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Optical soliton solutions of the NLSE with quadratic-cubic-Hamiltonian perturbations and modulation instability analysis
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical soliton solutions of the NLSE with quadratic-cubicHamiltonian perturbations and modulation instability analysis
T
Ebru Cavlak Aslan, Mustafa Inc Science Faculty, Department of Mathematics, Firat University, 23119 Elazig, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: NLSE Jacobi elliptic ansatz method Optical soliton
The aim of this paper is to present optical soliton solutions of the nonlinear Schrödinger equation. Here, the quadratic-cubic nonlinearity in the existence of Hamiltonian perturbations is considered. The Jacobi elliptic ansatz method is applied to obtain the optical soliton solutions. As a result, dark, bright and w-shaped optical soliton are obtained. Additional, the modulation instability analysis is given.
1. Introduction Nonlinear Schrödinger equation (NLSE) is the most famed model in the field of nonlinear science [1–5]. It appears in nonlinear optics and the study of optical solitons have attracted many researchers interest [6–16]. In many studies investigated soliton solutions of NLSE with difference nonlinearities such as anti-cubic [17], cubic-quartic [18], quadratic-cubic [19,20], quintic [21], cubic [22], cubic-quintic [23,24], and other nonlinearities [25–31]. In this work, we analyzed the NLSE with quadratic-cubic nonlinearity in the existence of Hamiltonian perturbation and acquired dark, bright and w- shaped optical solitons by the Jacobi elliptic ansatz method [32,33]. The equation of interest is given by
iq t + aq xx + (b1 |q| + b2 |q|2 ) q = i {αqx + λ (|q|2 q)x + θ (|q|2 )x q}.
(1.1)
Here, the dependent variable q(x, t) represents the wave profile where x and t are spatial and temporal independent variables. The parameter a is the group velocity. The two terms with b1 and b2 are from quadratic and cubic nonlinear terms, respectively. Also, α represents the inter-modal dispersion, λ represents the self-steeping term and finally θ gives the nonlinear dispersion effect. The two terms with b1 and b2 are from quadratic and cubic nonlinear terms, respectively [34]. In this work, the chapters are organized as follows. In Section 2, we have obtained to dark, bright and w-shaped optical solitons by the Jacobi ansatz method. In Section 3 considers the phenomenon of modulation instability. Finally, the conclusion is given in Section 4. Also, the optical solitons obtained for the different states are plotted. 2. Soliton solutions To solve Eq. (1.1) by the Jacobi elliptic functions, the initial assumption is
q (x , t ) = P (x , t ) eiϕ (x , t ), ϕ (x , t ) = −κx + wt + θ
(2.1)
where, κ and w are the frequency and the wave number of soliton and θ is the phase constant. Substituting Eq. (2.1) into Eq. (1.1) and
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.04.008 Received 14 May 2018; Received in revised form 14 February 2019; Accepted 1 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
equating the real and imaginary parts yield
aP′ ′ + (b2 − λκ ) P 3 + b1 P 2 − (w + ακ + aκ 2) P = 0
(2.2)
v + α + 2aκ + (3λ + 2θ) P 2 = 0.
(2.3)
and
In Eq. (2.3), v is the velocity and from Eq. (2.3) as
v = −α − 2aκ
(2.4)
3λ + 2θ = 0.
(2.5)
and
From the real part given by Eq. (2.2) will be investigated optical solitons. Case 1: The form of the soliton solution is
P (x , t ) = μ0 + μ1 sn p1 (ξ , l)
(2.6)
ξ = B1 (x − vt)
(2.7)
with
where B1 is the inverse width of soliton and l is the modulus of the new Jacobi elliptic function. Substituting Eq. (2.6) into Eq. (2.2) yield
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 + aμ1 p1 (p1 − 1) B12 sn p1 − 2 + (−ap12 B12 (l 2 + 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ1 sn p1 + ap1 (p1 + 1) l 2μ1 B12 sn p1 + 2 + (3(b2 − λκ ) μ0 + b1) μ12 sn2p1 + (b2 − λκ ) μ13 sn3p1 = 0.
(2.8)
From Eq. (2.8), equating (p1 + 2, 3p1) leads to
p1 = 1.
(2.9) pi+j
As a result, the coefficients of sn
w = −aB1
(l 2
terms to zero, we obtain
+ 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.10)
or
w = aκ 2 + ακ + b1 μ0 + (b2 − λκ ) μ02 .
(2.11)
−(b2 − λκ ) ⎞1/2 μ1 B1 = ⎛ . 2a ⎠ l ⎝
(2.12)
Also,
Consequently, the soliton solution of Eq. (1.1) is
q (x , t ) = (μ0 + μ1 sn p1 (ξ , l)) eiϕ (x , t ).
(2.13)
In case of l → 1, dark soliton solution is given by
q (x , t ) = (μ0 + μ1 tanh[B1 (x − vt), l]) eiϕ (x , t ).
(2.14)
Furthermore, while l → 1
w=
3 b1 μ0 + 2(b2 − λκ ) μ02 − aB1 2
(2.15)
and
−(b2 − λκ ) ⎞1/2 B1 = ⎛ μ1 2a ⎠ ⎝
(2.16)
The necessary condition for soliton presence is
(b2 − λκ ) a < 0.
(2.17)
In Fig. 1, the dark optical soliton obtained by Eqs. (2.14)–(2.16) are presented for b2 = −0.5, μ0 = 5, μ1 = 10 and a = 1. In Fig. 2, comparison of the dark optical solitons for different values of a and b are presented. Also, the contour plots of dark optical soliton are given by Fig. 3. Case 2: In another Jacobi elliptic function solution is 2
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
Fig. 1. The dark optical solution for Eqs. (2.14) and (2.16).
Fig. 2. Comparisons of |q(x, t)|2 when t = 0.1, λ = 2,θ = −3 for different values of a and b.
Fig. 3. The contour plot of dark optical soliton.
3
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
P (x , t ) = μ0 + μ 2 cn p2 (ξ , l)
(2.18)
ξ = B2 (x − vt)
(2.19)
with
where B2 is the inverse width of soliton and l is the modulus of the new Jacobi elliptic function. Substituting Eq. (2.18) into Eq. (2.2) yield
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 − aμ 2 p2 (p2 − 1)(l 2 − 1) B22 cn p2 − 2 + (−ap22 B22 (1 − 2l 2) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ 2 cn p2 − ap2 (p2 + 1) l 2μ 2 B22 cn p2 + 2 + (3(b2 − λκ ) μ0 + b1) μ22 cn2p2 + (b2 − λκ ) μ23 cn3p2 = 0.
(2.20)
From here, equating (p2 + 2, 3p2) leads to
p2 = 1.
(2.21) pi+j
Since p2 = 1, the coefficients of cn
w = −a (1 −
2l 2) B22
terms to zero, we obtain
+ 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.22)
and
b − λκ ⎞1/2 μ 2 B2 = ⎛ 2 . l ⎝ 2a ⎠
(2.23)
So, the soliton solution of Eq. (1.1) is
q (x , t ) = (μ0 + μ 2 cn p2 (ξ , l)) eiϕ (x , t ).
(2.24)
In case of l → 1, bright soliton solution is given by
q (x , t ) = (μ0 + μ 2 sec h [B2 (x − vt), l]) eiϕ (x , t ).
(2.25)
From this place, while l → 1
w = aB22 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.26)
b − λκ ⎞1/2 B2 = ⎛ 2 μ2 ⎝ 2a ⎠
(2.27)
and
Similarity to Case 1, the necessary condition for soliton presence is (2.28)
(b2 − λκ ) a > 0.
In Fig. 4, the bright optical soliton obtained by Eqs. (2.25)–(2.27) is presented for b2 = 0.5, μ0 = 5, μ2 = −1 and a = 1. In Fig. 5, comparison of the bright optical solitons for different values of a and b are presented. Also, the contour plots of bright optical soliton are given by Fig. 6. Case 3: To construct the soliton, we have the hypothesis of the form
P (x , t ) = μ0 + μ1 sn p1 (ξ , l) + μ 2 cn p2 (ξ , l)
(2.29)
with
Fig. 4. The bright optical solution for Eqs. (2.25) and (2.27). 4
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
Fig. 5. Comparisons of |q(x, t)|2 when t = 0.1, λ = 2,θ = −3 for different values of a and b.
Fig. 6. The contour plot of bright optical soliton.
ξ = B3 (x − vt) or ξ = B4 (x − vt)
(2.30)
where B3 and B4 are the inverse widths of soliton and l is the modulus of the new Jacobi elliptic function. If we substitute Eq. (2.29) into Eq. (2.2) is
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 + (aμ1 p1 (p2 − 1) B32 sn p1 − 2 + (−ap1 (p1 − 1) B32 (l 2 + 1) − ap1B32 (l 2 + 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ1 sn p1 + ap1 (p1 + 1) l 2μ1 B32 sn p1 + 2 + (3(b2 − λκ ) μ0 + b1) μ12 sn2p1 + (b2 − λκ ) μ13 sn3p1 + (ap22 B42 (1 − 2l 2) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ 2 cn p2 − ap2 (p2 + 1) l 2μ 2 B42 cn p2 + 2 − aμ 2 p2 (p2 − 1)(l 2 − 1) B42 cn p2 − 2 + (3(b2 − λκ ) μ0 + b1) μ22 cn2p2 + (b2 − λκ ) μ23 cn3p2 + (6(b2 − λκ ) μ0 + 2b1) μ1 μ 2 sn p1 cn p2 + 3(b2 − λκ ) μ12 μ 2 sn2p1 cn p2 + 3(b2 − λκ ) μ1 μ22 sn p1 cn2p2 = 0
(2.31)
and, equating (p1 + 2, 3p1 ; p2 + 2, 2p2) leads to
p1 = 1 and p2 = 2.
(2.32) pi+j
In this, the coefficients of sn
and cn
pi+j
terms to zero, we obtain as above w, B3 and B4 .The soliton solution is 5
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
Fig. 7. The w-shaped optical soliton for Eqs. (2.34) and (2.37).
q (x , t ) = (μ0 + μ1 sn p1 (ξ , l) + μ 2 cn p2 (ξ , l)) eiϕ (x , t ).
(2.33)
In case of l → 1
q (x , t ) = (μ0 + μ1 tan h [B3 (x − vt), l] + μ 2 sec h [B4 (x − vt), l]) eiϕ (x , t ).
(2.34)
and while l → 1
w = −2aB32 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.35)
w = 4aB24 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2.
(2.36)
and
Additionaly,
3(b2 − λκ ) μ0 μ 2 + b1 μ 2 ⎞1/2 −(b2 − λκ ) ⎞1/2 . B3 = ⎛ μ1 and B4 = ⎛ 6a 2a ⎠ ⎝ ⎝ ⎠ ⎜
⎟
(2.37)
In Fig. 7, the w-shaped optical soliton obtained by Eqs. (2.33)–(2.37) is presented for b1 = −5, b2 = −1, μ0 = 5, μ1 = 1, μ2 = −20 and a = 0.01. The contour plots of w-shaped optical soliton are given by Fig. 8.
Fig. 8. The contour plot of combined optical soliton. 6
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
3. Modulation instability In this section, modulation instability investigated of Eq. (1.1). Many nonlinear systems exhibit an instability that emerge of the steady state as a result of an interaction between the nonlinear and dispersive effects [35–37]. We investigate Eq. (1.1) by using the linear stability analysis. Eq. (1.1) is easily solved to obtain
q=
P0 eiη (t ),
η (t ) = (P0 β + γεP02) t
(3.1)
where P0 is the optical power [38,39]. Thus, we consider the development of perturbation with Eq. (3.1)
q (x , t ) = ( P0 + ψ (x , t )) eiη (t ).
(3.2)
In this equation, examine evolution of the perturbation Ψ(x, t) using a linear stability analysis [40]. From this idea, substituting Eq. (3.2) into Eq. (1.1) occurs to the linearized equation (3.3)
i Ψt + a Ψxx − i (α + 3λP0 + 2θP0)Ψx + ΔΨ = 0 where Ω = P0 β +
γεP02
and Δ = iΩ + 2b1 P0
eiη
+ 3b2 P0 . In Eq. (3.3), Ψ(x, t) as given
ψ (x , t ) = ψ1 eiλ + ψ2 e−iλ , λ = wx − kt.
(3.4)
Here, w and k are the wave number and the frequency of perturbation, respectively. By putting Eq. (3.4) into Eq. (3.3) are obtained two homogeneous equations for Ψ1 and Ψ2. From this homogeneous equations, the relation between k and w given as
k = ± aw 2 − w (α + 3λP0 + 2θP0) ± Δ
(3.5)
If the frequency of perturbation is real for all values of the wave number, the soliton solution is stable. 4. Conclusion In this work, we have used the nonlinear Schrödinger equation with quadratic-cubic and Hamiltonian perturbation terms. We have obtained the dark, bright and w-shaped optical solitons solutions by the Jacobi elliptic ansatz method. From in this solutions, we obtained the necessary conditions for the presence of optical solitons. We have plotted the shapes of optical soliton solutions when λ = 2, θ = −3, κ = 0.1 and α = −1. Additionally, we have achieved modulation instability analysis for Eq. (1.1). So, from this analysis, we have obtained equations that express the relation of frequency and wave number of perturbation. References [1] X.B. Wang, S.F. Tian, T.T. Zhang, Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 146 (8) (2018) 3353–3365. [2] S.F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method, J. Differ. Equ. 262 (1) (2017) 506–558. [3] S.F. Tian, T.T. Zhang, Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition, Proc. Amer. Math. Soc. 146 (2018) 1713–1729. [4] S.F. Tian, Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval, Commun. Pure Appl. Anal. 17 (3) (2018) 923–957. [5] L.L. Feng, T.T. Zhang, Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation, Appl. Math. Lett. 78 (2018) 133–140. [6] M. Inc, A.I. Aliyu, A. Yusuf, Soliton and conservation laws to the resonance nonlinear Schrödinger's equation with both spatio-temporal and inter-modal dispersions, Optik 142 (2017) 509–522. [7] Q. Zhou, Dark optical solitons in quadratic nonlinear media with spatio-temporal dispersion, Nonlinear Dyn. 81 (2015) 733–738. [8] Q. Zhou, Q. Zhu, A.H. Bhrawy, L. Moraru, A. Biswas, Optical soliton perturbation with time-and space- dependent dissipation (or gain) and nonlinear dispersion in Kerr and non-Kerr media, Optik 124 (2013) 2368–2372. [9] A. Bansal, A. Biswas, M.F. Mahmood, Q. Zhou, M. Mirzazadeh, et al., Optical soliton perturbation with Radhaskrishnan-Kundu-Lakskmann equation by Lie group analysis, Optik 163 (2017) 137–141. [10] H. Triki, A. Biswas, S.P. Moshokoa, M. Belic, Dipole solitons in optical materials with Kerr law nonlinearity, Optik 128 (2017) 71–76. [11] H. Triki, A. Biswas, Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities, Math. Methods Appl. Sci. 34 (2011) 958–962. [12] K.U. Tariq, A.R. Seadawy, Optical soliton solutions of higher order nonlinear Schrödinger equation in monomode fibers and its applications, Optik 154 (2018) 785–798. [13] A.R. Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, J. Electromagn. Waves Appl. 31 (14) (2017) 1353–1362. [14] A.R. Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions, Optik 139 (2017) 31–43. [15] A.R. Seadawy, D. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability, Results Phys. 7 (2017) 43–48. [16] B. Younas, M. Younis, M.O. Ahmed, S.T.R. Rizvi, Chirped optical solitons in nanofibers, Mod. Phys. Lett. B 32 (26) (2018) 1850320. [17] E.C. Aslan, M. Inc, D. Baleanu, Optical solitons and stability analysis of the NLSE with anti-cubic nonlinearity, Superlattices Microstruct. 109 (2017) 784–793. [18] E.C. Aslan, Solitary solutions of modulation instability analysis of the nonlinear Schrödinger equation with cubic-quartic nonlinearity, J. Adv. Phys. 6 (2017) 579–585. [19] E.C. Aslan, Mustafa Inc, Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis, Waves Random Complex Media (2017), https://doi. org/10.1080/17455030.2017.1286060. [20] K. Ali, S.T.R. Rizvi, A. Khalil, M. Younis, Chirped and dipole soliton in nonlinear negative-index materials, Optik 172 (2018) 657–661. [21] G.L. Alfimov, V.V. Konotop, P. Pacciani, Stationary localized modes of the quintic nonlinear Schrödinger equation with a periodic potential, Phys. Rev. A 75 (2) (2007) 023624.
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8
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical soliton solutions of the NLSE with quadratic-cubicHamiltonian perturbations and modulation instability analysis
T
Ebru Cavlak Aslan, Mustafa Inc Science Faculty, Department of Mathematics, Firat University, 23119 Elazig, Turkey
A R T IC LE I N F O
ABS TRA CT
Keywords: NLSE Jacobi elliptic ansatz method Optical soliton
The aim of this paper is to present optical soliton solutions of the nonlinear Schrödinger equation. Here, the quadratic-cubic nonlinearity in the existence of Hamiltonian perturbations is considered. The Jacobi elliptic ansatz method is applied to obtain the optical soliton solutions. As a result, dark, bright and w-shaped optical soliton are obtained. Additional, the modulation instability analysis is given.
1. Introduction Nonlinear Schrödinger equation (NLSE) is the most famed model in the field of nonlinear science [1–5]. It appears in nonlinear optics and the study of optical solitons have attracted many researchers interest [6–16]. In many studies investigated soliton solutions of NLSE with difference nonlinearities such as anti-cubic [17], cubic-quartic [18], quadratic-cubic [19,20], quintic [21], cubic [22], cubic-quintic [23,24], and other nonlinearities [25–31]. In this work, we analyzed the NLSE with quadratic-cubic nonlinearity in the existence of Hamiltonian perturbation and acquired dark, bright and w- shaped optical solitons by the Jacobi elliptic ansatz method [32,33]. The equation of interest is given by
iq t + aq xx + (b1 |q| + b2 |q|2 ) q = i {αqx + λ (|q|2 q)x + θ (|q|2 )x q}.
(1.1)
Here, the dependent variable q(x, t) represents the wave profile where x and t are spatial and temporal independent variables. The parameter a is the group velocity. The two terms with b1 and b2 are from quadratic and cubic nonlinear terms, respectively. Also, α represents the inter-modal dispersion, λ represents the self-steeping term and finally θ gives the nonlinear dispersion effect. The two terms with b1 and b2 are from quadratic and cubic nonlinear terms, respectively [34]. In this work, the chapters are organized as follows. In Section 2, we have obtained to dark, bright and w-shaped optical solitons by the Jacobi ansatz method. In Section 3 considers the phenomenon of modulation instability. Finally, the conclusion is given in Section 4. Also, the optical solitons obtained for the different states are plotted. 2. Soliton solutions To solve Eq. (1.1) by the Jacobi elliptic functions, the initial assumption is
q (x , t ) = P (x , t ) eiϕ (x , t ), ϕ (x , t ) = −κx + wt + θ
(2.1)
where, κ and w are the frequency and the wave number of soliton and θ is the phase constant. Substituting Eq. (2.1) into Eq. (1.1) and
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.04.008 Received 14 May 2018; Received in revised form 14 February 2019; Accepted 1 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
equating the real and imaginary parts yield
aP′ ′ + (b2 − λκ ) P 3 + b1 P 2 − (w + ακ + aκ 2) P = 0
(2.2)
v + α + 2aκ + (3λ + 2θ) P 2 = 0.
(2.3)
and
In Eq. (2.3), v is the velocity and from Eq. (2.3) as
v = −α − 2aκ
(2.4)
3λ + 2θ = 0.
(2.5)
and
From the real part given by Eq. (2.2) will be investigated optical solitons. Case 1: The form of the soliton solution is
P (x , t ) = μ0 + μ1 sn p1 (ξ , l)
(2.6)
ξ = B1 (x − vt)
(2.7)
with
where B1 is the inverse width of soliton and l is the modulus of the new Jacobi elliptic function. Substituting Eq. (2.6) into Eq. (2.2) yield
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 + aμ1 p1 (p1 − 1) B12 sn p1 − 2 + (−ap12 B12 (l 2 + 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ1 sn p1 + ap1 (p1 + 1) l 2μ1 B12 sn p1 + 2 + (3(b2 − λκ ) μ0 + b1) μ12 sn2p1 + (b2 − λκ ) μ13 sn3p1 = 0.
(2.8)
From Eq. (2.8), equating (p1 + 2, 3p1) leads to
p1 = 1.
(2.9) pi+j
As a result, the coefficients of sn
w = −aB1
(l 2
terms to zero, we obtain
+ 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.10)
or
w = aκ 2 + ακ + b1 μ0 + (b2 − λκ ) μ02 .
(2.11)
−(b2 − λκ ) ⎞1/2 μ1 B1 = ⎛ . 2a ⎠ l ⎝
(2.12)
Also,
Consequently, the soliton solution of Eq. (1.1) is
q (x , t ) = (μ0 + μ1 sn p1 (ξ , l)) eiϕ (x , t ).
(2.13)
In case of l → 1, dark soliton solution is given by
q (x , t ) = (μ0 + μ1 tanh[B1 (x − vt), l]) eiϕ (x , t ).
(2.14)
Furthermore, while l → 1
w=
3 b1 μ0 + 2(b2 − λκ ) μ02 − aB1 2
(2.15)
and
−(b2 − λκ ) ⎞1/2 B1 = ⎛ μ1 2a ⎠ ⎝
(2.16)
The necessary condition for soliton presence is
(b2 − λκ ) a < 0.
(2.17)
In Fig. 1, the dark optical soliton obtained by Eqs. (2.14)–(2.16) are presented for b2 = −0.5, μ0 = 5, μ1 = 10 and a = 1. In Fig. 2, comparison of the dark optical solitons for different values of a and b are presented. Also, the contour plots of dark optical soliton are given by Fig. 3. Case 2: In another Jacobi elliptic function solution is 2
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
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Fig. 1. The dark optical solution for Eqs. (2.14) and (2.16).
Fig. 2. Comparisons of |q(x, t)|2 when t = 0.1, λ = 2,θ = −3 for different values of a and b.
Fig. 3. The contour plot of dark optical soliton.
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P (x , t ) = μ0 + μ 2 cn p2 (ξ , l)
(2.18)
ξ = B2 (x − vt)
(2.19)
with
where B2 is the inverse width of soliton and l is the modulus of the new Jacobi elliptic function. Substituting Eq. (2.18) into Eq. (2.2) yield
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 − aμ 2 p2 (p2 − 1)(l 2 − 1) B22 cn p2 − 2 + (−ap22 B22 (1 − 2l 2) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ 2 cn p2 − ap2 (p2 + 1) l 2μ 2 B22 cn p2 + 2 + (3(b2 − λκ ) μ0 + b1) μ22 cn2p2 + (b2 − λκ ) μ23 cn3p2 = 0.
(2.20)
From here, equating (p2 + 2, 3p2) leads to
p2 = 1.
(2.21) pi+j
Since p2 = 1, the coefficients of cn
w = −a (1 −
2l 2) B22
terms to zero, we obtain
+ 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.22)
and
b − λκ ⎞1/2 μ 2 B2 = ⎛ 2 . l ⎝ 2a ⎠
(2.23)
So, the soliton solution of Eq. (1.1) is
q (x , t ) = (μ0 + μ 2 cn p2 (ξ , l)) eiϕ (x , t ).
(2.24)
In case of l → 1, bright soliton solution is given by
q (x , t ) = (μ0 + μ 2 sec h [B2 (x − vt), l]) eiϕ (x , t ).
(2.25)
From this place, while l → 1
w = aB22 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.26)
b − λκ ⎞1/2 B2 = ⎛ 2 μ2 ⎝ 2a ⎠
(2.27)
and
Similarity to Case 1, the necessary condition for soliton presence is (2.28)
(b2 − λκ ) a > 0.
In Fig. 4, the bright optical soliton obtained by Eqs. (2.25)–(2.27) is presented for b2 = 0.5, μ0 = 5, μ2 = −1 and a = 1. In Fig. 5, comparison of the bright optical solitons for different values of a and b are presented. Also, the contour plots of bright optical soliton are given by Fig. 6. Case 3: To construct the soliton, we have the hypothesis of the form
P (x , t ) = μ0 + μ1 sn p1 (ξ , l) + μ 2 cn p2 (ξ , l)
(2.29)
with
Fig. 4. The bright optical solution for Eqs. (2.25) and (2.27). 4
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
E.C. Aslan
Fig. 5. Comparisons of |q(x, t)|2 when t = 0.1, λ = 2,θ = −3 for different values of a and b.
Fig. 6. The contour plot of bright optical soliton.
ξ = B3 (x − vt) or ξ = B4 (x − vt)
(2.30)
where B3 and B4 are the inverse widths of soliton and l is the modulus of the new Jacobi elliptic function. If we substitute Eq. (2.29) into Eq. (2.2) is
(b2 − λκ ) μ03 + b1 μ02 − (w + ακ + aκ 2) μ0 + (aμ1 p1 (p2 − 1) B32 sn p1 − 2 + (−ap1 (p1 − 1) B32 (l 2 + 1) − ap1B32 (l 2 + 1) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ1 sn p1 + ap1 (p1 + 1) l 2μ1 B32 sn p1 + 2 + (3(b2 − λκ ) μ0 + b1) μ12 sn2p1 + (b2 − λκ ) μ13 sn3p1 + (ap22 B42 (1 − 2l 2) + 3(b2 − λκ ) μ02 + 2b1 μ0 − (w + ακ + aκ 2)) μ 2 cn p2 − ap2 (p2 + 1) l 2μ 2 B42 cn p2 + 2 − aμ 2 p2 (p2 − 1)(l 2 − 1) B42 cn p2 − 2 + (3(b2 − λκ ) μ0 + b1) μ22 cn2p2 + (b2 − λκ ) μ23 cn3p2 + (6(b2 − λκ ) μ0 + 2b1) μ1 μ 2 sn p1 cn p2 + 3(b2 − λκ ) μ12 μ 2 sn2p1 cn p2 + 3(b2 − λκ ) μ1 μ22 sn p1 cn2p2 = 0
(2.31)
and, equating (p1 + 2, 3p1 ; p2 + 2, 2p2) leads to
p1 = 1 and p2 = 2.
(2.32) pi+j
In this, the coefficients of sn
and cn
pi+j
terms to zero, we obtain as above w, B3 and B4 .The soliton solution is 5
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
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Fig. 7. The w-shaped optical soliton for Eqs. (2.34) and (2.37).
q (x , t ) = (μ0 + μ1 sn p1 (ξ , l) + μ 2 cn p2 (ξ , l)) eiϕ (x , t ).
(2.33)
In case of l → 1
q (x , t ) = (μ0 + μ1 tan h [B3 (x − vt), l] + μ 2 sec h [B4 (x − vt), l]) eiϕ (x , t ).
(2.34)
and while l → 1
w = −2aB32 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2
(2.35)
w = 4aB24 + 3(b2 − λκ ) μ02 + 2b1 μ0 − ακ − aκ 2.
(2.36)
and
Additionaly,
3(b2 − λκ ) μ0 μ 2 + b1 μ 2 ⎞1/2 −(b2 − λκ ) ⎞1/2 . B3 = ⎛ μ1 and B4 = ⎛ 6a 2a ⎠ ⎝ ⎝ ⎠ ⎜
⎟
(2.37)
In Fig. 7, the w-shaped optical soliton obtained by Eqs. (2.33)–(2.37) is presented for b1 = −5, b2 = −1, μ0 = 5, μ1 = 1, μ2 = −20 and a = 0.01. The contour plots of w-shaped optical soliton are given by Fig. 8.
Fig. 8. The contour plot of combined optical soliton. 6
Optik - International Journal for Light and Electron Optics 196 (2019) 162661
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3. Modulation instability In this section, modulation instability investigated of Eq. (1.1). Many nonlinear systems exhibit an instability that emerge of the steady state as a result of an interaction between the nonlinear and dispersive effects [35–37]. We investigate Eq. (1.1) by using the linear stability analysis. Eq. (1.1) is easily solved to obtain
q=
P0 eiη (t ),
η (t ) = (P0 β + γεP02) t
(3.1)
where P0 is the optical power [38,39]. Thus, we consider the development of perturbation with Eq. (3.1)
q (x , t ) = ( P0 + ψ (x , t )) eiη (t ).
(3.2)
In this equation, examine evolution of the perturbation Ψ(x, t) using a linear stability analysis [40]. From this idea, substituting Eq. (3.2) into Eq. (1.1) occurs to the linearized equation (3.3)
i Ψt + a Ψxx − i (α + 3λP0 + 2θP0)Ψx + ΔΨ = 0 where Ω = P0 β +
γεP02
and Δ = iΩ + 2b1 P0
eiη
+ 3b2 P0 . In Eq. (3.3), Ψ(x, t) as given
ψ (x , t ) = ψ1 eiλ + ψ2 e−iλ , λ = wx − kt.
(3.4)
Here, w and k are the wave number and the frequency of perturbation, respectively. By putting Eq. (3.4) into Eq. (3.3) are obtained two homogeneous equations for Ψ1 and Ψ2. From this homogeneous equations, the relation between k and w given as
k = ± aw 2 − w (α + 3λP0 + 2θP0) ± Δ
(3.5)
If the frequency of perturbation is real for all values of the wave number, the soliton solution is stable. 4. Conclusion In this work, we have used the nonlinear Schrödinger equation with quadratic-cubic and Hamiltonian perturbation terms. We have obtained the dark, bright and w-shaped optical solitons solutions by the Jacobi elliptic ansatz method. From in this solutions, we obtained the necessary conditions for the presence of optical solitons. We have plotted the shapes of optical soliton solutions when λ = 2, θ = −3, κ = 0.1 and α = −1. Additionally, we have achieved modulation instability analysis for Eq. (1.1). So, from this analysis, we have obtained equations that express the relation of frequency and wave number of perturbation. References [1] X.B. Wang, S.F. Tian, T.T. Zhang, Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 146 (8) (2018) 3353–3365. [2] S.F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method, J. Differ. Equ. 262 (1) (2017) 506–558. [3] S.F. Tian, T.T. Zhang, Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition, Proc. Amer. Math. Soc. 146 (2018) 1713–1729. [4] S.F. Tian, Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval, Commun. Pure Appl. Anal. 17 (3) (2018) 923–957. [5] L.L. Feng, T.T. Zhang, Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation, Appl. Math. Lett. 78 (2018) 133–140. [6] M. Inc, A.I. Aliyu, A. Yusuf, Soliton and conservation laws to the resonance nonlinear Schrödinger's equation with both spatio-temporal and inter-modal dispersions, Optik 142 (2017) 509–522. [7] Q. Zhou, Dark optical solitons in quadratic nonlinear media with spatio-temporal dispersion, Nonlinear Dyn. 81 (2015) 733–738. [8] Q. Zhou, Q. Zhu, A.H. Bhrawy, L. Moraru, A. Biswas, Optical soliton perturbation with time-and space- dependent dissipation (or gain) and nonlinear dispersion in Kerr and non-Kerr media, Optik 124 (2013) 2368–2372. [9] A. Bansal, A. Biswas, M.F. Mahmood, Q. Zhou, M. Mirzazadeh, et al., Optical soliton perturbation with Radhaskrishnan-Kundu-Lakskmann equation by Lie group analysis, Optik 163 (2017) 137–141. [10] H. Triki, A. Biswas, S.P. Moshokoa, M. Belic, Dipole solitons in optical materials with Kerr law nonlinearity, Optik 128 (2017) 71–76. [11] H. Triki, A. Biswas, Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities, Math. Methods Appl. Sci. 34 (2011) 958–962. [12] K.U. Tariq, A.R. Seadawy, Optical soliton solutions of higher order nonlinear Schrödinger equation in monomode fibers and its applications, Optik 154 (2018) 785–798. [13] A.R. Seadawy, Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, J. Electromagn. Waves Appl. 31 (14) (2017) 1353–1362. [14] A.R. Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions, Optik 139 (2017) 31–43. [15] A.R. Seadawy, D. Lu, Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability, Results Phys. 7 (2017) 43–48. [16] B. Younas, M. Younis, M.O. Ahmed, S.T.R. Rizvi, Chirped optical solitons in nanofibers, Mod. Phys. Lett. B 32 (26) (2018) 1850320. [17] E.C. Aslan, M. Inc, D. Baleanu, Optical solitons and stability analysis of the NLSE with anti-cubic nonlinearity, Superlattices Microstruct. 109 (2017) 784–793. [18] E.C. Aslan, Solitary solutions of modulation instability analysis of the nonlinear Schrödinger equation with cubic-quartic nonlinearity, J. Adv. Phys. 6 (2017) 579–585. [19] E.C. Aslan, Mustafa Inc, Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis, Waves Random Complex Media (2017), https://doi. org/10.1080/17455030.2017.1286060. [20] K. Ali, S.T.R. Rizvi, A. Khalil, M. Younis, Chirped and dipole soliton in nonlinear negative-index materials, Optik 172 (2018) 657–661. [21] G.L. Alfimov, V.V. Konotop, P. Pacciani, Stationary localized modes of the quintic nonlinear Schrödinger equation with a periodic potential, Phys. Rev. A 75 (2) (2007) 023624.
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