conditions: (i) The energy can propagate in the form of localized packets without dissipation or gain; (ii) The solitary waves after a collision have exactly the same form as before the collision. [18,19] So we use the phrase solitary wave in this paper. It is also difficult to solve a fourth-order derivative equation with quintic terms by a direct integration process used in Refs. [15] and [20]. Therefore, the necessity for new theoretical methods to be developed increases. In this paper, we derive analytic periodic sn wave and cnoidal wave solutions and fundamental bright and dark solitary wave solutions for Eq. (1) by using the generalized sine-gordon equation expansion method. [21] And the properties of the obtained solutions are shown graphically. The virtue of this method is twofold: first, without much complicated calculations, we circumvent integration to directly get both bright and dark solitary waves simultaneously, which is simpler than the direct ansatz approach in Ref. [22]. Compared with the traditional direct integration methods, this method can be used to easily identify the solution pattern of a nonlinear wave equation. Meanwhile, we do not have to give any specific value to the coefficient of each term of equation while we are solving it. The second feature of this method is that it is independent of the integrability of nonlinear equations. To seek exact travelling wave solutions to Eq. (1), we make the transformation u(z, t) = A(ξ) exp(iθ), ξ = pz t, θ = kz ct, (2)

3 No. 4 Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schrödinger Equation 723 where the amplitude A(ξ) is real. The inverse pulse width p, the phase shift k, and the frequency shift c are all undetermined constants. Substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts leads to (β 3 β 4 c)a + (6p 6β 2 c 3β 3 c 2 + β 4 c 3 )A + (18α α 2 )A 2 A + 6α 1 ca 3 = 0, (3) β 4 A (4) + (12β β 3 c 6β 4 c 2 )A + (24k 12β 2 c 2 4β 3 c 3 + β 4 c 4 )A 24(γ 1 α 1 c)a 3 24γ 2 A 5 24(3α 1 + 2α 3 )A 2 A = 0. (4) Inserting the differential of Eq. (3) into Eq. (4) has [ β 4 (6p 6β 2 c 3β 3 c 2 + β 4 c 3 ) (β 3 β 4 c)(12β β 3 c 6β 4 c 2 )]A (β 3 β 4 c) [(24k 12β 2 c 2 4β 3 c 3 + β 4 c 4 )A 24(γ 1 α 1 c)a 3 24γ 2 A 5 24(3α 1 + 2α 3 )A 2 A ] + β 4 (18α α 2 )(2AA 2 + A 2 A ) + 18β 4 α 1 ca 2 A = 0. (5) For Eq. (5), in the generalized sine-gordon equation expansion method the crucial step is that the solutions we are looking for can be expressed in the form n A(ξ) = a 0 + cos i 1 ω(ξ)[a i sin ω(ξ) + b i cos ω(ξ)], (6) with i=1 dω dξ = µ 1 m 2 sin 2 ω(ξ), µ = ±1, (7) sin ω(ξ) = sn(ξ; m), cos ω(ξ) = cn(ξ; m), (8) where a 0, a i, b i (i = 1,..., n) are undetermined real constants. By balancing the highest-order derivative term with the nonlinear term in Eq. (5), one can find n = 1 in Eq. (6). Substituting Eq. (6) with n = 1 and Eq. (7) into Eq. (5) and setting the coefficients of individual term ω s (ξ) sin l ω(ξ) cos m ω(ξ) (s = 0, 1, l = 0, 1, m = 0,..., 5) to zero, with the aid of Maple, we generate eighteen real algebraic equations for the free parameters. We do not present them in detail here, but discuss their solutions. Among these eighteen real algebraic equations, seven equations are not compatible with other equations. Therefore, from these seven equations one can get the constraint to the following solutions of Eq. (1). with Solution 1 4(3α 1 + 2α 3 )(β 4 c β 3 ) 3β 4 α 1 c = 0 (9) sn-function cnoidal wave solution (3α 2 + 2α 4 )β 4 u 1 = ±m sn(pz t) exp[i(kz ct)], (10) p = {9β 3 β 4 γ 2 c(3β 3 c + 2β 2 ) 2c 2 γ 2 β 4 2 (3β 2 c β 3 c 3 ) + [9(1 + m 2 )(3α 2 + 2α 4 ) c(3α 2 + 2α 4 )(γ 1 cα 1 )]β β 4 3 c 4 γ 2 [12β 3 (3α 2 + 2α 4 )(γ 1 cα 1 )]β 4 12β 3 2 γ 2 (β 3 c + β 2 )}/[6γ 2 β 4 (β 4 c β 3 )], (11) k = {3β 2 4(3α 2 + 2α 4 )[3(1 + m 4 )(3α 2 + 2α 4 ) + 4c(1 + m 2 )(γ 1 α 1 c) + 2m 2 (3α 2 + 2α 4 )] + 6c 4 γ 2 β 2 4(2β 2 + β 3 c) c 6 β 3 4γ 2 3c 3 β 3 β 4 γ 2 (8β 2 + 3β 3 c) + 12β 3 β 4 (1 + m 2 )(α 1 c γ 1 )(3α 2 + 2α 4 ) + 4c 2 β 2 3γ 2 (3β 2 + β 3 c)}/[24γ 2 (β 3 β 4 c) 2 ], (12) where c is an arbitrary constant. When the Jacobian elliptic modulus m 1, the solution (10) degenerates as the dark solitary wave solution u (3α 2 + 2α 4 )β 4 1 = ± tanh(pz t) exp[i(kz ct)], (13) where p, k are given by Eqs. (11) and (12) with m = 1. Solution 2 cn-function cnoidal wave solution u 2 = ±m (3α 2 + 2α 4 )β 4 cn(pz t) exp[i(kz ct)] (14)

4 724 ZONG Feng-De, DAI Chao-Qing, and ZHANG Jie-Fang Vol. 45 with p = { 9β 3 β 4 γ 2 c(3β 3 c + 2β 2 ) + 2c 2 γ 2 β 4 2 (3β 2 c β 3 c 3 ) + [9(2m 2 1)(3α 2 + 2α 4 ) 2 12c(3α 2 + 2α 4 )(γ 1 cα 1 )]β 4 2 5β 4 3 c 4 γ 2 + [12β 3 (3α 2 + 2α 4 )(γ 1 cα 1 )]β β 3 2 γ 2 (β 3 c + β 2 )}/[6γ 2 β 4 (β 4 c β 3 )], (15) k = {3β 2 4(3α 2 + 2α 4 )[8m 2 (m 2 1)(3α 2 + 2α 4 ) 4c(1 + 2m 2 )(γ 1 α 1 c) + 3(3α 2 + 2α 4 )] + 6c 4 γ 2 β 2 4(2β 2 + β 3 c) c 6 β 3 4γ 2 3c 3 β 3 β 4 γ 2 (8β 2 + 3β 3 c) + 12β 3 β 4 (1 2m 2 )(α 1 c γ 1 )(3α 2 + 2α 4 ) + 4c 2 β 2 3γ 2 (3β 2 + β 3 c)}/[24γ 2 (β 3 β 4 c) 2 ], (16) where c is an arbitrary constant. When the Jacobian elliptic modulus m 1, the solution (14) degenerates as the bright solitary wave solution u (3α 2 + 2α 4 )β 4 2 = ± sech(pz t) exp[i(kz ct)], (17) where p, k are given by Eqs. (15) and (16) with m = 1. Now the exact bright and dark solitary wave solution are derived. With the help of these exact solutions, the phenomena modelled by this fourth-order dispersive NLSE (1) can be better understood. They can help to analyze the stability of these solutions and to check numerical analysis for the fourth-order dispersive NLSE (1). For the constraint condition (9), when α 1 = 0, α 3 must be equal to zero because β 4 c β 3 0 from Eqs. (10) and (14) or Eqs. (13) and (17). That is to say, when the combined effect between nonlinear gain and/or absorption processes and nonresonant is omitted, the self-frequency shift effect must be ignored, which is similar to the case that Blow et al. [23] have employed the bandwidth-limited gain to suppress the SFS and use an adiabatic perturbation theory as well as numerical simulations to determine the laser pulse form. This result offers engineers an alternative scheme to devise appropriate optical fiber in all-optical ultrafast soliton communication. If the effects such as TOD, FOD, SFS, and CE are coincident with Eq. (9), then equation (1) has the bright and dark solitary wave solutions (13) and (17). When α 1 = α 3 = 0, the solutions (13) and (17) are the bright and dark solitary wave solutions obtained in Ref. [17]. From Eqs. (13) and (17), one can find that the formation conditions of the bright and dark solitary waves are opposite, i.e. the formation condition of the bright solitary wave is and for the dark solitary wave is β 4 γ 2 (3α 2 + 2α 4 )(β 4 c β 3 ) < 0, (18) β 4 γ 2 (3α 2 + 2α 4 )(β 4 c β 3 ) > 0. (19) From Eqs. (18) and (19), the bright and dark solitary waves exist both for the anomalous-dispersion fiber and the normal-dispersion fiber if only the relations (18) and (19) are satisfied respectively, which is different to the existence of the bright soliton in anomalous-dispersion fiber and dark soliton in the anomalous-dispersion fiber for classic NLSE. It is interesting that the bright and dark solitary waves in Eq. (1) exist even if there is not effect of the GVD and SPM (β 2 0, γ 1 0). From Eqs. (9), (13), and (17), we can see the formation of the solitary waves is decided by α 1,...,4, β 3, β 4, γ 2, namely, the right terms in Eq. (1). If the SS and the nonlinear dispersion (ND) are both neglected (α 2 0, α 4 0), both the bright and dark solitary waves disappear, but either the SS or the ND exists while β 4 0, the bright and dark solitary waves do not disappear. When the FOD is omitted (β 4 0), the bright and dark solitary waves also disappear. Either the quintic nonlinearity γ 2 0 or (β 4 c β 3 ) 0, both the bright and dark solitary waves are destroyed, but the bright and dark solitary waves exist when other conditions are satisfied. Therefore, under certain parameter conditions (9), (18), and (19), although the higher-order terms are considered, the solitary wave solutions can be achieved. Those disadvantaged results caused by higher-order terms, such as the asymmetrical broadening in the time domain caused by TOD and the asymmetrical spectral broadening of the pulse made by self-steepening and even the redshift of the spectrum caused by the self-frequency shift, can be removed. Namely, there exists a balance of the cooperation of these effects: TOD, FOD, SS, SFS, quintic nonlinear, ND, and CE, which lead to the formation of the bright and dark solitary waves. Figure 1(a) shows the intensity profile of the dark solitary wave solution (13) for different model coefficients which satisfy the constraints (9) and (19). The asymptotic value of the dark solitary wave is non-zero as the time variable approaches infinity ( t ). In general, as the higher-order dispersive terms, i.e. β 3 and β 4 increase, the amplitude of the solitary wave increases, while the pulse width gets narrower. And the corresponding spatiotemporal plot is

5 No. 4 Optical Solitary Waves in Fourth-Order Dispersive Nonlinear Schrödinger Equation 725 depicted in Fig. 1(b). As shown in Fig. 2(a), the intensity u 2 2 is dark solitary wave solution at different distances z = 1, 0, 1 for α 1 = α 2 = 0.1, β 2 = 1, γ 1 = γ 2 = 1, β 3 = 0.9, β 4 = 0.5, c = 1 in Eq. (17), and the corresponding spatiotemporal plot is depicted in Fig. 2(b). Fig. 1 (a) The intensity of dark solitary wave U = u 1 2 for different model coefficients which satisfy the constraints (9) and (19) at z = 0.2 for curve 1 (β 3 = 0.9, β 4 = 0.5), curve 2 (β 3 = 0.8, β 4 = 0.4), and curve 3 (β 3 = 0.7, β 4 = 0.3) with α 1 = α 2 = 0.1, β 2 = 1, γ 1 = γ 2 = 1, c = 1 in Eq. (13). (b) The corresponding spatiotemporal plot with the same parameters of curve 2. Fig. 2 (a) The intensity of bright solitary wave U = u 2 2 which satisfies the constraints (9) and (18) at different distances z = 1, 0, 1 for α 1 = α 2 = 0.1, β 2 = 1, γ 1 = γ 2 = 1, β 3 = 0.9, β 4 = 0.5, c = 1 in Eq. (17). (b) The corresponding spatiotemporal plot with the same parameters of (a). In conclusion, by utilizing the generalized sine-gordon equation expansion method, we find cnoidal periodic wave solutions and fundamental bright and dark optical solitary wave solutions for the fourth-order dispersive NLSE with the quintic nonlinearity, self-steepening and self-frequency shift and nonlinear dispersive terms, which always exists provided that certain relations between the parameters are fulfilled. Moreover, we discuss the formation conditions of the bright and dark solitary waves (18) and (19), respectively. The specific significance of these results lies in their potential application to the design of optical pulse compressors and solitary wave based communications links. It will offer the guarantee for the transmission of communication in theory, and provide the developmental direction for searching for the next generation carrier of communication. These solutions in this paper cannot be degenerated into the bright and dark solitary waves in Ref. [16]. This suggests that the HNLSE (1) possesses other rich possible solutions, which is worth while further studying. Although these solutions represent only a small subset of the large variety of possible solutions admitted by this higher-order nonlinear Schrödinger equation (1), those presented here are the first examples of exact analytic solutions found so far. The properties of the solitary wave solutions for Eq. (1) are shown in Figs. 1 and 2. Since the propagation of ultrashort light pulses in optical fibers is of particular interest because of the common expectation that solitary waves may be of extensive use in telecommunication and even revolutionize it,

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