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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 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].
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