Sequel to highly dispersive optical soliton perturbation with cubic-quintic-septic refractive index by semi-inverse variational principle

doi:10.1016/j.ijleo.2019.163451

Optik

Volume 203, February 2020, 163451

https://doi.org/10.1016/j.ijleo.2019.163451 Get rights and content

Abstract

The current work is a generalization of previously reported results on highly dispersive optical solitons with cubic-quintic-septic law of refractive index that was studied with perturbation terms having cubic nonlinearity. The present paper considers perturbation terms with full nonlinearity and the results collapse to former ones upon setting the full nonlinearity parameter to unity.

Introduction

One of the most interesting and essential features of optical soliton propagation is the existence and sustainment of the very necessary delicate balance between dispersion and nonlinearity. However, the engineering of optical fibers can lead to the dominance of dispersion terms, in which case it would lead to highly dispersive optical solitons [1], [2], [3], [4]. Thus, in addition to the usual group velocity dispersion (GVD), additional linear dispersive effects stemming from inter-modal dispersion (IMD), third-order dispersion (3OD), fourth-order dispersion (4OD), fifth-order dispersion (5OD) and sixth-order dispersion dominate. These lead to the concept of highly dispersive optical solitons. This paper will address such kind of optical solitons when the refractive index is of cubic-quintic-septic (CQS) type. The perturbation terms that are included are all of Hamiltonian type and appear with full nonlinearity. Thus, the results of the current paper collapse to the results of the previous work when the full nonlinearity parameter is set to unity [4]. The integration algorithm adopted in this paper is semi-inverse variational principle (SVP) that has been successfully applied to study problems from nonlinear optics [5], [6], [7], [8], [9], [10]. The details of the algorithm and the derivation of the results are jotted down in the rest of the paper.

The perturbed NLSE with dispersion terms of all orders and having CQS nonlinearity with perturbation terms is given by: $\begin{matrix} {iq}_{t} + {ia}_{1} q_{x} + a_{2} q_{xx} + {ia}_{3} q_{xxx} + a_{4} q_{xxxx} + {ia}_{5} q_{xxxxx} + a_{6} q_{xxxxxx} + (b_{1} {|q|}^{2} + b_{2} {|q|}^{4} + b_{3} {|q|}^{6}) q \\ = i [λ {({|q|}^{2 m} q)}_{x} + θ {({|q|}^{2 m})}_{x} q + μ {|q|}^{2 m} q_{x}] . \end{matrix}$ Thus in (1), q(x, t) is the complex-valued wave profile where x and t represent the spatial and temporal variables respectively. Then, the first term denotes linear temporal evolution with $i = \sqrt{- 1}$ . Next, a_j for 1 ≤ j ≤ 6 are the coefficients of IMD, GVD, 3OD, 4OD, 5OD and 6OD respectively. Finally b_j for 1 ≤ j ≤ 3 gives CQS law of refractive index. From the perturbation terms on the right hand side, we have λ is the coefficient of self-steepening term for short pulses, while θ and μ account for nonlinear dispersions. Furthermore, the full nonlinearity parameter is indicated by m. If m = 1, Eq. (1) reduces to the model studied earlier [4]. Eq. (1) will now be handled by SVP to retrieve highly dispersive bright 1-soliton solution.

Section snippets

Preliminaries

The starting hypothesis to handle Eq. (1) would be: $q (x, t) = g (s) e^{i ϕ (x, t)}$ where $s = x - vt$ with $v$ as the speed of soliton. From the phase part, $ϕ (x, t) = - κ x + ω t + Θ_{0} .$ Here, κ is the frequency, ω is the wave number and Θ₀ is the phase center. After putting (2) into (1) imaginary part equation yields: $\begin{matrix} [v - a_{1} + 2 a_{2} κ + 3 a_{3} κ^{2} - 4 a_{4} κ^{3} - 5 a_{5} κ^{4} + 6 a_{6} κ^{5} + \{(2 m + 1) λ + 2 m θ + μ\} g^{2}] g^{'} \\ - (a_{3} - 4 a_{4} κ - 10 a_{5} κ^{2} + 20 a_{6} κ^{3}) g^{'''} - (a_{5} - 6 a_{6} κ) g^{(v)} = 0 . \end{matrix}$ From Eq. (5), the conditions come out as: $a_{3} - 4 a_{4} κ - 10 a_{5} κ^{2} + 20 a_{6} κ^{3} = 0$ $a_{5} = 6 a_{6} κ$ and the soliton velocity is $v = a_{1} - 2 a_{2} κ - 3 a_{3} κ^{2} + 4 a_{4} κ^{3} + 5 a_{5}$

Application of SVP

The stationary integral falls out to be $J = \int_{- \infty}^{\infty} [3 b_{3} g^{8} + 4 b_{2} g^{6} + 6 b_{1} g^{4} + 12 P_{1} g^{2} + 12 P_{2} {(g^{'})}^{2} - 12 P_{3} {(g^{″})}^{2} + 12 a_{6} {(g^{″'})}^{2} - \frac{12 κ (λ + μ)}{m + 1} g^{2 m + 2}] dx$ once multiplying (10) by g′ and integrating. For CQS form of nonlinearity, the choice is [1], [2], [3], [4]: $g (s) = A sech (Bs) = A sech [B (x - vt)]$ where A is the soliton amplitude and B is its inverse width. The stationary integral therefore condenses to: $\begin{matrix} J & = \frac{A^{2}}{315 (2 m + 1) (m + 1) B Γ (m + \frac{1}{2})} [(2 m + 1) (m + 1) Γ (m + \frac{1}{2}) (36 b_{3} A^{6} + 56 b_{2} A^{4} \\ + 105 b_{1} A^{2} + 315 P_{1} + 105 P_{2} B^{2} - 147 P_{3} B^{4} + 465 a_{6} B^{6}) - κ (λ + μ) A^{2 m} Γ (\frac{1}{2}) Γ (m)] . \end{matrix}$ It is pointed out by SVP

Conclusions

Today's paper discusses perturbed highly dispersive optical solitons with CQS law of refractive index. The perturbation terms appear with full nonlinearity thus keeping the results on a generalized setting as compared to the ones reported earlier. The analytical results recovered here cannot be recovered by any other scheme, to the best of our knowledge. Thus, SVP is a very indispensable algorithm that aids in securing analytical solutions to perturbed problems when the solution to the

Conflict of interest

The authors also declare that there is no conflict of interest.

Acknowledgements

The research work of fourth author (QZ) was supported by the National Natural Science Foundation of China (Grant Nos. 11705130 and 1157149); this author was also sponsored by the Chutian Scholar Program of Hubei Government in China. The research work of the seventh author (MRB) was supported by the grant NPRP 11S-1126-170033 from QNRF and he is thankful for it.

References (10)

A. Biswas et al.
Highly dispersive optical solitons with cubic-quintic-septic law by exp-expansion
Optik
(2019)
A. Biswas et al.
Highly dispersive optical solitons with cubic-quintic-septic law by extended Jacobi's elliptic function expansion
Optik
(2019)
A. Biswas et al.
Highly dispersive optical solitons with cubic-quintic-septic law by F-expansion
Optik
(2019)
R.W. Kohl et al.
Highly dispersive optical soliton perturbation with cubic?quintic?septic refractive index by semi-inverse variational principle
Optik
(2019)
R.W. Kohl et al.
Cubic-quartic optical soliton perturbation by semi-inverse variational principle
Optik
(2019)

There are more references available in the full text version of this article.

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Original research articleSequel to highly dispersive optical soliton perturbation with cubic-quintic-septic refractive index by semi-inverse variational principle

Abstract

Introduction

Section snippets

Preliminaries

Application of SVP

Conclusions

Conflict of interest

Acknowledgements

Optik

Optik

Optik

Optik

Optik

Original research article
Sequel to highly dispersive optical soliton perturbation with cubic-quintic-septic refractive index by semi-inverse variational principle