Aliyu Isa Aliyu, Mustafa Inc, Abdullahi Yusuf, and Dumitru Baleanu

1Department of Mathematics,Science Faculty,Firat University,Elazig 23000,Turkey

2Department of Mathematics,Science Faculty,Federal University Dutse,Jigawa 7156,Nigeria

3Department of Mathematics,Science Faculty,Cankaya University,Ankara 06010,Turkey

4Institute of Space Sciences,Magurele 077126,Romania

AbstractThis paper addresses the ring-cavity fiber laser system.A class of gray and black soliton solutions of the model are reported by adopting an appropriate envelope ansatz.Further more,the modulation instability(MI)of the equation is studied using the linear-stability analysis(LSA)technique and the MI gain spectrum is derived.Some physical interpretations and analysis of the results obtained are also presented.

Key words:complex envelope ansatz solution,ring-cavity fiber laser system,linear stability analysis,optical solitons

1 Introduction

Carbon materials do have unique geometrical features,different photoconductive and photoelectric properties with fast recovery time and high nonlinearity.[1?3]As such,they are classified photoconductive fiber media.[4?5]Nanoscale carbon materials are applied as saturated absorbers(SA)in a mode-locked erbium-doped fiber.[6?9]A lot of studies on fiber lasers have concentrated on applying CNTs with erbium(Er) fiber lasers to produce short optical pulse waves.[10?17]Most of the previous studies on CNTs in ring-cavity fiber laser have been the experimental types.It is therefore relevant to study the model analytically in order to provide an insight on the principles for a wide range scopes.[12]Generally,the theory is modelled based on the various nonlinear Schr?dinger equation(NLSE).The NLSE can be solved and investigated either numerically or analytically.Triki et al.[18]put forward an ansatz for deriving black and gray solitons.Several concepts have been utilized in the last few decades to study NLSEs.[19?33]

The system is modelled as the following NLSE[19]

where ψ = ψ(x,t)represents the optical pulse amplitude energy,t is the time,x the propagation distance,α is a coefficient that considers into account the material losses in the cavity,δ represents the second-order dispersion coefficient,σSAis the absorbtion parameter,? is the two-photon parameter,represents saturation power,γ represents self-phase modulation(SPM)parameter,represents frequency-dependent gain and finallyrepresents saturation power of SA.[19]Equation(1)has two coefficients

To study Eq.(1),we begin by introducing the notations given by[19]

Thus Eq.(1)becomes

In the present paper,we will investigate the existence of gray and dark optical solitons to the ring-cavity fiber laser system(see Fig.1)using a suitable complex envelope ansatz solution.[18]Then the MI analysis of the model will be studied using the LSA technique.[31?32]

Fig.1 Optical pulse propagation system.[19]

2 Gray and Dark Optical Solitons

To investigate the exact solitary wave solutions of the model under consideration,we consider a soliton transformation of the form[18]

where

In Eq.(6),A(x,t)is a function and ? is the phase shift,k is wave number,ω is the frequency and ν represents phase constant.

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In order to investigate localized gray and dark optical solitons of Eq.(5),we use the ansatz given by[18]

where η is the pulse width and v is the velocity.The amplitude of A(x,t)is given by:

and phase shift is

If β=0 in Eq.(9),a black soliton is derived.[18]But ifin Eq.(9),then a gray soliton is retrieved.[18]Putting Eq.(6)into Eq.(5),we acquire:

Inserting Eq.(8)into Eq.(11),we obtain

Performing all necessary algebraic computations,we get:

Constants:

sech2(τ):

sech2(τ)tanh(τ):

tanh(τ):

2.1 Gray Optical Soliton

From Eqs.(13)–(16),we obtain an expression for the gray soliton with the following parameters

which will in turn give the gray soliton represented by

The intensity is

and the phase shift is represented by

2.2 Black Optical Soliton

From Eqs.(13)–(16),it is possible to obtain the following soliton parameters when β=0

which will in turn give the black soliton represented by

The intensity is given by

3 Physical Expressions and Discussion

In this section,we provide an insight of the physical meaning of the obtained solutions.We observe that some of the solutions in this manuscript are newly constructed solutions.The solutions are related to the physical features of optical soliton solutions.The gray and black solitons Eqs.(18)and Eq.(22)are to the best of our knowledge new.Although,some hyperbolic and complexiton solutions have been reported in Ref.[19].To understand the physical nature and evolution of the optical soliton solutions.We considered only two optical soliton solutions in the analysis,namely;the gray optical soliton Eq.(16)and the black soliton Eq.(20).This is because the parameter β does not actually affect the physical nature of the solitons in the instance when β =0.Figures 1–3 are shown below:

Fig.2 (a)3D and(b)2D of the gray soliton solution Eq.(18)by choosing the values of β =1,λ =0.5,η =0.2,v=0.7.

Fig.3 Surface pro files of(a)The contour plot and(b)The contour plot in spherical coordinates of the gray soliton solution Eq.(18)by choosing the values of β =3,λ =0.2,η =0.1,v=5.

4 Modulation Instability Analysis

We use the LSA technique[31?32]to study Eq.(5).Suppose Eq.(5)has a steady-state solution given by

where P0is the incident power and ρ(x,t)perturbation function.Putting Eq.(24)into Eq.(5)and linearizing we obtain

We seek for

to find the solution of the linearized Eq.(25).In Eq.(26),K is the wave number and ? represents the frequency.Inserting Eq.(26)into Eq.(25),we get

From Eqs.(27)and(28),we obtain a matrix in a1and a2.The matrix has a nontrivial solution of determinant given by

From Eq.(29),we obtain

It can be observed from Eq.(30)that MI will always exists since.The MI gain is given by

5 Concluding Remarks

This paper successfully derived the gray and black optical solitons to the ring-cavity fiber laser system.By utilizing the concept of LSA,the MI is studied and the MI gain spectrum is reported.It has been shown that MI will always exist.Physical interpretation of the acquired result have been shown in Figs.2 and 3.