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ORIGINAL RESEARCH article

Front. Phys., 14 March 2019 | https://doi.org/10.3389/fphy.2019.00028

Dark-Bright Optical Soliton and Conserved Vectors to the Biswas-Arshed Equation With Third-Order Dispersions in the Absence of Self-Phase Modulation

  • 1Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
  • 2Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey
  • 3Department of Mathematics, Cankaya University, Ankara, Turkey
  • 4Institute of Space Sciences, Magurele, Romania
  • 5Department of Computer Engineering, Istanbul Gelisim University, Istanbul, Turkey

The form-I version of the new celebrated Biswas-Arshed equation is studied in this work with the aid of complex envelope ansatz method. The equation is considered when self-phase is absent and velocity dispersion is negligibly small. New Dark-bright optical soliton solution of the equation emerge from the integration. The acquired solution combines the features of dark and bright solitons in one expression. The solution obtained are not yet reported in the literature. Moreover, we showed that the equation possess conservation laws (Cls).

1. Introduction

The study of dynamical systems in non-linear physical models plays an important role in optical fibers, electrical transmission lines, plasma physics, mathematical biology, and many more [1]. This is motivated by the capacity to model the behavior of these systems and other under different physical conditions [2]. These systems are represented by non-linear equations. Seeking the exact solutions of non-linear evolution equations has been an interesting topic in mathematical physics, and the solutions of corresponding models are the ways to well describe their dynamics. Several results have been reported in the last few decades [324]. The main principle for the existence of solitons in metamaterials, optical fibers, and crystals is the existence of a balance between non-linearity and dispersion. It is obvious that some situations may lead to. Recently, Biswas and Arshed [3] put forward a new model for soliton transmission in optical fibers in the event when self-phase modulation is neglible in the absence of non-linearity.

The third-order model in the absence of self-phase modulation that will be studied in this paper is given by [3, 4]:

iψt+αψxx+γψxt+i[σψxxx+δψxxt]-i[Ω(|ψ|2ψ)x+μ(|ψ|2)xψ+θ|ψ|2ψx]=0.    (1)

The function ψ(x, t) representing the dependent involving t an x which denotes the temporal and spatial components. The first term represents the temporal evolution of the wave, γ represents the(STD) coefficient, α is the coefficient of GVD and σ is the coefficient of the third order dispersion, δ is the coefficient of spatio-temporal 3OD (ST-3OD), Ω is the effect of self-steepening. Finally, μ and θ provide the effect of non-linear dispersion. Dark, bright, combo and singular soliton solutions of Equation (1) have been reported in Biswas and Arshed [3] and Ekici and Sonmezoglu [4]. But to the best of our knowledge, the dark-bright optical soliton and Cls of the equation have not been reported. In this work, this special solution combining the features of dark and bright optical soliton in one expression will be recovered by applying a suitable ansatz. The Cls of the equation will be derived using the multiplier method [8, 9].

2. Dark-Bright Optical Soliton

In order derive the dark-bright soliton solution of the equation, we consider the ansatz solution given by Li et al. [5]:

ψ(x,t)=A(x,t)×eiΨ(x,t),    (2)

with

Ψ(x,t)=-kx+ωt+ν.    (3)

In Equation (2), Ψ denotes the phase shift, k denotes the wave number, ω represents the frequency and ν is the phase constant. We now utilize the ansatz put forward from Li et al. [5]:

A(x,t)=iβ+λtanh[η(x-vt)]+iρsech[η(x-vt)],    (4)

where v represents the velocity and η is the pulse width. In the even when λ → 0 or ρ → 0, the Equation (4) transforms to a bright or dark soliton solution. The intensity of A(x, t) is given by

|A(x,t)|={λ2+β2+2βρsech[η(xvt)]+(ρ2λ2)sech2[η(xvt)]}12.    (5)

The non-linear phase shift ψNL is represented by

ψNL=arctan[β+ρsech[η(x-vt)]λtanh[η(x-vt)]].    (6)

Putting Equation (2) into Equation (1) leads to

 -A(ω+k(k(α+kσ)-(γ+kδ)ω)+k(θ+μ+Ω)|A|2+2iΩAAx)- i((-1+k(γ+kδ))At+(k(2α+3kσ)-(γ+2kδ)ω+(θ+μ+Ω)|A|2)Ax+i((γ+2kδ)Axt+(α+3kσ-δω)Axx+i(δAxxt+σAxxx)))=0.    (7)

Now, putting Equation(4) into Equation(7), expanding the result and equating the combination of coefficients of sech(τ) and tanh(τ), we acquire the independent parametric equations represented by:

Constants:

 -iβ(ω+k(-γω+k(α+kσ-δω)+(β2+λ2)(θ+μ+Ω)))=0,    (8)

sech(τ) :

-iρ(ω+k(-γω+k(α+kσ-δω)+(3β2+λ2)(θ+μ+Ω)))=0,    (9)

sech2(τ) :

i(v(-1+k(γ+kδ))ηλ-3k2ηλσ-ηλ(-γω+(β2+λ2)(θ+μ+Ω))+k(-2αηλ+2δηλω+β(λ2-3ρ2)(θ+μ+Ω)))=0,    (10)

sech3(τ) :

iρ(-αη2+v(γ+2kδ)η2-2βηθλ+kθλ2-2βηλμ+kλ2μ-kθρ2-kμρ2-3kη2σ+δη2ω-2βηλΩ+kλ2Ω-kρ2Ω)=0,    (11)

sech4(τ) :

iηλ(2η2(vδ-σ)+(λ-ρ)(λ+ρ)(θ+μ+Ω))=0,    (12)

tanh(τ) :

-λ(ω+k(-γω+k(α+kσ-δω)+(β2+λ2)(θ+μ+Ω)))=0,    (13)

tanh(τ)sech(τ) :

ρ(v(-1+k(γ+kδ))η-3k2ησ-2k(αη-δηω+βλ(θ+μ+Ω))-η(-γω+λ2(θ+μ+Ω)+β2(θ+μ+3Ω)))=0,    (14)

tanh(τ)sech2(τ) :

(-2αη2λ+2v(γ+2kδ)η2λ+kθλ3+kλ3μ-2βηθρ2-kθλρ2-2βημρ2-kλμρ2-6kη2λσ+2δη2λω+(λ2(2βη+kλ)-(6βη+kλ)ρ2)Ω)=0,    (15)

tanh(τ)sech3(τ) :

ηρ(5η2(vδ-σ)+(λ-ρ)(λ+ρ)(θ+μ+3Ω))=0,    (16)

tanh2(τ)sech(τ) :

-iηρ(η(-α+v(γ+2kδ)-3kσ+δω)-2βλΩ)=0,    (17)

tanh2(τ)sech2(τ) :

-2iηλ(2η2(vδ-σ)+(λ-ρ)(λ+ρ)Ω)=0,    (18)

tanh3(τ)sech(τ) :

η3ρ(-vδ+σ)=0,    (19)

where τ = η(xvt). From the solution of Equations(8)–(19), we observed that β = 0. but, for a dark-bright optical soliton to exist, we require both ρ ≠ 0 and λ ≠ 0. For the sake of compatibility, we considered the case when ρ = λ from the solutions of Equations(8)–(19). We acquire the velocity as

v=-ρ2(θ+μ+Ω),    (20)

the wave number is represented by

k=-ωρ2(θ+μ+Ω).    (21)

We also acquire the value of δ and α as

δ=-σρ2(θ+μ+Ω),    (22)
α=-γρ2(θ+μ+Ω).    (23)

The dark-bright optical soliton to the model reads:

ψ(x,t)={iλsech[η(x+tλ2(θ+μ+Ω))]+λtanh[η(x+tλ2(θ+μ+Ω))]}×ei(θ+tω+xωλ2(θ+μ+Ω)).    (24)

and the intensity gives

|ψ(x,t)|2=λ2.    (25)

The phase shift is represented by

ψNL=arctan[sech[η(x+tλ2(θ+μ+Ω)]tanh[η(x+tλ2(θ+μ+Ω)]].    (26)

The dark-bright soliton Equation (24) represents a soliton combining the features of dark and bright solitons in one expression. The constant β = 0 implies a pronounced “platform” underneath the soliton under non-zero boundary conditions and its asymptotic value approaches λ as |t| → ∞. To analyze the dynamics behavior of the soliton solution Equation (24), we have made numerical evolutions for some perturbations to show the evolution of the dark-bright optical soliton solution. Figures 1-3 shows the profiles surfaces of the dark-bright soliton Equation (24). The obtained soliton Equation (24) possesses the structure of the physical properties of dark and bright optical solitons in the same expression. These solitons appear temporal solitons observed in optical fibers.

FIGURE 1
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Figure 1. 3D surface of solution Equation(24) by selecting the parameter values of η = 0.1, λ = 0.1, θ = 1, Ω = 0.1.

FIGURE 2
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Figure 2. The contour surface of solution Equation (24) by selecting the parameter values of η = 0.1, λ = 0.1, θ = 1, Ω = 0.1.

FIGURE 3
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Figure 3. contour plot in spherical coordinates of solution Equation (24) by selecting the parameter values of η = 0.1, λ = 0.1, θ = 1, Ω = 0.1.

3. Conservation Laws

In this part, we will utilize the multiplier to derive the Cls [8, 9]. To achieve this aim, we apply

ψ(x,t)=u(x,t)+iυ(x,t),    (27)

to transform Equation (1) to a system of PDEs. Putting Equation (27) into Equation (1), we acquire:

{-υt+2(μ+Ω)uυux+(θ+Ω)u2υx+(θ+2μ+3Ω)υ2υx+γuxt+αuxx-δυxxt-συxxx=0. ut+(-θ-2μ-3Ω)u2ux+(-θ-Ω)υ2ux-2(μ+Ω)uυυx+γυxt+αυxx+δuxxt+σuxxx=0.    (28)

Applying the formula for determining equations in [9], we acquires the multipliers of zeroth-order Λ1(x, t, u, υ), Λ2(x, t, u, υ) for Equation (1)

Λ1=c1u,Λ2=c1υ,    (29)

where c1 is a constant.

1. If c1 = 1 in Equation(29), then we have the following multipliers:

Λ1=u,Λ2=υ.    (30)

Subsequently, we acquire the fluxes given by:

Tx=-δ(uuxx-υυxx)σ,Tt=u3tut(3Ω+2μ+θ)+σ(uuxx+υυxx)σ.    (31)

4. Concluding Remarks

In this article, we have explored a suitable ansatz solution to derive a dark-bright soliton solution of the new celebrated Biswas-Arshed equation. observing the solutions derived in Biswas and Arshed [3] and Ekici and Sonmezoglu [4], we observed that the solution of the equation acquired in this manuscript is new. The method used here has been proved to be efficient in investigating the combined soliton solution of non-linear models. We finally showed that the equation has conservation laws and we reported the conserved vectors. We hope to apply other techniques to extract additional new forms of solutions of the new model in the future.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Funding

This study was Funded by Cankaya University.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: complex envelope ansatz, dark-bright optical soliton, Biswas-Arshed equation, conserved vectors, multiplier, numerical simulations

Citation: Aliyu AI, Inc M, Yusuf A, Baleanu D and Bayram M (2019) Dark-Bright Optical Soliton and Conserved Vectors to the Biswas-Arshed Equation With Third-Order Dispersions in the Absence of Self-Phase Modulation. Front. Phys. 7:28. doi: 10.3389/fphy.2019.00028

Received: 04 December 2018; Accepted: 18 February 2019;
Published: 14 March 2019.

Edited by:

Jesus Martin-Vaquero, University of Salamanca, Spain

Reviewed by:

Emrullah Yasar, Uludaǧ University, Turkey
Haci Mehmet Baskonus, Harran University, Turkey

Copyright © 2019 Aliyu, Inc, Yusuf, Baleanu and Bayram. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Mustafa Inc, minc@firat.edu.tr