Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model

Khalil Salim Al-Ghafri

doi:10.1515/phys-2021-0074

Article Open Access

Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model

Khalil Salim Al-Ghafri

Published/Copyright: November 15, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 19 Issue 1

Abstract

In the present work, we investigate soliton structures in optical fiber communications. The medium is described by the Kundu–Mukherjee–Naskar model. With the aid of the ansatz approach, the exact solutions are constructed. Consequently, distinct wave structures including W-shaped, bright and dark solitons are derived. These new soliton solutions are retrieved under certain parametric conditions. Besides, it is found that the bright soliton has two different types in a particular limit. Optical solitons are displayed graphically to shed light on their behaviors.

Keywords: optical solitons; Kundu–Mukherjee–Naskar model; ansatz approach; W-shaped; bright and dark solitons

1 Introduction

Information transmission in optical communication channels is based on soliton propagation [1,2,3]. Thus, the dynamic of solitons in the field of fiber optics has been under remarkable attention for over a decade [4,5]. One of the important models that describe the soliton in optics and optical fibers is the Kundu–Mukherjee–Naskar (KMN) equation. This model is considered as an extension to the nonlinear Schrödinger equation containing mixed types of nonlinear effect in reference to Kerr and non-Kerr law nonlinearities.

The model of KMN equation discussed in the present work is given by

(1) i Ψ t + a Ψ x y + i b Ψ ( Ψ Ψ x ∗ − Ψ ∗ Ψ x ) = 0 ,

where the dependent variable Ψ ( x , y , t ) stands for the optical soliton profile. In equation (1), the first term indicates the temporal evolution of the pulse, while the second term with the coefficient of a stands for the dispersion term. The nonlinear effect given by the coefficient of b represents current-like nonlinearity arising from chirality.

Equation (1) was introduced by the Kundu and Mukherjee in 2013 [6]. It is basically originated as a two-dimensional nonlinear Schrödinger equation which is derived from the basic hydrodynamic equations. This model can be used to describe wave propagation in optical fiber and oceanic rogue waves as well as ion-acoustic wave in a magnetized plasma [7,8, 9,10]. Unlike most two-dimensional models, it is classified as an integrable equation which can be solved by the inverse scattering transform. Moreover, many studies are devoted to equation (1) to examine soliton propagation into an optical fiber. Therefore, several mathematical tools are implemented to extract exact analytic solutions [11,12,13, 14,15].

The present study is devoted to investigating new forms of optical solitons with nontrivial phase component possessing a form of nonlinear function, in contrast to previous studies in the literature which scrutinized KMN equation. The traveling wave reduction of the model is derived and then it is dealt with soliton ansatz having new functional form in terms of the hyperbolic secant and tangent functions.

2 Mathematical analysis of the model

Now, we intend to obtain the traveling wave reduction of equation (1) so as to derive the optical soliton solutions. Thus, we introduce the traveling wave transformation of the form

(2) Ψ ( x , y , t ) = ψ ( ξ ) e i ϕ ( x , y , t ) ,

where ψ ( ξ ) accounts for the amplitude of the soliton while ϕ ( x , y , t ) denotes the phase component. The wave variable ξ is given by

(3) ξ = κ x + λ y + ν t ,

and the phase component is presented as

(4) ϕ ( x , y , t ) = α x + β y + ω t + θ ( ξ ) ,

where κ and λ are the inverse width of the soliton along x - and y -directions, respectively, whereas ν is the velocity of the soliton. The parameters α and β indicate the soliton frequencies in the x - and y -directions, respectively, and ω represents the soliton wave number. The function θ ( ξ ) is defined as a nonlinear phase shift.

Applying the transformation (2) to equation (1) and separating the real and imaginary parts lead to a pair of equations having the form

(5) ( ν + a β κ + a α λ ) ψ ′ + 2 a κ λ ψ ′ θ ′ + a κ λ ψ θ ″ = 0 ,

(6) a κ λ ψ ″ − ( ω + a α β ) ψ + 2 b α ψ 3 − ( ν + a β κ + a α λ ) ψ θ ′ − a κ λ ψ θ ′ 2 + 2 b κ ψ 3 θ ′ = 0 ,

where the prime denotes the derivative with respect to ξ . Once we multiply equation (5) by ψ and integrate it, we arrive at the first integral given by

(7) θ ′ = C ψ − 2 a κ λ − ( ν + a β κ + a α λ ) 2 a κ λ ,

where C is the constant of integration. Substituting equation (7) into equation (6) gives rise to

(8) ψ ″ + A 1 ψ + A 2 ψ 3 − A 3 ψ − 3 = 0 .

Multiplying equation (8) by ψ ′ and integrating with respect to ξ , we find

(9) ψ ′ 2 + A 1 ψ 2 + A 2 2 ψ 4 + A 3 ψ − 2 + 2 A 0 = 0 ,

where A 0 is an arbitrary constant of integration and the rest of constants are defined by

(10) A 1 = 8 b κ C − 4 a κ λ ( ω + a α β ) + ( ν + a β κ + a α λ ) 2 4 a 2 κ 2 λ 2 ,

(11) A 2 = 2 a b α κ λ − b κ ( ν + a β κ + a α λ ) a 2 κ 2 λ 2 ,

(12) A 3 = C 2 a 2 κ 2 λ 2 .

To avoid the complexity, we make use of the variable transformation given by

(13) ψ 2 = V .

Hence, equation (9) reduces to

(14) V ′ 2 + 4 A 1 V 2 + 2 A 2 V 3 + 4 A 3 + 8 A 0 V = 0 .

Differentiating equation (14), for convenience, yields

(15) V ″ + 4 A 0 + 4 A 1 V + 3 A 2 V 2 = 0 .

The solution to equation (15) along with the relations (2) and (13) constructs the general form of exact solutions for equation (1) addressed as

(16) Ψ ( x , y , t ) = V 1 / 2 e i [ α x + β y + ω t + θ ( ξ ) ] ,

where the phase variable θ ( ξ ) can be turned up through integrating equation (7) with respect to ξ as

(17) θ ( ξ ) = C a κ λ ∫ d ξ V − ( ν + a β κ + a α λ ) 2 a κ λ ( κ x + λ y + ν t ) + θ 0 ,

where θ 0 is a constant phase.

3 Optical soliton solutions

In this section, soliton ansatz is applied to extract the exact solutions of equation (1). Various solutions describing W-shaped, bright and dark solitons are retrieved.

We assume that equation (15) has an exact optical soliton solution in the form

(18) V ( ξ ) = l + m sech 2 ( p ξ ) 4 − [ 1 − tanh ( p ξ ) ] 2 + n sech 4 ( p ξ ) ( 4 − [ 1 − tanh ( p ξ ) ] 2 ) 2 ,

where l , m , n and p are constants to be determined. As one can see that the expression (18) has a constant term and two terms in the hyperbolic secant and tangent functions with degree two and four. This combination of terms may enable us to generate different forms of soliton solutions.

Substituting equation (18) into equation (15) and equating all coefficients having the same order of sech ( p ξ ) tanh ( p ξ ) to zero give us a set of algebraic equations that induce different cases of solutions presented as follows.

Case I. W-shaped soliton

l = − 4 A 1 3 A 2 , m = 16 A 1 A 2 , n = − 32 A 1 A 2 , p = A 1 ,

with neglecting integration constant in equation (15), i.e., A 0 = 0 . Making use of these findings leads to the formation of optical soliton solution to equation (1) given as

(19) Ψ ( x , y , t ) = − 4 A 1 3 A 2 1 − 12 sech 2 ( A 1 ξ ) 4 − [ 1 − tanh ( A 1 ξ ) ] 2 + 24 sech 4 ( A 1 ξ ) ( 4 − [ 1 − tanh ( A 1 ξ ) ] 2 ) 2 1 2 e i ϕ ( x , y , t ) ,

provided that A 1 > 0 and A 2 < 0 . It is clear from Figure 1(a) that the plot of solution (19) takes the shape of W.

$Figure 1 W-shaped, bright (type I), bright (type II) and dark soliton solutions given by (19)–(22) with a = b = β = ν = ω = κ = 0.5 a=b=\beta =\nu =\omega =\kappa =0.5 , α = − 0.5 \alpha =-0.5 , λ = C = t = 1 \lambda =C=t=1 ; for (b) and (c) b = − 0.5 b=-0.5 .$

Figure 1

W-shaped, bright (type I), bright (type II) and dark soliton solutions given by (19)–(22) with a = b = β = ν = ω = κ = 0.5 , α = − 0.5 , λ = C = t = 1 ; for (b) and (c) b = − 0.5 .

Case II. Bright soliton (type I)

l = 0 , m = − 16 A 1 A 2 , n = 32 A 1 A 2 , p = − A 1 .

with the constraint A 0 = 0 . These results give rise to exact soliton solution to equation (1) of the form

(20) Ψ ( x , y , t ) = − 16 A 1 A 2 sech 2 ( − A 1 ξ ) 4 − [ 1 − tanh ( − A 1 ξ ) ] 2 − 2 sech 4 ( − A 1 ξ ) ( 4 − [ 1 − tanh ( − A 1 ξ ) ] 2 ) 2 1 2 × e i ϕ ( x , y , t ) ,

which demands A 1 < 0 and A 2 > 0 . The behavior of solution (20) describes a bright soliton pulse as shown in Figure 1(b).

Case III. Bright soliton (type II)

l = − 6 A 1 11 A 2 , m = − 32 A 1 11 A 2 , n = 64 A 1 11 A 2 , p = − 2 A 1 11 ,

under the constraint condition 121 A 0 A 2 = 39 A 1 2 . By virtue of the obtained outcomes, an exact optical soliton solution to equation (1) is acquired as

(21) Ψ ( x , y , t ) = − 2 A 1 11 A 2 3 + 16 sech 2 − 2 A 1 11 ξ 4 − 1 − tanh − 2 A 1 11 ξ 2 − 32 sech 4 − 2 A 1 11 ξ 4 − 1 − tanh − 2 A 1 11 ξ 2 2 1 2 × e i ϕ ( x , y , t ) ,

provided that A 1 < 0 and A 2 > 0 . As one can see, Figure 1(c) illustrates the evolution of solution (21) which characterizes a bright soliton wave.

Case IV. Dark soliton

l = − 26 A 1 33 A 2 , m = 32 A 1 11 A 2 , n = − 64 A 1 11 A 2 , p = 2 A 1 11 ,

under the restriction 121 A 0 A 2 = 39 A 1 2 . From these results, one can secure exact soliton solution to equation (1) of the form

(22) Ψ ( x , y , t ) = − 2 A 1 33 A 2 13 − 48 sech 2 2 A 1 11 ξ 4 − 1 − tanh 2 A 1 11 ξ 2 + 96 sech 4 2 A 1 11 ξ 4 − 1 − tanh 2 A 1 11 ξ 2 2 1 2 × e i ϕ ( x , y , t ) ,

provided that A 1 > 0 and A 2 < 0 . The behavior of this soliton solution is depicted in Figure 1(d) and it obviously presents the pulse profile for dark soliton.

4 Discussion and conclusion

The current study has dealt with the soliton solutions of KMN equation. Based on soliton ansatz method, new types of soliton structures are revealed with phase component having a form of nonlinear function which is different from the one addressed in the previous studies. Among them, the W-shaped and bright (type I) optical solitons are derived due to the absence of integration constant in equation (15). Additionally, bright (type II) and dark solitons are extracted under specific relation between the physical parameters. The validity conditions for the existence of all optical solitons are given. The structures of obtained solitons are clearly illustrated by selecting suitable values of the model parameters. The results obtained for the KMN model are new and can be benefited in the field of optical fiber.

As the KMN model is in its infancy, it can be studied further in future to examine its applications in various physical areas such as optical couplers, meta-optics and magneto-optic waveguides. Hence, there are still many powerful analytical and numerical techniques to be implemented so as to generate new forms of solutions.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Doran N, Blow K. Solitons in optical communications. IEEE J Quantum Elect. 1983;19(12):1883–8. 10.1109/JQE.1983.1071806Search in Google Scholar

[2] Haus HA, Wong WS. Solitons in optical communications. Rev Mod Phys. 1996;68(2):423. 10.1103/RevModPhys.68.423Search in Google Scholar

[3] Bigo S, Leclerc O, Desurvire E. All-optical fiber signal processing and regeneration for soliton communications. IEEE J Sel Top Quant. 1997;3(5):1208–23. 10.1109/2944.658596Search in Google Scholar

[4] Al-Ghafri KS. Soliton behaviours for the conformable space-time fractional complex Ginzburg-Landau equation in optical fibers. Symmetry. 2020;12(2):219. 10.3390/sym12020219Search in Google Scholar

[5] Al-Ghafri K, Krishnan E, Biswas A. W-shaped and other solitons in optical nanofibers. Results Phys. 2021;23:103973. 10.1016/j.rinp.2021.103973Search in Google Scholar

[6] Kundu A, Mukherjee A. Novel integrable higher-dimensional nonlinear Schroedinger equation: properties, solutions, applications. arXiv:1305.4023v1 [nlinSI]. 2013Search in Google Scholar

[7] Kundu A, Mukherjee A, Naskar T. Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents. Proc Roy Soc A-Math Phy. 2014;470(2164):20130576. 10.1098/rspa.2013.0576Search in Google Scholar PubMed PubMed Central

[8] Wen X. Higher-order rational solutions for the (2+1)-dimensional KMN equation. Proc Romanian Acad A. 2017;18(3):191–8. Search in Google Scholar

[9] Mukherjee A, Kundu A. Novel nonlinear wave equation: Regulated rogue waves and accelerated soliton solutions. Phys Lett A. 2019;383(10):985–90. 10.1016/j.physleta.2018.12.023Search in Google Scholar

[10] Mukherjee A, Janaki M, Kundu A. A new (2+1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma. Phys Plasmas. 2015;22(7):072302. 10.1063/1.4923296Search in Google Scholar

[11] Ekici M, Sonmezoglu A, Biswas A, Belic MR. Optical solitons in (2+1)-Dimensions with Kundu-Mukherjee-Naskar equation by extended trial function scheme. Chinese J Phys. 2019;57:72–7. 10.1016/j.cjph.2018.12.011Search in Google Scholar

[12] Yıldırım Y. Optical solitons to Kundu-Mukherjee-Naskar model with trial equation approach. Optik. 2019;183:1061–5. 10.1016/j.ijleo.2019.02.117Search in Google Scholar

[13] Yıldırım Y. Optical solitons to Kundu-Mukherjee-Naskar model with modified simple equation approach. Optik. 2019;184:247–52. 10.1016/j.ijleo.2019.02.135Search in Google Scholar

[14] Kudryashov NA. General solution of traveling wave reduction for the Kundu–Mukherjee–Naskar model. Optik. 2019;186:22–7. 10.1016/j.ijleo.2019.04.072Search in Google Scholar

[15] Biswas A, Vega-Guzman J, Bansal A, Kara AH, Alzahrani AK, Zhou Q, et al. Optical dromions, domain walls and conservation laws with Kundu-Mukherjee-Naskar equation via traveling waves and Lie symmetry. Results Phys. 2020;16:102850. 10.1016/j.rinp.2019.102850Search in Google Scholar

Received: 2021-08-12

Revised: 2021-10-01

Accepted: 2021-10-08

Published Online: 2021-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2021-0074

Keywords for this article

optical solitons; Kundu–Mukherjee–Naskar model; ansatz approach; W-shaped; bright and dark solitons

Creative Commons

BY 4.0