Optical solitons of a time-fractional higher-order nonlinear Schrödinger equation

doi:10.1016/j.ijleo.2020.164574

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Volume 209, May 2020, 164574

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Abstract

A time-fractional higher-order nonlinear Schrödinger equation with Kerr law, power law and log law of nonlinearity is studied, and bright, dark soliton, Jacobian elliptic function solutions and combined solutions are found by applying the Riccati equation approach, Jacobian elliptic function approach and double function approach. Dynamical evolution of these fractional solitons is discussed, and two kinds of dark solitons and two kinds of bright solitons are found.

Introduction

The nonlinear Schrodinger equation (NLSE) as a control model governs the soliton dynamics in optical fibers [[1], [2], [3]]. Optical solitons have become the subjects of universal study owing to their extensive applications in ultrafast optics and nonlinear optics [[4], [5], [6]].

Recently, when studying the soliton problem in optical fiber, one presented the fractional NLSE (FNLSE) with the dispersion of non-parabola applying the fractional Laplacian operator [[7], [8], [9]]. The FNLSE applying the fractional operator is more suitable for describing the real dispersion than the integer operator of NLSE [10,11].

More recently, higher-order FNLSE were proposed to describe fractional optical solitons [[12], [13], [14]]. Rich methods were used to get exact solutions of higher-order FNLSE [[12], [13], [14], [15], [16]]. The fractional sinh-Gordon equation expansion method was constructed to study the higher-order FNLSE with exponential-parameter nonlinearity [12]. The exp-function method was proposed to study the higher-order FNLSE with higher-order dispersion [13]. The extended direct algebraic method was applied to study the higher-order FNLSE with self-steepening perturbation and nonlinear dispersion [14].The sine-Gordon equation method was used to study the higher-order FNLSE with higher-order dispersion and full nonlinearity effects [15]. The F-expansion method was utilized to study the higher-order FNLSE with nonlinearities of Kerr law and anticubic law [16].

In Ref. [17], the extended trial equation scheme was put forward to study the higher-order FNLSE with Kerr law, power law and log law of nonlinearity, and some solutions in terms of Jacobian elliptic functions were derived. There still exists some interesting questions. For such, whether other methods can be applied to study this higher-order FNLSE with Kerr law, power law and log law of nonlinearity? Whether other forms of exact solutions can be found for this model?

In this paper, we try to give the answer to the above two questions. We will find bright, dark soliton, rational function solutions, Jacobian elliptic function solutions and combined solutions for the higher-order FNLSE with Kerr law, power law and log law of nonlinearity by applying Riccati equation approach (REA), Jacobian elliptic function approach (JEFA) and double function approach(DFA).

Section snippets

Mathematical models and their solutions

The time fractional perturbed NLSE [17] is given as $i \frac{\partial^{k} h}{\partial t^{k}} + \frac{\partial^{2} h}{\partial x^{2}} + κ (F ({|h|}^{2}) h) + i κ_{1} \frac{\partial^{3} h}{\partial x^{3}} + i κ_{2} F ({|h|}^{2}) \frac{\partial h}{\partial x} + i κ_{3} \frac{\partial}{\partial x} (F ({|h|}^{2})) h = 0$ Where the function $h$ is a complex-valued envelop with time t, spatial coordinate x and fractional order 0 < k ≤ 1, $κ$ and $κ_{1}$ are coefficients of nonlinearity and third order dispersion, $κ_{2}$ and $κ_{3}$ are coefficients of nonlinear dispersions. Here the Jumarie^’s modified Riemann–Liouville derivative for the fractional order $λ$ has the definition [18] $\frac{\partial^{k} f (t)}{\partial t^{k}} = \{\begin{cases} \frac{1}{Γ (- k)} \int_{0}^{t} {(t - τ)}^{- k - 1} [f (τ) - f (0)] d τ, \underset{}{} \underset{}{} \underset{}{} \underset{}{} \underset{}{} k < \end{cases}$

Conclusions

In short, we study a time-fractional higher-order NLSE with Kerr law, power law and log law of nonlinearity, and get bright, dark soliton, Jacobian elliptic function solutions and combined solutions by applying the REA, JEFA and DFA. We discuss dynamical evolution of these fractional solitons. We find two kinds of dark solitons, that is, when the parameter a adds, the width of dark soliton decreases and the amplitude is unchanged in Fig. 1, however, the width of dark solitons decreases and yet

Declaration of Competing Interest

The authors have declared that they have no known competing financial interest/personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR20A050001), and the science and technology innovation program for college students of Zhejiang Provincial Education Department.

References (22)

S.J. Chen et al.
Soliton solutions and their stabilities of three (2+1)-dimensional PT-symmetric nonlinear Schrödinger equations with higher-order diffraction and nonlinearities
Optik
(2019)
A. Biswas
Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities
Optik
(2019)
Y.F. Hua et al.
Interaction behavior associated with a generalized (2 + 1)-dimensional Hirota bilinear equation for nonlinear waves
Appl. Math. Model.
(2019)
H. Rezazadeh et al.
New optical solitons of nonlinear conformable fractional Schrodinger-Hirota equation
Optik
(2018)
E.A. Yousif et al.
On the solution of the space-time fractional cubic nonlinear Schrödinger equation
Res. Phys.
(2018)
A. Esen et al.
Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrodinger equation
Optik
(2018)
G.Z. Wu et al.
Fractional optical solitons of the space-time fractional nonlinear Schrodinger equation
Optik
(2020)
M. Foroutan et al.
New explicit soliton and other solutions for the conformable fractional Biswas–Milovic equation with Kerr and parabolic nonlinearity through an integration scheme
Optik
(2018)
M.A. Abdou et al.
New exact travelling wave solutions for space-time fractional nonlinear equations describing nonlinear transmission lines
Res. Phys.
(2018)
N. Sajid et al.
Optical solitons with full nonlinearity for the conformable spacetime fractional Fokas–Lenells equation
Optik
(2019)

E. Yaşar et al.

New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method

Res. Phys.

(2018)

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Original research articleOptical solitons of a time-fractional higher-order nonlinear Schrödinger equation

Abstract

Introduction

Section snippets

Mathematical models and their solutions

Conclusions

Declaration of Competing Interest

Acknowledgements

Optik

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Appl. Math. Model.

Optik

Res. Phys.

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Res. Phys.

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Res. Phys.

Original research article
Optical solitons of a time-fractional higher-order nonlinear Schrödinger equation