# Modulational Instability of the Higher-Order Nonlinear Schrodinger¨ Equation with Fourth-Order Dispersion and Quintic Nonlinear Terms

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Modulational Instability of the Higher-Order Nonlinear Schrodinger¨ Equation with Fourth-Order Dispersion and Quintic Nonlinear Terms

Woo-Pyo Hong Department of Electronics Engineering, Catholic University of Daegu, Hayang, Gyongsan, Gyungbuk 712-702, South Korea Reprint requests to Prof. W.-P. H.; E-mail: [email protected]

Z. Naturforsch. 61a, 225 – 234 (2006); received March 20, 2006

The modulational instability of the higher-order nonlinear Schr¨odinger equation with fourth-order dispersion and quintic nonlinear terms, describing the propagation of extremely short pulses, is inves- tigated. Several types of gains by modulational instability are shown to exist in both the anomalous and normal dispersion regimes depending on the sign and strength of the higher-order nonlinear terms. The evolution of the modulational instability in both the anomalous and normal dispersion regimes is numerically investigated and the effects of the higher-order dispersion and nonlinear terms on the formation and evolution of the solitons induced by modulational instability are studied. – PACS numbers: 42.65.Tg, 42.81Dp, 42.65Sf

Key words: Higher-Order Nonlinear Schr¨odinger Equation; Modulational Instability; Optical Gain; Optical Soliton; Numerical Simulation.

1. Introduction dispersion regimes and to study the dynamics of the MI induced solitons in the context of an extended higher- Modulational instability (MI), occuring as a result order NLS equation. of an interplay between the nonlinearity and dispersion As the extended higher-order NLS equation we con- (or diffraction, in the spatial domain), is a fundamental sider the generalized NLS with a third- and fourth- and ubiquitous process that appears in most nonlinear order dispersion and cubic-quintic nonlinear terms, de- wave systems in nature such as ﬂuid dynamics [1, 2], scribing the propagation of intense femtosecond opti- nonlinear optics [3, 4], and plasma physics [5]. As a cal pulses in a medium which exhibits a parabolic non- result, a continuous-wave (CW) or quasi-CW radiation linearity law [6, 9 – 12], in the form of propagating in a nonlinear dispersive medium may suf- fer the instability with respect to weak periodic modu- ∂A i ∂3A 1 ∂3A i ∂4A α + β − β − β + A = lations of the steady state and results in the breakup of ∂z 2 2 ∂τ2 6 3 ∂τ3 24 4 ∂τ4 2 CW into a train of ultrashort pulses [6]. In the context (1) γ ∂(| |2 ) ∂| |2 of ﬁber optics, the temporal MI has been experimen- 2 4 1 A A A iγ1|A| A+iγ2|A| A − − iγ1TRA . tally veriﬁed for a single pump wave propagating in ω0 ∂τ ∂τ a standard non-birefringence ﬁber, which can be mod- eled by the nonlinear Schr¨odinger (NLS) equation, and Here A(z,τ) is the slowly varying envelope of the elec- it was found that the MI only occurs in the anomalous tromagnetic ﬁeld, τ ≡ t − β1z is the retarded time, β ≡ ( 2 / ω2) = , , group-velocity dispersion (GVD) regime with a posi- j d k j d ω=ω0 ( j 2 3 4) are the GVD coefﬁ- tive cubic nonlinear term [6, 7]. On the other hand, in cients evaluated at the carrier frequency ω0, the atten- the context of the vector NLS equations which can de- uation term corresponding to the ﬁber loss α, and γ 1 scribe two or more optical ﬁelds copropagating inside and γ2 are the nonlinear coefﬁcients. The term propor- the highly birefringence optical ﬁbers, MI can also ex- tional to γ1/ω0 results from the intensity dependence ist in the normal dispersion regime as the result of a of the group velocity and causes self-steepening and novel phenomenon called the cross-phase modulation shock formation at the pulse edge. The last term rep- (XPM) [8]. The purpose of the present work is to show resents the intrapulse Raman scattering, which causes that MI can also occur in both anomalous and normal a self-frequency shift, where TR is related to the slope

0932–0784 / 06 / 0500–0225 $ 06.00 c 2006 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 226 W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation of the Raman gain [6]. Some analytic bright solitary- 2. Gain by the Modulational Instability wave solutions of (1) in the anomalous and normal dis- persion regimes in the absence of the self-steepening In order to investigate how weak and time- and the Raman scattering terms were obtained under dependent perturbations evolve along the propagation various constraints among the coefﬁcients, and the ef- distance, we consider the following linear stability fects of the third- and fourth-order dispersion terms on analysis. The steady-state solution of (1) is given by √ the dynamics of solitary-waves were numerically stud- A¯(z,τ)= Pexp(iφ ), (2) ied [9, 10]. NL

Shagalov [13] has investigated the effects of the where the nonlinear phase shift φNL is related to the op- third- and fourth-order dispersion terms, in the absence tical power P and the propagation distance z as φNL = γ = 2 of the quintic nonlinear term ( 2 0) and the self- (iα/2−γ1P−γ2P )z. The linear stability of the steady- steepening and the Raman terms of (1), on the MI state can be examined by introducing a perturbed ﬁeld and showed that modulationally unstable waves evolve of the form to soliton-type or turbulent states depending on the √ α strength of the higher-order dispersion terms. Recently, A(z,τ)=[ P+ε(z,τ)]exp[− z−i(γ P+γ P2)z], (3) 2 1 2 the author [14] has obtained the exact analytic expres- √ sion for the gain by the MI and numerically studied the where the complex ﬁeld |ε(z,τ)| << P. Thus, if the dynamics of the solitons induced by the MI of (1) in the perturbed ﬁeld grows exponentially, the steady-state absence of the self-steepening and Raman terms. It was becomes unstable. By substituting (3) into (1) and col- shown that solitons exist both for the anomalous and lecting terms in ε, we obtain the linearized equation the normal dispersion regimes and in particular the role as of the third-order dispersion term β on the evolution 3 γ P γ P of the MI was analyzed. More recently, in the context ε − ( 1 + γ )ε − ( 1 + γ )ε∗ z 2 ω i 1TRP t ω i 1TRP t of (1) without the quintic nonlinear term, Demircan 0 0 and Bandelow [12] have numerically veriﬁed the su- 1 1 1 (4) − iβ4εtttt + iβ2εtt − β3εttt percontinuum generation by the MI in the anomalous 24 2 6 2 ∗ as well as in the normal dispersion region, which has + i(γ1P + 2γ2P )(ε + ε )=0, been experimentally observed [15]. They showed that the MI dominates higher-order effects such as third- where ∗ denotes complex conjugates. We assume a and fourth-order dispersion, self-steepening and stim- general solution of the form ulated Raman scattering. However, the self-consistent ε( ,τ)= [ ( −Ωτ)]+ [− ( −Ωτ)], investigation on the MI of (1) and the dynamics of the z U exp i Kz Vexp i Kz (5) solitons induced by the MI have not been performed, where K and Ω represent the wave number and the which will be pursued in the present work. frequency of the modulation [6], respectively. Insert- The paper is organized as follows. In Section 2, we ing (5) to (4), we obtain the dispersion relation obtain the analytic expressions for the gain by MI and 2 8 6 4 2 show that the inclusion of the fourth-order dispersion K +CkK +Φ8Ω +Φ6Ω +Φ4Ω +Φ2Ω = 0, (6) and quintic nonlinear terms leads to a more compli- cated gain spectrum than those of [13, 14, 16, 17]. Sub- where sequently, we show that the MI gain can also occur γ Ω ≡ + = −1 β Ω 3 + 1P + γ Ω, even in the normal dispersion regime, depending on Ck Ck,r iCk,i 3 4 2i 1TRP 3 ω0 the sign and strength of the higher-order dispersion and 1 the nonlinear terms. In Section 3, we then numerically Φ = − β 2, 8 576 4 investigate the dynamics of the initial steady CW in the anomalous and normal dispersion regimes under a 1 1 2 Φ6 = − β4β2 + β3 , weak modulational ﬁeld with a modulation frequency 24 36 β γ obtained from the MI gain spectra. In particular, the 1 2 2 3 1P Φ4 ≡ Φ4,r + iΦ4,i = − β2 − effect of the third-order dispersion and cubic-quintic 4 3 ω0 nonlinear terms on the evolution of MI is studied. The 1 1 1 +( γ P + γ P2)β − iβ γ PT , conclusions follow in Section 4. 12 1 6 2 4 3 3 1 R W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation 227 2 Φ2 ≡ Φ2,r + iΦ2,i =(2γ2P + γ1P)β2 has previously derived the MI gain as (7) |β Ω| β β γ 2P2 2γ 2P2T (Ω)= 4 − (Ω 2 + 2 ) Ω 4 + 2 Ω 2 + 3 1 + i 1 R . g 12 β 12 β ω 2 ω 12 4 4 0 0 (11) (γ + γ ) 1/2 The dispersion relation has the solution + P 1 2 2P , 48 β 1 1 1 √ 4 K = − C , − iC , ± Ξ + i∆, 2 k r 2 k i 2 from which four types of MI gain spectra as functions of γ , γ , β , and P have been classiﬁed. Those MI gain Ξ = 2 − 2 − Ω 2Φ − Ω 4Φ 1 2 j Ck,r Ck,i 4 2,r 4 4,r spectra are shown to exist both for the anomalous dis- (8) β < β > 6 8 persion ( 2 0 with 4 0) and for the normal disper- − 4Φ6Ω − 4Φ8Ω , sion (β2 > 0 with β4 < 0) showing four MI peaks which 2 4 resemble to the cross-phase modulation term [14]. It is ∆ =(2C , C , − 4Ω Φ , − 4Ω Φ , ) k r k i 2 i 4 i also shown that the power is related to the cubic and or quintic nonlinear terms. α Now we turn to the characteristics of the MI gain 1 1 1 2 2 1/4 K = − Ck,r − iCk,i ± (Ξ +∆ ) cos spectrum under some speciﬁc physical conditions. In 2 2 2 2 general, the nonlinear coefﬁcients γ (i = 1,2) are sen- α i + i(Ξ 2+∆ 2)1/4 sin , sitive to the properties of optical ﬁbers, which can be 2 estimated from α ∆ (9) π = √ , 2 n j sin γi = , (12) 2 2Ξ 2 + 2∆ 2 + 2Ξ Ξ 2 + ∆ 2 λA eff = , α 1 Ξ where n j ( j 2 4) are the nonlinear refractive in- cos = 2 + 2 √ . dex coefﬁcients, λ is the typical wavelength for op- 2 2 Ξ 2 + ∆ 2 tical ﬁber (λ ≈ 1.55 µm), and Aeff is the effective The steady-state solution becomes unstable when- ﬁber core area (see [6] for detail) which varies from µ 2 µ 2 ever K has an imaginary part since the perturbation 20 m to 100 m . As examples, in the following then grows exponentially with the intensity given by analysis we take the typical values for n2 and n4 as: = × −17 2/ = − × −31 4/ 2 the growth rate or the MI gain deﬁned as g(Ω) ≡ n2 2 10 m W and n4 5 10 m W for =( −17 ∼ −18) 2/ 2Im(K) [6] as an AlGaAs waveguide; n2 10 10 m W −28 −29 4 2 and n4 = −(10 − 10 ) m /W for semiconduc- ∆(Ξ 2 + ∆ 2)1/4 tor doped and double-doped optical ﬁbers [18]. Thus, (Ω)= + . g Ck,i √ (10) we adopt |γ | =(1 − 100) W−1/km and |γ | =(1 × 2Ξ 2 + 2∆ 2 + 2Ξ Ξ 2 + ∆ 2 1 2 10−4 −5×10−7) W−2/km, where the absolute denotes We note that the MI gain does not depend on the ﬁber the possibility of either positive or negative signs of loss term α in (1) and the third-order dispersion term γ j depending on the properties of optical ﬁbers. As an β3. In fact, any odd-order β j term does not contribute example, if the nonlinear coefﬁcients γ1 and γ2 have to the MI gain, however, it can inﬂuence the velocity the same sign, then the ﬁfth-order nonlinearity only and the propagation direction of the soliton induced by supports the self-focusing or defocusing effect of the the MI [14]. It is interesting to ﬁnd that the gain in (10) medium. However, in the case γ1γ2 < 0, the role of the 4 is proportional to Ω due to the presence of the fourth- γ2 nonlinearity can be essential for the physical fea- order dispersion term in (1). Thus, we expect a more tures and the stability of the optical soliton propaga- complex gain spectrum in comparison with the higher- tion [19]. Finally, we assume the typical values for β j 2 3 order nonlinear Schr¨odinger equation only containing as β2 ≈±(20 − 60) ps /km, β3 ≈ (0.1 − 1) ps /km, −3 −1 4 the second and third dispersion terms. Under the condi- and β4 ≈±(10 − 10 ) ps /km. In the following tion that the nonlinear response of the medium is non- analysis, we omit the units of the coefﬁcients. resonant and instantaneous so that we can ignore other Since it is difﬁcult to analytically investigate the phenomena appearing like self-steepening and the self- characteristics of the MI gain function (10), we cal- frequency shifting through the Raman effect, i.e, ignor- culate K in (9) numerically for various model coef- ing the second and third terms in (1), the author [14] ﬁcients. Figure 1 shows the MI gain as functions of 228 W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation

Fig. 1. Gain spectra as functions of β4 and Ω for different power, dispersion, and nonlinear coefﬁcients. (a) Four gain peaks exist for the normal dispersion regime when both β2β4 < 0 and γ1γ2 > 0 are satisﬁed. For β2β4 > 0, only two gain peaks remain. Note that the peak to peak separation increases for β4 → 0. A small fourth-order dispersion term results in two additional gain peaks. (b) Only two gain peaks appear if β2β4 < 0, γ2 > 0 and γ4 > 0 are satisﬁed. (c) Similarly four peaks are shown for the anomalous dispersion regime when β2β4 < 0 and γ1γ2 < 0 are satisﬁed. (d) Similar to (c) but having stronger dispersion coefﬁcient and power.

β4 and Ω with ﬁxed β2 and P. The gain is sensitive Fig.1b when β2β4 > 0, while non-zero gain exists re- to the sign and strength of the dispersion and nonlin- gardless of the signs of β2 and β4 as shown in Fig- ear coefﬁcients. Figures 1a and 1b show the spectra ure 1a. Figures 1c and 1d plot the gain spectra for for the cases of the normal dispersion regime β 2 = 20 the anomalous dispersion regime for β2 = −20 and and β2 = 40, respectively. Four gain peaks are shown β4 = −40, respectively. Four similar gain peaks as in in Figure 1a as β4 → 0 when γ1 < 0 and γ2 < 0. Inter- Fig. 1a show up if β2 → 0 in case γ1γ2 < 0 is satis- estingly, the fact that the gain peaks move away from ﬁed. On the other hand, a ﬂat-top gain peak appears in the center and the peaks at lower frequencies disappear Fig. 1d as β4 → 0.4. Note that the gain slope disappears resembles the XPM induced gain spectrum in the nor- in comparison with Figs. 1a and 1b when γ1γ2 < 0is mal dispersion regime (see [6] for details), occuring satisﬁed. when two beams with orthogonal polarizations propa- Figure 2 displays the MI gain as functions of β2 gate simultaneously [6, 8]. On the other hand, only two and Ω with ﬁxed β4 and P. Two gain peaks appear gain peaks appear for γ1 > 0 and γ2 > 0 as shown in (Fig. 2a) if β2β4 > 0 is satisﬁed, which separate as Figure 1b. We note that the gain in Figs. 1a and 1b in- β2 → 0 and disappear in the β2β4 < 0 regime regard- creases linearly as the modulation frequency increases, less of the signs of γ1 and γ2. In Fig. 2b, we ﬁnd the i.e, the gain plane has a slope, when γ1γ2 > 0 is sat- appearance of the two gain peaks of the dispersion isﬁed. We also ﬁnd that the gain peak disappears in regime satisfying β2β4 < 0 for γ1 < 0 and γ2 < 0, and W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation 229

Fig. 2. Gain spectra as functions of β2 and Ω for different power, dispersion, and nonlinear coefﬁcients. (a) The gain peaks separate as β2 →−0.5 in the case of both negative nonlinear coefﬁcients. (b) Four gain peaks appear for −1.5 ≤ β2 ≤−0.5 if both nonlinear coefﬁcients are positive. Note that for β2β4 > 0 the gain vanishes. (c) Similar to (b) except γ1γ2 < 0 and β4 < 0. Gain peak separation disappears as β2 →−3. (d) Similar to (c) except γ1γ2 < 0 and β4 < 0. Gain peak separation disappears as β2 → 0. four gain peaks at −2 ≤ β2 < 0. The gain spectrum dispersion for the MI in the case of normal dispersion in Fig. 2c is similar to Fig. 2b except that non-zero has also been investigated in [12]. The purpose of the gain exists if β2β4 > 0 and γ1γ2 < 0. Finally, four gain present work is to investigate the dynamics of the soli- peaks appear in Fig. 2d as β2 → 2 while β2β4 < 0is tons induced by the MI in the regimes of the anoma- satisﬁed. lous dispersion (β2 < 0) with β4 > 0 and the normal dispersion (β2 > 0) with β4 < 0. 3. Numerical Simulations of the Modulational We consider an incident ﬁeld at the launch plane Instability z = 0 into the nonlinear medium of the form In order to understand the dynamics of a CW beam A(0,τ)= P0[1 + εm cos(Ωmτ)], (13) propagation under the MI, (1) is solved utilizing the split-step Fourier method under periodic boundary where εm is the normalized modulation amplitude and condition [20]. Recently, Pitois and Milot [11] have Ωm is the angular frequency of a weak sinusoidal mod- experimentally investigated the inﬂuence of the fourth- ulation imposed on the CW beam, which can be deter- order dispersion on the onset of scalar spontaneous MI mined from the above gain spectrum when γ j and β j in a single-mode ﬁber and demonstrated the evidence are ﬁxed. In the following simulations, we set the ﬁber of a new MI spectral window due to a β4 effect in the loss term α = 0 since it does not contribute to the MI normal dispersion regime. The role of the fourth-order gain. 230 W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation

Fig. 3. Development of the MI in the anomalous√ dispersion regime. (a) Evolution of the relative power, i.e., P(z,t)/P0, for the given perturbed initial proﬁle A(0,τ)= P0[1+εmcos(Ωmτ)], where the modulation frequency Ωm = 1 THz, εm = 0.01, −3 β2 = −20,β3 = 0.04, β4 = 0.005, γ1 = −20, γ2 = 4 × 10 , and TR = 0.03. Note the periodic occurrences of the solitons induced by the MI. (b) The contour plot shows a symmetry about τ = 0 as the consequence of the periodic boundary condition. (c) Snapshots of the relative power due to the MI. The initial perturbed proﬁle develops to the soliton with the sidebands at z = 1 km. (d) Q(z) and R(z) represent the changes of the energy and mass, respectively, along the evolution distance z. The peaks (solid curve) or dips (dot-dashed curve) are the distances where the solitons are induced by the MI.

3.1. Anomalous Dispersion (β2 < 0) with β4 > 0 states. The reason for the existence of the soliton in this case is a complicated interplay between the non- The results of the numerical simulations of the MI linear and higher-order dispersion terms. As shown in in the anomalous dispersion regime are presented in the contour plot of Fig. 3b, the soliton evolution shows τ = Figs. 3 and 4. As an example, by choosing β 2 = a perfect symmetry about 0 as a consequence of −3 −20,β3 = 0.04,β4 = 0.005,γ1 = −20, γ2 = 4 × 10 , the imposed periodic boundary condition. The snap- and TR = 0.03 fs, which belong to the coefﬁcients of shots of the relative peak in Fig. 3c show the initial the MI spectrum in Fig. 1c, with Ωm = 1 THz and perturbed CW developing into a soliton with sidebands εm = 0.01, we plot in Fig. 3a the evolution of the rel- at z = 1 km, which is similar to the result investigated 2 ative peak power P(z,τ)/P0, where P(z,τ) ≡|A(z,τ)| in the context of the supercontinuum generation by the 2 and P0 ≡|A(z,0)| . The typical periodic occurrence of MI [12]. To better understand the dynamical behav- stable soliton-like pulses (hereafter we denote them as ior of the soliton along the evolution distance z,we solitons) induced by the MI along the evolution dis- calculate the total energy and mass (or the area un- ∞ 2 tance is observed, even in the presence of the higher- der |A(z,τ)|) deﬁned as E(z) ≡ −∞ |A(z,τ)| dτ and ∞ order dispersion terms, in contrast to the numerical re- M(z) ≡ −∞ |A(z,τ)|dτ, respectively, and plot Q(z) ≡ sults of [13] where non-zero third- and fourth-order E(z)/E(0) and R(z) ≡M(z)/M(0) in Figure 3d. An dispersion terms lead the initial waves to turbulent interesting feature is that the peaks in Q(z) (solid W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation 231

Fig. 4. Evolutions of the solitons induced by the MI for the same coefﬁcients as in Fig. 3 but varying γ1 and γ2 or the intrapulse Raman scattering term TR. (a) – (c) As both strength of γ1 and γ2 increase, more solitons interacting with each other along the evolution distance are shown. Note γ1 < 0 and γ2 > 0. (d) In comparison with Fig. 3a, the stronger intrapulse Raman scattering term does not alter the dynamics of the solitons. curve) or the dips in R(z) (dot-dashed curve) represent to the fact that the TR term does not affect the super- the regions at which the soliton is induced by the MI continuum generation [12]. and they occur at the same evolution distance. In par- Q(z) ≈ R(z) ≈ . z . ticular, we ﬁnd 1 0at 0 22 km and 3.2. Normal Dispersion (β2 > 0) with β4 < 0 z 1.0 km, where the system returns to the initial per- turbed state. The simulated evolutions of the MI in the normal We now investigate the evolution of the MI with the dispersion regime are displayed in Figs. 5 and 6. As an same coefﬁcients as in Fig. 3 but varying γ1 and γ2 or example, we set β2 = 30, β3 = 0.3, β4 = −0.002, γ1 = −3 TR. Figures 4a and 4c show the effects of the nonlin- 20, γ2 = 4 × 10 , and TR = 0.03, which belong to the ear terms on the behavior of the injected CW. As both coefﬁcients of the MI spectrum type in Fig. 1b, with strength of γ1 and γ2 increase, we observe the appear- Ωm = 1 THz and εm = 0.01. In contrast to the results ance of more MI-induced solitons which subsequently in Figs. 3a and b, due to the stronger β3 term, the soli- undergo mutual interaction with each other along the ton in Fig. 5a travels towards the right, which can be evolution distance, as shown in Figures 4a and 4c. As more clearly demonstrated in the asymmetric contour the strength of the nonlinear terms further increases, plot in Figure 5b. This shows that even though the β 3 i.e, |γ1|≥120 and |γ2|≥0.0001, the initial CW even- term does not contribute to the MI gain, it can alter the tually turns into a chaotic state, which has been numer- propagation direction of the soliton in the presence of ically conﬁrmed. However, in comparison with Fig. 3a, both the higher-order nonlinear and dispersion terms. the stronger intrapulse Raman scattering term does not The fact that the solitons attract each other and col- alter the dynamics of the solitons, which can be related lide periodically along the evolution distance, which is 232 W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation

Fig. 5. Development of the MI in the normal dispersion regime. (a) Evolution of the relative power for β2 = 30, β3 = 0.3, β4 = −3 −0.002, γ1 = 20, γ2 = 4×10 , and TR = 0.03 with Ωm = 1 and εm = 0.01. Due to the strong β3 term, the solitons in Fig. 5a travel towards the right in comparison with the result in Fig. 3a. The collision of the solitons along the evolution distance is shown. (b) The contour plot shows asymmetry about τ = 0. (c) Snapshots of the solitons for various evolution distances. The snapshot at z = 1 km shows two interacting solitons (in the middle). (d) The peaks in Q(z) (solid curve), which are anticorrelated with the dips in R(z) (dot-dashed curve), represent the distance at which the soliton is induced by the MI. similar to the collision of two solitons in-phase case Figs. 4a and b, due to the presence of the strong non- in the context of the standard nonlinear Schr¨odinger linear terms. On the other hand, for the case of fur- equation [6], is demonstrated. As the consequence of a ther increased γ1 but γ2 values as in Fig. 4c under the stronger β3 term than that in Fig. 3, we ﬁnd the snap- constraint that γ1 > 0 and γ2 < 0, the dynamical com- shot of the two interacting solitons (in the middle) at plexity of the solitons increases by showing a chaotic z = 1 km as shown in Fig. 5c, instead of the formation behavior. However, for both negative nonlinear coefﬁ- of the sidebands. Figure 5d shows the peaks in Q(z) cients in Fig. 4d, the initial CW decays to a constant (solid curve), which are anticorrelated with the dips in background. R(z) (dot-dashed curve), representing the distances at which the solitons are produced by the MI. However, 4. Conclusions the energy of the system is more rapidly decreasing In this paper we have derived an analytic expres- . β above z 0 65 km due to the stronger 2 term in com- sion of the MI gain for the generalized nonlinear parison with that of Figure 3d. Schr¨odinger equation with a third- and fourth-order The simulations of the initial CW with the same co- dispersion and cubic-quintic nonlinear terms in (1), de- efﬁcients as in Fig. 3 but with varying γ1 and γ2 are scribing the propagation of intense femtosecond opti- presented in Figure 6. As both γ1 > 0 and γ2 > 0 in- cal pulses in a medium which exhibits a parabolic non- crease, we observe some highly peaked solitons which linearity law [6, 9, 10 – 12]. We have shown in Figs. 1 undergo more active mutual interaction as found in and 2 that the MI gain can exist even in the normal W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation 233

Fig. 6. Evolutions of the solitons induced by the MI for the same coefﬁcients as in Fig. 3 but varying γ1 and γ2. (a), (b) As both γ1 > 0 and γ2 > 0 increase, higher peaked solitons with active mutual interactions appear. (c) For further increased values with γ1 > 0 and γ2 < 0 constraint, the dynamical complexity of the solitons increases by showing a chaotic behavior. (d) For both negative nonlinear coefﬁcients in Fig. 4d, the initial CW is shown to decay to a constant background. dispersion medium, depending on the strength of the sults of [13] in which the non-zero third- and fourth- higher-order dispersion and nonlinear terms, in con- order dispersion terms lead the initial waves to turbu- trast to the NLS equation in which the MI gain occurs lent states, due to a complicated interplay between the only in the anomalous dispersion regime. In particu- nonlinear and the higher-order dispersion terms. An in- lar, the gain spectra in Fig. 2c in the anomalous dis- teresting feature in Fig. 3d is that the peaks in Q(z) persion regime for the nonlinear coefﬁcients (γ 1 > 0 (solid curve) or the dips in R(z) (dot-dashed curve) and γ2 < 0) may be experimentally testable since the represent the distances at which the solitons are pro- nonlinear medium has been recently realized in or- duced by the MI. It was found that as the strength of the ganic materials such as polydiacetylene paratoluene nonlinear terms increases, more MI-induced solitons sulfonate (PTS) [21]. appear which undergo mutual interactions with each Numerical simulations have been performed to in- other along the evolution distance as demonstrated in vestigate the effects of the higher-order nonlinear and Figs. 4a – c, while the intrapulse Raman scattering term dispersion terms on the evolution of the steady-state does not alter the dynamics of the soliton as shown in initial CW under the modulationally perturbed ﬁeld. In Figure 4d. Figs. 3 and 4, we have simulated the evolutions of the In Figs. 5 and 6, we have shown the evolutions of the MI in the anomalous dispersion regime (β2 < 0) with MI in the normal dispersion regime (β2 > 0) with β4 < β4 > 0. The typical periodic occurrences of the soli- 0. For a strong β3 term, the solitons in Fig. 5a were tons induced by the MI along the evolution distance shown to travel towards the right, even though it does were found in Fig. 3 even in the presence of the higher- not contribute to the MI gain. We have found that as order dispersion terms, in contrast to the numerical re- both γ1 > 0 and γ2 > 0 increase highly peaked solitons 234 W.-P. Hong · Higher-Order Nonlinear Schr¨odinger Equation with more active mutual interaction appear as shown Acknowledgements in Figs. 4a and 4b, due to the presence of the strong nonlinear terms. However, for both negative nonlinear This work was supported by Catholic University of coefﬁcients in Fig. 4d, the initial CW was shown to Daegu in 2006. decay to a constant background.

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