Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

doi:10.1016/j.optlastec.2013.05.031

Optics & Laser Technology

Volume 54, 30 December 2013, Pages 265-273

https://doi.org/10.1016/j.optlastec.2013.05.031 Get rights and content

Highlights

•
Bright and dark soliton solutions for the HNLSE are obtained.
•
The parametric conditions for the formation of soliton pulses are determined.
•
Conserved quantities have been calculated for Hirota and Sasa-Satsuma cases.
•
Periodic solutions obtained by reducing HNLSE into quartic anharmonic oscillator equation and Liénard equation.

Abstract

In this paper, we have obtained the optical solitary wave solutions for the nonlinear Schrödinger equation which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method was used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses were determined. Using the 1-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa-Satsuma cases. We have also constructed some periodic wave solutions of the higher order nonlinear Schrödinger equation by reducing it to quartic anharmonic oscillator equation and by using projective Ricatti equations. Moreover by using He's semi-inverse method, variational formulation was established to obtain exact soliton solutions. The 1-soliton solutions of time dependent form of this equation was also obtained. To visualize the propagation characteristics of dark-bright soliton solutions, few numerical simulations are given.

Introduction

Optical solitons have been the subjects of extensive theoretical and experimental studies in recent years because of promising potential to become principal information carriers in telecommunication due to their capability of propagating long distance without attenuation and changing their shapes. These special types of optical wave packets appearing as a result of interplay between dispersion and nonlinearity are good information carriers for high-bit-rate optical transmission systems [1]. The waveguides used in the picosecond optical pulse propagation in nonlinear optical communication systems are usually of Kerr type and consequently the dynamics of light pulses are described by nonlinear Schrödinger family of equations with cubic nonlinear terms. The propagation of picosecond optical solitons in a monomode optical fiber is well described by the celebrated nonlinear Schrödinger equation (NLSE) ${iE}_{x} + a_{1} E_{tt} + a_{2} | E |^{2} E = 0,$ where $E (x, t)$ is a complex envelope of electrical field in a comoving frame, t is the retarded time, x represents the distance along the direction of propagation, a₁ is the group velocity dispersion (GVD) parameter and a₂ specifies the strength of Kerr nonlinearity [2]. Depending on the sign of GVD, the NLSE have two distinct localized solutions, bright and dark soliton solutions, which are, respectively, existent in the anomalous and normal dispersion regimes [3]. Optical solitons have been regarded as the next generation technology for high-capacity optical communications, mainly because of its promise to transmit signals over long distances while resisting chromatic dispersion [4] and high frequency of optical carrier make possible high bit rate transmission and to increase the bit rate further it is desirable to use shorter femtosecond pulses. Generalization of the NLSE is necessitated to take into account higher order dispersion, self-steepening of the pulse due to the dependence of the slowly varying part of the nonlinear polarization on time and the delayed effect of Raman response for describing optical pulse propagation in the femtosecond domain [5]. For HNLS equations, to our knowledge, only a few kinds of these equations satisfy certain proportion relations between model coefficients that are completely integrable by inverse scattering-like methods. A thorough discussion on these situations has been given in [6].

In recent years, many authors have analyzed the HNLS equation from different points of view and some interesting results have also been obtained [7], [8], [9], [10], [11], [12], [13]. In this work, we analytically derive both bright and dark solitary wave solutions of HNLS equation under some parametric conditions. To date, most applications are designed for bright solitons because they are relatively easy to generate in low-loss and low dispersion optical fibers, however, it is well known that dark solitons have some advantages over their bright counterparts. Compared with the bright solitons, they have better stability against various perturbations such as fiber loss, mutual interaction between neighboring pulses, the Raman effect, and the superposition of noise emitted from optical amplifiers [14], [15], [16].

The paper is organized as follows. In Section 2, we obtain the exact bright and dark optical soliton solutions of HNLS equation under some physical constraints on the model coefficients. In Section 3, we compute some integrals of motion of HNLS equation for Hirota and Sasa-Satsuma cases. In Section 4, we find some periodic wave solutions of HNLS equation. Next, in Section 5, we obtain the topological 1-soliton solution of time-dependent HNLS equation under the parametric restrictions. The conclusions are given in Section 6.

Section snippets

Bright and dark soliton solutions of HNLS equation

The HNLS equation describes the propagation of femtosecond optical pulses in optical fibers and can be written as [2], [17] ${iE}_{x} + a_{1} E_{tt} + a_{2} | E |^{2} E + i [a_{3} E_{ttt} + a_{4} {(| E |^{2} E)}_{t} + a_{5} E {(| E |^{2})}_{t}] = 0 .$ Here E represents the complex envelope of the electric field, x is the normalized distance along the fiber, t is the normalized time with the frame of the reference moving along the fiber at the group velocity. The subscripts x and t denote the spatial and temporal partial derivatives respectively and the coefficients $a_{i} (i$ $=$

Integrals of motion of HNLS system

An intrinsic property of HNLS equation is that in the absence of higher-order terms ( $a_{3} = a_{4} =$ $a_{5} = 0$ ), it possesses an infinite number of conserved quantities also known as integrals of motion. But in the presence of higher order effects, NLSE does not hold conservation laws except for energy conservation, unless the higher order-terms are of unique type. Here, we computed, the first three conserved quantities of the HNLS equation for Hirota and Sasa-Satsuma cases [26].

(i) Hirota case [ $a_{3} : a_{4} :$ $(a_{4} + a_{5}) =$

Periodic solutions of HNLS equation

To seek the periodic traveling wave solution of Eq. (2), we make use of gauge transformation $E (x, t) = A (ξ) \exp [i (kx - ω t)],$ where $ξ = β t -$ $λ x + ξ_{0}$ and $β$ , k, $ω$ , $λ$ are constants to be determined later. Substituting Eq. (30) into Eq. (3) yields a set of coupled equations. $β^{2} (a_{1} - 3 a_{3} ω) A_{ξ ξ} + (a_{3} ω^{3} - a_{1} ω^{2} - k) A + (a_{2} - a_{4} ω) A^{3} = 0,$ $β^{3} a_{3} A_{ξ ξ ξ} + (2 β a_{1} ω - 3 β a_{3} ω^{2} + λ) A_{ξ} + β (3 a_{4} + 2 a_{5}) A^{2} A_{ξ} = 0 .$ Note that Eq. (32) has only first and third order derivatives. It is possible to integrate Eq. (32) and resulting integrated equation can be written as $A_{ξ ξ} + 2$

1-Soliton solutions of HNLS equation with time-dependent coefficients

The study of optical solitons in a Kerr law media is an important area of study. It is governed by the NLSE. It studies the propagation of solitons through optical fibers for trans-continental and trans-oceanic distances. Thus the dynamics of solitons governed by the NLSE is well understood and well known in this context. When inhomogeneities of the media and nonuniformity of the boundaries are taken into account in various real physical situations, the variable-coefficient NLSE provides more

Conclusion

In the present work, we have found exact bright, dark and cnoidal wave soliton solutions to the HNLS equation describing propagation of femtosecond light pulses in an optical fiber under certain parametric conditions. Using soliton ansatz, both dark and bright solitons are constructed. The obtained solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher order nonlinear Schrödinger model equation. Note

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Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

Highlights

Abstract

Introduction

Section snippets

Bright and dark soliton solutions of HNLS equation

Integrals of motion of HNLS system

Periodic solutions of HNLS equation

1-Soliton solutions of HNLS equation with time-dependent coefficients

Conclusion

Physics Reports

Optics & Laser Technology

Optics & Laser Technology

Physics Letters A

Physica D

Chaos, Solitons & Fractals

Communications in Nonlinear Science and Numerical Simulation

Communications in Nonlinear Science and Numerical Simulation

Communications in Nonlinear Science and Numerical Simulation

Optics & Laser Technology

Solitons in optical communications

Nonlinear fiber optics

Soviet Physics JETP

IEEE Journal of Quantum Electronics

Physical Review E

Physical Review E

Physical Review E

Physical Review Letters