Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis

Accepted Manuscript Title: Soliton solutions of the nonlinear Schr¨odinger equation with the dual power law nonlinearity and resonant nonlinear Schr¨odinger equation and their modulation instability analysis Author: Asghar Ali Aly R. Seadawy Dianchen Lu PII: DOI: Reference:

S0030-4026(17)30833-1 http://dx.doi.org/doi:10.1016/j.ijleo.2017.07.016 IJLEO 59412

To appear in: Received date: Accepted date:

18-5-2017 6-7-2017

Please cite this article as: Asghar Ali, Aly R. Seadawy, Dianchen Lu, Soliton solutions of the nonlinear Schrddotodinger equation with the dual power law nonlinearity and resonant nonlinear Schrddotodinger equation and their modulation instability analysis, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.07.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Soliton solutions of the nonlinear Schr¨ odinger equation with the dual power law nonlinearity and resonant nonlinear Schr¨ odinger equation and their modulation instability analysis Asghar Ali

1,2 ,

Aly R. Seadawy1,3 and Dianchen Lu1

1 Faculty

of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, P. R. China of Mathematics, University of Education, Multan Campus, Pakistan 3 Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia Mathematics Department, Faculty of Science, Beni-Suef University, Egypt

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Corresponding Authors:(Aly Seadway):E-mail:[email protected]

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2 Department

1

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Abstract: In optical fibers, the higher order nonlinear Schr¨odinger equations describes propagation of ultra-short pluse. The proposed modified simple equation method is employed to the nonlinear higher order Schr¨odinger equations for soliton solutions. The balance numbers in these Schr¨odinger equations are positive non integers. The obtain solitary solutions are also presented by graphically. The modulation instability analysis shows the stability and movement of the waves, which confirms that all obtained solutions are analytical and stable. This is a new and standardized method which is applicable to solve different kind of problems in mathematics and physics. Keywords: Nonlinear higher order schr¨ odinger equations, positive non integers balance numbers, modified simple equation method, solitons, Solitary wave solutions.

Introduction

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In various branches of mathematical physical sciences such as chemistry, biology, physics and engineering, the partial differential equations play as a basic tool for solving different kind of problems arises in these areas. In partial differential equations, the higher order nonlinear Schr¨odinger equations are essential module for nonlinear optics which explained the propagation especially short pulses in optical fibers and have a wide applications in ultrafast signal-routing, telecommunication system etc. The parameters involving in higher order Schr¨odinger equations are utilized to explain the pulse propagation in optical fibers. Optical solitons occur due the balance of group velocity dispersion and nonlinear effect. The phenomena of soliton was first described by author in [1] and the solution of solitons are acquired by means of the inverse scattering transform in [2]. In different aspect, solutions of nonlinear Schr¨odinger equations have been studied different authors in [3–11]. Many influential methods have been developed for the solitary waves solution of nonlinear partial equation such as the homogeneous balance method [12, 13], modified simple equation method [14–17], modified extended direct algebraic method [18], the tanh-sech method and the extended tanhcoth method [19–21], the soliton ansatz method [22–30], the Kudryashov method [31], the first integral method [32], the symmetry method [33], the (G0 /G)-expansion method [35] and many more. In the current work, we have employed proposed modified simple equation method on nonlinear higher order schr¨odinger equations with non integer balance number [34, 38] to obtain new solitary wave solutions. To the best of our knowledge, no work has been done in previous study by employing the current proposed method for solving such type of higher order nonlinear Schr¨odinger equations. The obtained solutions are useful in exploring nonlinear wave phenomena in mathematical physical sciences. The article structured as follows: The main steps of the proposed method are given in Section 2. In Section 3, we apply the present method to higher order nonlinear Schr¨odinger equations for constructing solitary wave solution. In section 4, we discuss the stability of higher order NLSEs. Lastly, the summary of the work is given in Section 5. Page 1 of 14

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2

Discription of Proposed Method

In this section, we describe the algorithm of modified simple equation method (MSEM) to obtain the solitary wave solutions of the nonlinear higher order Schr¨odinger equations. Consider the following nonlinear PDE: F (u, ut , ux , utt , uxx , utx , ...) = 0, (1)

ξ = kx + ωt,

us

by utilizing the above transformation , the Eq.(1) is reduces into ODE as
H U, U 0 , U 00 , U 000 , ... = 0,

(2)

cr

u(x, t) = U (ξ),

ip t

where F is a polynomial function of u(x, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. The basic key steps are as: Step 1: Consider travelling wave transformation

(3)

an

where H is a polynomial in U (ξ) its derivatives with respect to ξ. Step 2: Let us assume that the solution of Eq.(3) has the form: U (ξ) = a(ψ(ξ))M

(4)

M

where a is arbitrary constant, and the positive integer M can determined by using the homogeneous balance technique between the highest order derivatives and nonlinear terms come out from Eq.(3). Let ψ(ξ) satisfies the following new ansatz equation. ψ 0 (ξ) = b0 + b1 ψ + b2 ψ 2 + b3 ψ 3 ,

(5)

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where b0 , b1 , b2 , b3 are arbitrary constants. If b0 = 0, b3 = 0 , then simple ansstz equation (5)reduces to Bernoulli equation, which has the following solutions: b1 eb1 (ξ+ξ0 ) ψ(ξ) = , b1 > 0, (6) 1 − b2 eb1 (ξ+ξ0 ) −b1 eb1 (ξ+ξ0 ) , b1 < 0. (7) 1 + b2 eb1 (ξ+ξ0 ) If b1 = 0, b3 = 0, then the ansatz (5) reduces to Riccati equation, which has the following solutions: √ p b0 b2 ψ(ξ) = tan( b0 b2 (ξ + ξ0 )), b0 b2 > 0, (8) b2 √ p −b0 b2 ψ(ξ) = − tanh( −b0 b2 (ξ + ξ0 )), b0 b2 < 0. (9) b2 If b0 = 0, b2 = 0, then the ansatz equation (5) has the following solutions: √ −b1 ψ(ξ) = ± p , b1 < 0, (10) e−2(ξ+ξ0 )b1 + b3 √ e(ξ+ξ0 )b1 b1 ψ(ξ) = ± p , b1 > 0. (11) 1 − e2(ξ+ξ0 )b1 b3 ψ(ξ) =

Step 3: Substituting Eq.(4) along with Eq.(5) into Eq.(3), we obtained a system of algebraic equations in parameters b0 , b1 , b2 , b3 , k, ω. The system of algebraic equations are solved with the help of Mathematica and we get the values of parameters. Step 4: By substituting all these values of parameters and ψ(ξ) into Eq.(4), We obtained the required solutions of Eq.(1). Page 2 of 14

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3 3.1

Applications The nonlinear Schr¨ odinger equation with the dual power law nonlinearity: This equation [34, 35] has the form as 1 iut + (uxx + uyy ) + (|u|2n + m|u|4n )u = 0, 2

ip t

(12)

cr

where u(x, y, t) represents complex amplitude of the waveform, n is arbitrary constant, k represents the saturation non linear refractive index. This equation has wide application in different areas of physics, including nonlinear optics, plasma physics, super conductivity and quantum mechanics [36]. This equation also has been discussed by different authors by utilizing different methods [34, 35, 37]. Consider the traveling waves transformation

us

u(x, y, t) = V (ξ)eip , ξ = k1 x + k2 y − ωt, p = αx + βy + γt,

(13)

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where k1 , k2 , ω,α β, γ are arbitrary constants which can be determined. Now by substituting Eq.(13) into Eq.(12) and separating the real and imaginary parts, we get the following nonlinear differential equation:

k12 + K22 V 00 − α2 + β 2 + 2γ V + 2V 2n+1 + 2mV 4n+1 = 0, (14)

M

where ω = αk1 + βk2 , now applying the homogeneous balance principle on Equation on Eq.(14) 1 we obtained M = 2n , Consider solution of Eq.(14) is 1

V (ξ) = a(ψ(ξ)) 2n .

(15)

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Substituting Eq.(15) along Eq.(5) into Eq.(14), we get system of algebraic equation in paramerts, b0 , b1 , b2 , b3 , α, β, γ, k1 , k2 . The system of algebraic equations can be solved for these parameters, we have following family of solutions. Family 1: b0 = b3 = 0, p −2α2 m − 4γm − 2α2 mn2 − 4γmn2 − 4α2 mn − 8γmn − 2n − 1 2b1 m(n + 1)a2n √ √ β=− , , b2 = 2n + 1 2 mn2 + 2mn + m p −b21 k12 mn2 − 2b21 k12 mn − b21 k12 m − 4n3 − 2n2 p k2 = ± . (16) b21 mn2 + 2b21 mn + b21 m Substituting Eq.(16) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of Eq.(12) as:

u11 (ξ) = a

u12 (ξ) = a where

!

b1 eb1 (ξ+ξ0 )

1−

1 2n

eip , b1 > 0,

2n )eb1 (ξ+ξ0 ) ( 2b1 m(n+1)a 2n+1

−b1 eb1 (ξ+ξ0 ) 2n

1 + ( 2b1 m(n+1)a )eb1 (ξ+ξ0 ) 2n+1

!

(17)

1 2n

eip , b1 < 0,

(18)

ξ = k1 x + k2 y − (αk1 + βk2 )t, p = αx + βy + γt.

Page 3 of 14

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Figure 1: Exact solitary wave solutions of Eq.(17) and Eq.(18) with different shapes are plotted at: (1-a), (1-b) are solitary wave of u11 and (1-c), (1-d) are solitary wave of u12 by choosing different values of parameters. Family 2: b0 = b3 = 0, s p −b21 k12 m(n + 1)2 − 2(2n + 1)n2 −2m (n2 + 2n + 1) (α2 + 2γ) − 2n − 1 p β= , k2 = ± , 2 2m (n + 2n + 1) b21 m(n + 1)2 b2 =

2b1 m(n + 1)(a2n ) . 2n + 1

(19)

Substituting Eq.(19) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of (12) as: !1 2n b1 eb1 (ξ+ξ0 ) u13 (ξ) = a eip , b1 > 0. (20) 2b1 m(n+1)a2n b1 (ξ+ξ0 ) 1−( )e 2n+1

u14 (ξ) = a

−b1 eb1 (ξ+ξ0 ) 1+

2n ( 2b1 m(n+1)a )eb1 (ξ+ξ0 ) 2n+1

!

1 2n

eip b1 < 0,

(21)

where ξ = k1 x + k2 y − (αk1 + βk2 )t, p = αx + βy + γt. Family 3:

b1 = b3 = 0, Page 4 of 14

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Figure 2: Exact solitary wave solutions of Eq.(20) and Eq.(21) with different shapes are plotted at: (2-a)and (2-b) are solitary wave of u13 and u14 by choosing different values of parameters.

p m = 0, β = ± −α2 − 2γ , k2 = ik1 .

(22)

√

1

 2n p b0 b2 tan( b0 b2 (ξ + ξ0 ) eip b2

b0 b2 > 0,

(23)

d

u15 (ξ) = a

M

Substituting Eq.(22) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of Eq.(12) as:

(24)

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1  2n  √ p −b0 b2 eip b0 b2 < 0, u16 (ξ)) = a − tanh( −b0 b2 (ξ + ξ0 ) b2 where ξ = k1 x + k2 y − (αk1 + βk2 )t, p = αx + βy + γt

Figure 3: Exact solitary wave solutions of Eq.(23) and Eq.(24) with different shapes are plotted at: (3-a)and (3-b) are solitary wave of u15 and u16 by choosing different values of parameters. Page 5 of 14

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Family 4:

b1 = b3 = 0, p m = 0, β = ± −α2 − 2γ , k2 = −ik1 .

(25)

Substituting Eq.(25) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of Eq.(12) as: u17 (ξ) = a

1

 2n p b0 b2 tan( b0 b2 (ξ + ξ0 ) eip b2

, b0 b2 > 0

(27)

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cr

1  2n  √ p −b0 b2 tanh( −b0 b2 (ξ + ξ0 ) eip , b0 b2 < 0, u18 (ξ) = a − b2 where ξ = k1 x + k2 y − ωt, p = αx + βy + (αk1 + βk2 )t.

b0 = b2 = 0,

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Family 5:

(26)

ip t

√

Figure 4: Exact solitary wave solutions of Eq.(26) and Eq.(27) with different shapes are plotted at: (4-a)and (4-b) are solitary wave of u17 and u18 by choosing different values of parameters. p m = 0, β = − −α2 − 2γ , k2 = ±ik1 .

(28)

Substituting Eq.(28) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of equation (12) as: √

!

−b1

1 2n

eip

u19 (ξ) = a ± p e−2(ξ+ξ0 )b1 + b3 √ e(ξ+ξ0 )b1 b1

u20 (ξ) = a ± p 1 − e2(ξ+ξ0 )b1 b3

!

b1 < 0,

(29)

1 2n

eip

b1 > 0,

(30)

where ξ = k1 x + k2 y − ωt, p = αx + βy + (αk1 + βk2 )t. Family:6

b0 = b2 = 0, Page 6 of 14

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m = 0, β =

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Figure 5: Exact solitary wave solutions of Eq.(29) and Eq.(30) with different shapes are plotted at: (5-a)and (5-b) are solitary wave of u19 and u20 by choosing different values of parameters.

p −α2 − 2γ , k2 = ±ik1 .

(31)

d

M

Substituting Eq.(31) into Eq.(15) along with the solutions of Eq.(5), the following solitary wave solutions are constructed of equation (12) as: !1 √ 2n −b1 u21 (ξ) = a ± p eip b1 < 0, (32) −2(ξ+ξ )b 0 1 e + b3

te

√ e(ξ+ξ0 )b1 b1

!

1 2n

eip

u22 (ξ) = a ± p 1 − e2(ξ+ξ0 )b1 b3

b1 > 0,

(33)

3.2

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where ξ = k1 x + k2 y − ωt, p = αx + βy + (αk1 + βk2 )t.

The resonant nonlinear Schr¨ odinger equation: This equation [35, 38] has the form as,; 2m

iut + αuxx + (β|u|

+ γ|u|

4m

 )u + δ

(|u|)xx |u|

 u = 0, m > 0,

(34)

which describes the propagation of optical pulses in nonlinear optical fibers, where u(x, t) is a normalized complex amplitude of the pulse envelop in nonlinear optical fiber, where as x is called normalized propagation distance, t is called the retarded time, α, β,γ and δ are constants. Eq.(34) has been discussed different authors by applying different methods [35,42,43]. Now we solve Eq.(34) by current propsed method in section 2. Consider the traveling transformation u(x, t) = φ(ξ)eiq , ξ = kx − ωt,

q = −px + ct,

(35)

where φ(ξ) is a real function, k, ω, p and c are constants. By substituting Eq.(35) into Eq.(34) and separating the real and imaginary parts, we have ω = −2αkp and the equation
k 2 (α + δ)φ00 − c + αp2 φ + βφ2m+1 + γφ4m+1 = 0, (36) Page 7 of 14

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Figure 6: Exact solitary wave solutions of Eq.(32) and Eq.(33) with different shapes are plotted at: (6-a)and (6-b) are solitary wave of u21 and u22 by choosing different values of parameters. 1 2m .

Let us (37)

M

Now by applying the homogeneous balance principle on Eq.(36), we obtained get M = consider the solution of Eq.(36) is 1 φ(ξ) = a(ψ(ξ)) 2m .

d

Now substituting Eq.(37) along with Eq.(5) into Eq.(36), we get system of algebraic equation in paramerts, b0 , b1 , b2 , b3 , α, β, γ, δ, k, c, p. The system of algebraic equations can be solved for these parameters, we have following solutions.

te

Family 1: p

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β 2 (−m2 ) (2m + 1) β 2 (2m + 1) 2b1 γ(m + 1)a2m k=p 2 , c = −αp2 − , b2 = . 2 4γ(m + 1) β(1 + 2m) b1 γ(m + 1)2 (α + δ)

(38)

Substituting Eq.(38) into Eq.(37) along with the solution of Eq.(5), the following solitary wave solutions are constructed of Eq.(34) as: 

u11 (ξ) = a 

1−



u12 (ξ) = a 



b1 eb1 (ξ+ξ0 )

1+

γ(m+1)a2m b1 (ξ+ξ0 ) ( 2b1β(1+2m) )e

eb1 (ξ+ξ0 )

1 2m





−b1  2b1 γ(m+1)a2m b1 (ξ+ξ0 ) ( β(1+2m) )e

eiq ,

b1 > 0;

(39)

1 2m

eiq ,

b1 < 0;

(40)

where ξ = kx + (2αkp)t, q = −px + ct.

Family 2: p β 2 (−m2 ) (2m + 1) β 2 (2m + 1) 2b1 γ(m + 1)a2m k = −p 2 , c = −αp2 − , b2 = . 2 4γ(m + 1) β(1 + 2m) b1 γ(m + 1)2 (α + δ)

(41)

Page 8 of 14

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Figure 7: Exact solitary wave solutions of Eq.(39) and Eq.(40) with different shapes are plotted at: (7-a)and (7-b) are solitary wave of u11 and u12 by choosing different values of parameters. Substituting Eq.(41) into Eq.(37) along with the solution of Eq.(5), the following solitary wave solutions are constructed of Eq.(34) as: 

 1+

1 2m





−b1  2b1 γ(m+1)a2m b1 (ξ+ξ0 ) ( β(1+2m) )e

eiq ,

b1 > 0;

(42)

eiq ,

b1 < 0;

(43)

1 2m

ξ = kx + (2αkp)t, q = −px + ct.

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where

eb1 (ξ+ξ0 )

te

u14 (ξ) = a 

M

1−



γ(m+1)a2m b1 (ξ+ξ0 ) ( 2b1β(1+2m) )e

d

u13 (ξ) = a 

b1 eb1 (ξ+ξ0 )

Figure 8: Exact solitary wave solutions of Eq.(42) and Eq.(43) with different shapes are plotted at: (8-a)and (8-b) are solitary wave of U13 and U14 by choosing different values of parameters.

Page 9 of 14

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4

Modulation instability

Many nonlinear systems demonstrate an instability which leads to modulation of the steady state that describes the relations between the nonlinear and dispersive effects. We apply the standard linear stability analysis [39–41] to derive the modulation instability of the generalized NLSE.

Modulation instability of NLSE with the dual power law nonlinearity: The generalized higher order NLS equation has the steady state solution √

 P + ψ(x, y, t) eiφ(t) ,

φ(t) = (P β + γP 2 )t,

cr

u(x, y, t) =

ip t

4.1

(44)

an

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where P represent the normalized optical power.we examine evolution of the perturbation ψ(x, y, t) by using a linear stability analysis. By putting Eq.(44) into Eq.(12) and linearizing in ψ(x, y, t), we get the following form as:  
∂ψ 1 ∂ 2 ψ ∂ 2 ψ n 2n 2 i + + + ψ (2n + 1)P + m(4n + 1)P − (P β + γP ) . (45) ∂t 2 ∂x2 ∂y 2 Consider the solution of Eq.(45) in the form

M

ψ(x, y, t) = α1 ei(k1 x+k2 y−ωt) + α2 e−i(k1 x+k2 y−ωt) ,

(46)


1 −k12 − k22 + 2mP 2n + 8mnP 2n + 4nP n + 2P n − 2βp − 2γP 2  . 2

te

ω=±

d

where ω are is the frequency of perturbation and k1 , k2 normalized wave numbers. The dispersion relation determines how time oscillations eik1 x , eik2 y are linked to spatial oscillations eiωt of a wave number, substituting Eq.(46) into Eq.(45), we obtain the following dispersion relation as: (47)

4.2

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The dispersion relation in Eq.(47) shows the value of frequency ω is real for all values of k1 , k2 which implies that steady state solution is stable.

Modulation instability of resonant NLSE: The generalized higher order NLS equation has the steady state solution

u(x, t) =

√

 P + ψ(x, t) eiφ(t) ,

φ(t) = (P β + γP 2 )t,

(48)

where P is represent the normalized optical power. We check evolution of the perturbationψ(x, t) by using a linear stability analysis. By putting Eq.(48) into Eq.(34) and linearizing in ψ(x, t), we get i

∂ψ ∂2ψ + (α + δ) 2 + ψ ∂t ∂x




βP (P m−1 (2m + 1) − 1) + γP 2 (P 2m−2 (4m + 1) − ) .

(49)

Consider the solution of Eq.(49) in the form ψ(x, t) = α1 ei(kx−ωt) + α2 e−i(kx−ωt) ,

(50)

where ω and k are the frequency of perturbation and normalized wave number. The dispersion relation k = k(ω) of a constant coefficient linear evolution equation determines how time oscillations eikx are Page 10 of 14

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linked to spatial oscillations eiωt of a wave number, subsituting Eq.(50) into Eq.(49), we obtain the following dispersion relation as: ω = ±(αk 2 + δk 2 − 4γmP 2m − γP 2m − 2βmP m − βP m + γP 2  + βP ).

(51)

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ip t

The above dispersion relation in Eq.(51)also shows the value of frequency ω is real for all values of k which implies that steady soultion is stable

5

Conclusion

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Figure 9: The dispersion relation among frequency (ω) and wave numbers (k1 , k2 ) of (47) is shown in (9 − a) and The dispersion relation among frequency and wave number of (51) is shown in (9 − b).

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In this paper, proposed modified simple equation method has been employed to construct new soliton solutions of the higher order Schr¨odinger equations which describes propagation of ultrashort pluse in optical fibers. These solitary solutions have wide applications in telecommunication system and also helps to understand the physical phenomena of these equations. These solutions are also presented by graphically. We apply the modulation instability analysis for the stability of these solutions and the movement of the waves are also analyzed which shows that all constructed solutions are exact and stable. This method is helpful to solve different kind of problems which arises in mathematics and physics.

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cr

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