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On multi-graded-index soliton solutions for the Boussinesq-Burgers equations in optical communications
Optics Communications 384 (2017) 7–10
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
On multi-graded-index soliton solutions for the Boussinesq-Burgers equations in optical communications
crossmark
⁎
H.I. Abdel-Gawad, M. Tantawy
Department of Mathematics, Faculty of Science, Cairo University, Egypt
A R T I C L E I N F O
A BS T RAC T
Keywords: Boussinesq-Burgers equations (BBEs) Traveling wave solutions(TWS) Multi-soliton Optical communications
Very recently, multi-solitary long waves for the homogeneous Boussinesq-Burgers equations (BBEs) were studied. Here its found that the time dependent coefficients (BBEs), shows multi-graded-index solitons waves, which are graded refractive index profile and can offer a new route for high-power lasers and transmission. They should increase data rates in low-cost telecommunications systems. Further, that (BBEs) show long periodic solitons waves in communications and television antennas.
Fk (x, t , ui1, ui2 , …ui1t , ui2 t …) = 0, k , i j , =1, 2…s
1. Introduction In recent years, much research focused on studies of nonautonomous solitons [1–7], breathers soliton [8,9], dissipative-soliton [10– 12], soliton with spatial dispersion [13–16], soliton perturbation in non-Kerr law media [17,18], with kerr law nonlinearity [19–25], chirped soliton [26,27], solitons on a cnoidal wave background [28,29], soliton through the nonlinear barrier [30–32]. The propagation of solitons waves play a fundamental role in optical waves, such as the stability of optical soliton propagation can be well know [33–35]. More recently, many powerful methods [36–38] are used in solitary waves of the nonlinear evolution equations. There are a various methods has been construct exact multi-soliton solutions for the Boussinesq-Burgers equations (BBEs) with constant coefficients, by a wide class of methods [39–46]. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean [47,48]. Also one of the important achievements in propagation of shallow water waves, with timedependent coefficients are given by the (BBEs) equations:
ut − vx + α (t ) uux = 0, vt + δ (t )(uv )x − γ (t ) uxxx = 0,
(1)
where u (x, t ) and v (x, t ) represent the horizontal velocity field and the height of the water surface above a horizontal level at the bottom. The coefficients α (t ) and δ (t ) are the nonlinearity gain (or loss ). γ (t ) is refer to the strength of fluid dispersion, γ (t ) < 0 . The Eq. (1) is have been used as the models to describe physical phenomena in portraying in nonuniform media with time dependent density gradients. Now, we consider the coupled evolution equations:
where Fk are polynomials in their argument. When x, y and t are missing in the Eqs. (2), then it has traveling wave solutions (TWS), we have
Gk (Ui1, Ui2, …Ui1 ′, Ui2 ′, …Ui1 ″, Ui2 ″…) = 0, k , i j = 1, 2, …sU ′ = η = κx +
dUi , dη
∫ ω (t ) dt.
(3)
Here, we use the unified method UM [49], to find the (TWS). By this method are classified to be polynomials or rational functions in an auxiliary function (namely φ (η)) that satisfies an appropriate auxiliary equation. By the notion of exact solubility of real valued partial differential equations (PDEs) of order k here mean that there exists of least one solution with k-free parameters. 2. The generalizing unified method In the present paper, we use the UM [49]. By this method are classified to be polynomials or rational functions solutions, which follow as: (i) Polynomial Solutions . To search for solutions of Eq. (1), the unified method suggests that the single (TWS) is given by nj
u j (η , t ) =
Corresponding author. E-mail address: [email protected] (M. Tantawy).
http://dx.doi.org/10.1016/j.optcom.2016.09.064 Received 8 August 2016; Received in revised form 28 September 2016; Accepted 29 September 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
j = pk
∑ ai j (t ) φi (η), j = 1, 2, …s (φ′(η, t )) pk = ∑ i =0
p = 1, 2,
⁎
(2)
cj (t ) φ j (η),
j =0
(4)
Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
3. Solutions of the(BBEs)
where φ (η) is the auxiliary function, and ai j (t ), cj (t ) are unknown parameters. In the Eq. (4) for nj and k is detrermined from the leading analysis and in this case the balance condition in the first and second Eq. (1) is reads n1 = (k − 1), and n2 = 2(k − 1) respectively. The consistency condition relates the number of equations obtained by substituting the Eq. (4) into (3), (namely n 2j ), the number of free parameters in polynomial and auxiliary functions (namely n1j ) and the integrability property of Eq. (3). As the Eq. (1) is completely integrable, then consistency condition reads n 2j − n1j ≤ mj . We mention that, when p=1, the solution of the auxiliary equation gives rise to (explicit or implicit) solutions in elementary functions. While when p=2 they give rise to explicit solutions in Jacobi-elliptic or periodic. (ii) Rational Solutions . To get the rational solutions of Eq. (3) we assume that n
u j (η , t ) =
The purpose of this section is to investigates the different geometrical structures of multi-soliton via rational solutions of the Eq. (1), and to the aim that the balance condition between the leading terms to exist, we use the transformation v (x, t ) = −ux (x, t ). When substituting by this condition into Eq. (1), then the first Eq. (1) (or in the second equation in (1) and integration with neglecting the constant of integration ) [50,51], reduce to:
3.1. Single-soliton
r
∑ pi j (t ) φi (η)/ ∑ qi j (t ) φi (η), j = 1, 2, i =0
Here, we find the exact (TWS) of Eq. (1) by using the UM. The propagation on (TWS) under more than varying dispersion or nonlinearity coefficients. Let us present, we use the transformations u (x, t ) = u (η1), and η1 = κ1 x + ∫ ω1 (t ) dt , where κ1−1 and ω1 (t ) are designate the characteristic wave lengths and frequency. Thus Eq. (8) becomes BBEs is becomes:
i =0 i = pk
∑ ci (t ) φi (η), p = 1, 2,
…s (φ′(η, t )) pk =
(5)
i =0
where pi j (t ), qi j (t ) are arbitrary parameters, n , r and k are detrermined from the leading analysis. It is worth to mention that the balance conditions in this case be obtained as in the case of polynomial solutions but n is replaced by n − r . Here a gain, the condition for the existence of the solutions in Eq. (3) is determined from the consistency equation. Indeed, when k=1 in the solution of the second equation in (5) was suggested to describe “a jet stream” or (wave pattern). Multi-waves solutions are found bythe generalizing unified method (GUM) that is by accounting for multi-auxiliary functions (with multiauxiliary equations ). In this case, solution are also given polynomial solutions or rational solutions. We bear in mind that, multi-wave polynomial solutions describe direct nonlinear interactions (DNLI) of basic waves, which are the solutions of the auxiliary equations. While multi-waves rational solutions describe indirect nonlinear interactions (IDNLI) of multi-waves (or wave patterns). The (GUM) suggests to get N-nonautonomous solitons (or N-waves) solutions in the auxiliary functions φi;i = 1, ⋯N + q − 1, where φi satisfy appropriate auxiliary equations. Here, we focus our attention to find N-nonautonomous solitons via rational solutions, namely.
u (φ1, φ2, …, φ) = PN (φi )/ QN (φi ),
ω1 (t ) u η1 − κ1 δ (t ) uu η1 − κ12 γ (t ) u η1η1 = 0,
∑
PN (φi ) = ai0 (t ) +
i1=1
u (η1, t ) = ( p1 (t ) φ (η1) + p0 (t ))/(q1 (t ) φ (η1) + q0 (t )), φη1 = c0 (t ) + c1 (t ) φ (η1),
u (x, t ) = (2κ1 c1 (t ) ρ (t )exp[λ (x, t )])/ a (c1 (t )(q1 (t )exp[λ (x, t )] + q0 (t )) − c0 (t ) q1 (t )), v (x, t ) = −ux (x, t ), v (x, t ) = 2κ12 c13 (t ) q1 (t ) ρ (t )exp[λ (x, t )]/ a (c1 (t )(c1 (t )(q1 (t )exp[λ (x, t )] + q0 (t )) − c0 (t ) q1 (t ))2 , λ (x, t ) = κ1 c1 (t )(x + κ
(6)
∑
ai2 (t ) φi1 φi2 N
∑
ai3 (t ) φi1 φi2 φi3+⋯+
i1≠ i2 ≠ i3
∑
∫ c1 (t ) γ (t ) dt, ρ (t ) = c0 (t ) q1 (t ) − c1 (t ) q0 (t ),
(11)
where qi (t ), and cj (t ), i , j = 0, 1 are arbitrary parameters. Under this condition, which is always supposed to be true in the similar circumstances thereafter, we have δ (t ) = a γ (t ), where a is arbitrary constant and ωj (t ) = κ j2 cj (t ) γ (t ), j = 1, 2, 3, for single, two and there solutions respectively.
i1≠ i2
N + q −1
+
(10)
by substituting from Eq. (10) into (9), we find that the solutions are
N + q −1
ai1 (t ) φi1 +
(9)
when p=1. By taking n=r (when k=1) and by using Eq. (5), the solution of Eq. (9) has the form
where PN is given by: N + q −1
(8)
ut + δ (t ) uux − γ (t ) uxx = 0.
ai(N −1) (t ) φi1 φi2 φi3 ⋯
i1≠ i2 ≠ i3≠⋯ i (N −1) N
+ aiN (t )
∏ φi , φi;′ = c0j (t ) + cj (t ) φi i =1
(7)
3.2. Two-soliton
By the same way the equation for QN (φi ) is found. In the case of the (1+1)-dimensional in the Eq. (7) , we haveq=1. Steps of computation: When substituting from Eq. (4) (or (5)) into the Eq. (3), we get the principle equations and the following steps are done.
For two-soliton solutions, we write transformation u (x, t ) = u (η1, η2 ), ηj = κj x + ∫ ωj (t ) dt and j = 1, 2 . By taking N = 2, q = 1, into the Eq. (6) , the solution of Eq. (8) with auxiliary equations is 2
1. 2. 3. 4.
Solving the principle equations. Solving the auxiliary equations. Finding the exact solution. We check that the solutions obtained satisfies the Eq. (3).
u (η1, η2 ) = PN (φi )/ QN (φi ), PN (φi ) = a 0 (t ) +
∑ ai (t ) φi + a3 (t ) φ1 φ2 φ1 ′(η1, i =1
t ) = c1 (t ) φ1 (η1) + c01 (t ), φ2 (η2 , t ) = c2 (t ) φ2 (η2 ) + c02 (t ),
(12)
and by same way QN (φi ) is given as in PN (φi ), but ai (t ) → qi (t ). By using a computer algebra system or otherwise, we find the solution Eq. (8) are
For convenience, we confine ourselves to find the solutions for one and multi-solution in the form of rational functions. 8
Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
In this case, we find that the we find the solution Eq. (8) are
u (x, t ) = P2 (φ1,2 )/ Q2 (φ1,2 ), v (x, t ) = ψ1i (x , t )/Q22 (φ1,2 ), i = 1, 2v (x, t ) = −ux (x, t ), P2 (φ1,2 ) = −2c1 (t ) c2 (t )(κ1 ρ1 (t )exp[λ2 (x,
u (x, t ) = P3 (φ1,2,3)/ Q3 (φ1,2,3),
t )] + κ2 ρ2 (t )exp[λ1 (x, t )]) Q2 (φ1,2 ) = ac1 (t ) ρ2 (t )exp[λ1 (x,
P3 (φ1,2,3) = −2c1 (t ) c2 (t ) c3 (t )(κ1 ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + exp[λ1 (x,
t )] + ac2 (t )exp[λ2 (x, t )](c1 (t )(q2 (t ) + q3 (t )exp[λ1 (x,
t )](κ2 ρ4 (t )exp[λ3 (x, t )] − κ3 ρ5 (t )exp[λ2 (x, t )]),
t )]) − c01 (t ) q3 (t )) ψ11 (x, t ) = −2c1 (t ) c2 (t )exp[λ1 (x, t ) + λ2 (x,
Q3 (φ1,2,3) = ac2 (t ) c3 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + ac1 (t )exp[λ1 (x,
t )](−κ22 c01 (t ) c2 (t )2 q3 (t ) + κ2 ρ2 (t ) c1 (t ) c2 (t )(2κ1 c01 (t ) q3 (t )
t )](c2 (t )exp[λ2 (x, t )](ρ5 (t ) + c3 (t ) q7 (t )exp[λ3 (x, t )]) c3 (t ) ρ4 (t )exp[λ3 (x,
+ κ2 c2 (t )(q2 (t ) q3 (t )exp[λ1 (x,
(16)
t )]),
t )])) + κ12 c13 (t ) q2 (t )(− c02 (t ) q3 (t ) + c2 (t )(q1 (t ) + q3 (t )exp[λ2 (x,
and
t )] )) + κ1 c12 (t )(−κ1 c01 (t ) c2 (t ) q3 (t )2 exp[λ2 (x, t )]) + (2κ2 c2 (t ) q2 (t ) + κ1 c01 (t ) q3 (t ))(−ρ2 (t )))) ψ12 (x ,
v = − u x , v (x , t ) =
t ) = a (c10 (t ) c2 (t ) q3 (t )exp[λ2 (x , t )]) + c1 (t )(c02 (t ) q3 (t )
ψ11 (x, t ) ψ12 (x, t ) + ψ21 (x, t ) ψ22 (x, t ) Q32 (φ1,2,3)
,
ψ11 (x, t ) = (ac2 (t ) c3 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + ac1 (t )exp[λ1 (x,
exp[λ1 (x , t )] q1 (t ) + exp[λ2 (x , t )](q2 (t ) + q3 (t )exp[λ1 (x , t )]))))2 ,
t )](c2 (t )exp[λ2 (x, t )](c3 (t ) q4 (t ) + q7 (t )(c3 (t )exp[λ3 (x,
(13)
t )] − c30 (t ))) + c3 (t ) ρ4 (t )exp[λ3 (x, t )])),
where
ψ12 (x, t ) = (−κ1 σ23 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + exp[λ1 (x,
ρ1 (t ) = c2 (t ) q2 (t ) − c01 (t ) q3 (t ), ρ2 (t ) = c2 (t ) q1 (t ) − c02 (t ) q3 (t ),
t )](κ3 σ23 (t ) ρ5 (t )exp[λ2 (x, t )] − κ2 σ13 (t ) ρ4 (t )exp[λ2 (x, t ) + λ3 (x, t )] ))
λj (x, t ) = κj cj (t )(x + κj
∫ cj (t ) dt, ), j = 1, 2.
ψ21 (x, t ) = exp[λ1 (x , t )](κ3 ρ5 (t )exp[λ2 (x , t )] − κ 4 ρ4 (t )exp[λ3 (x,
(14)
t )]) − κ1 ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] ψ22 (x, t ) = c3 (t )exp[λ3 (x,
And qi (t ), cj (t ) and c0j (t ), i , =0, 1, 2, 3, j = 1, 2 are arbitrary parameters. In this figures c0j (t ) = 1, j = 0, 1, 2 and qi (t ) = 1, i = 0, 1, 2, 3. These (Fig. 1 a and b) shows the graded-index for two-solitons to interconnect optical fibers and the long periodic zigzag (or antenna with a wide band of frequencies), for soliton similariton pairs [52,53] for the height wave v (x, t ).
t )](c2 (t ) σ23 (t ) ρ3 (t )exp[λ2 (x, t )] + c1 (t )exp[λ1 (x, t )](c2 (t ) q7 (t ) σ13 (t )exp[λ2 (x, t )] + σ12 (t ) ρ4 (t ))) − c1 (t ) c2 (t ) σ13 (t ) ρ4 (t )exp[λ1 (x, t ) + λ2 (x, t )],
(17)
where
ωj (t ) = κ j2 cj (t ) γ (t ), j = 1, 2, 3.ρ3 (t ) = c1 (t ) q5 (t ) − c01 (t ) q7 (t ),
3.3. Three-soliton
ρ4 (t ) = c2 (t ) q6 (t ) − c02 (t ) q7 (t ) ρ3 (t ) = c03 (t ) q7 (t ) − c3 (t ) q4 (t ),
In this case, we take N = 3, q = 1, into the Eq. (6), the solution of Eq. (8) with auxiliary equations is
λj (x, t ) = κj cj (t )(x + κj i ≠ j,
∫ cj (t ) dt ) σij (t ) = κi ci (t ) + κj cj (t ), i, j = 1, 2, 3, (18)
u (η1, η2 , η3) = PN (φi )/ QN (φi ), 3
PN (φi ) = a 0 (t ) +
and we take qi (t ) = 1, i = 4, 6, 7, but a7 and κi, i = 1, 2, 3 are arbitrary parameters. In this figures qi (t ) = 1, i = 4, 5, 6, 7 and c0j (t ) = 1, j = 1, 2, 3. These (Fig. 2 a and b) shows multi-graded-index and periodic zigzag propagations for the three- soliton collisions at v (x, t ).
∑ ai (t ) φi + a4 (t ) φ1 φ2 + a5 (t ) φ2 φ3 + a6 (t ) φ1 φ3 i =1
+ +a7 (t ) φ1 φ2 φ3, φηj = c0j (t ) + cj (t ) φi , i , j = 1, 2, 3. (15)
Fig. 1. a and b, the solutions of Eq. (13) are displayed against x and t. (a) c1 (t ) = 2, c2 (t ) = 5.6, γ (t ) = sn (t , 0.5) + tanh(t ), κ1 = 0.5, κ 2 = 0.4 and a = −1 and (b) c1 (t ) = 3.5, c2 (t ) = 4.5, γ (t ) = cos(t ) + sech (t ), κ1 = −0.5, κ 2 = 0.4 and a = −1.
9
Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
Fig. 2. a and b the solutions of Eq. (17) are displayed against x and t. (a) c1 (t ) = −10, c2 (t ) = 5, c3 (t ) = 4, γ (t ) = sn (t , 0.5) + tanh(t ), κ1 = −.35, κ 2 = 0.4, κ3 = 0.25 , and a = −1. (b) c1 (t ) = 8.5, c2 (t ) = 5, c3 (t ) = 3.2γ (t ) = cos(t ) + sech (t ), κ1 = −0.35, κ 2 = 0.3, κ 2 = 0.48 and a = −1. Optoelectron. Adv. Mater. 9 (2015) 384. [23] M. Savescu, A.A. Alshaery, E.M. Hilal, A.H. Bhrawy, Q. Zhou, A. Biswas, Optoelectron. Adv. Mater. 9 (2015) 14. [24] Q. Zhou, L. Liu, H. Zhang, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad, S. Moshokoa, A. Biswas, Opt. Appl. 46 (1) (2016) 79. [25] M. Mirzazadeh, M. Ekici, A. Sonomezoglu, M. Eslami, Q. Zhou, A.H. Kara, D. Milovic, F.B. Majid, A. Biswas, M. Belic, Nonlinear Dyn. 85 (3) (2016) 1979. [26] J. Wang, L. Li, S. Jia, Opt. Commun. 274 (2007) 223. [27] H. Kumar, F. Chand, Optik 125 (2014) 2949. [28] R. Murali, K. Senthilnathan, K. Porsezian, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 025401. [29] C.-Q. Dai, Y.-J. Xu, Opt. Commun. 311 (2013) 216. [30] C.-Q. Dai, R. Chen, J. Zhang, Chaos Solitons Fractals 44 (2011) 862. [31] M.S.M. Rajan, A. Mahalingam, A. Uthayakumar, K. Porsezian, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1410. [32] G. Yang, R. Hao, L. Li, Z. Li, G. Zhou, Opt. Commun. 260 (2006) 282. [33] J.-D. He, J. Zhang, Y. Zhang, C.-Q. Dai, Opt. Commun. 285 (2012) 755. [34] Y. Zheng, Y. Meng, Y. Liu, Opt. Commun. 315 (2014) 63. [35] J. Yang, D.J. Kaup, SIAM J. Appl. Math. 60 (2000) 967. [36] H.M. Baskonus, Nonlinear Dyn. 86 (1) (2016) 77. [37] H. Bulut, Abstr. Appl. Anal. 2013 (2013) 742643. [38] H.M. Baskonus, H. Bulut, Waves Random Complex 25 (4) (2015) 720. [39] X. Li, A. Chen, Phys. Lett. A 342 (2005) 413. [40] A. Chen, X. Li, Chaos Solitons Fractals 27 (2006) 43. [41] J. Mei, M. Zhangyun, Appl. Math. Comput. 219 (2013) 6163. [42] A.S. Abdel Rady, E.S. Osman, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1172. [43] P. Wang, W.-J. Bo Tian, X.L. Liu, Y. Jiang, Appl. Math. Comput. 218 (2011) 1726. [44] H.-Yi Wang, Appl. Math. Lett. 38 (2014) 100. [45] C.-C. Zhang, A.-H. Chen, Appl. Math. Lett. 58 (2016) 133. [46] G. Ebadi, N. Yousefzadeh, H. Triki, A. Yildirim, A. Biswas, Rom. Rep. Phys. 64 (4) (2012) 915. [47] D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450 (2007) 1054. [48] S. Birkholz, C. Bre, A. Demircan, G. Steinmeyer, Phys. Rev. Lett. 114 (2015) 213901. [49] H.I. Abdel-Gawad, J. Stat. Phys. 147 (2012) 506. [50] N.A. Kudryashov, M.B. Soukharev, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1765. [51] A.S. Abdel Rady, E.S. Osman, M. Khalfallah, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1172. [52] C.-Q. Dai, Y.J. Xu, R. Chen, S.Q. Zhu, Eur. Phys. J. D 59 (2010) 457. [53] C.-Q. Dai, Y. Wang, J. Zhang, Opt. Lett. 35 (2010) 1437.
4. Conclusion Its found that the (BBs) equations, with time-dependent coefficients leads to may useful interesting waves in science and communications namely optical fibers and long periodic waves for the telecommunications systems. Thus we think that the present in rich and may explore further new waves. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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10
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
On multi-graded-index soliton solutions for the Boussinesq-Burgers equations in optical communications
crossmark
⁎
H.I. Abdel-Gawad, M. Tantawy
Department of Mathematics, Faculty of Science, Cairo University, Egypt
A R T I C L E I N F O
A BS T RAC T
Keywords: Boussinesq-Burgers equations (BBEs) Traveling wave solutions(TWS) Multi-soliton Optical communications
Very recently, multi-solitary long waves for the homogeneous Boussinesq-Burgers equations (BBEs) were studied. Here its found that the time dependent coefficients (BBEs), shows multi-graded-index solitons waves, which are graded refractive index profile and can offer a new route for high-power lasers and transmission. They should increase data rates in low-cost telecommunications systems. Further, that (BBEs) show long periodic solitons waves in communications and television antennas.
Fk (x, t , ui1, ui2 , …ui1t , ui2 t …) = 0, k , i j , =1, 2…s
1. Introduction In recent years, much research focused on studies of nonautonomous solitons [1–7], breathers soliton [8,9], dissipative-soliton [10– 12], soliton with spatial dispersion [13–16], soliton perturbation in non-Kerr law media [17,18], with kerr law nonlinearity [19–25], chirped soliton [26,27], solitons on a cnoidal wave background [28,29], soliton through the nonlinear barrier [30–32]. The propagation of solitons waves play a fundamental role in optical waves, such as the stability of optical soliton propagation can be well know [33–35]. More recently, many powerful methods [36–38] are used in solitary waves of the nonlinear evolution equations. There are a various methods has been construct exact multi-soliton solutions for the Boussinesq-Burgers equations (BBEs) with constant coefficients, by a wide class of methods [39–46]. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean [47,48]. Also one of the important achievements in propagation of shallow water waves, with timedependent coefficients are given by the (BBEs) equations:
ut − vx + α (t ) uux = 0, vt + δ (t )(uv )x − γ (t ) uxxx = 0,
(1)
where u (x, t ) and v (x, t ) represent the horizontal velocity field and the height of the water surface above a horizontal level at the bottom. The coefficients α (t ) and δ (t ) are the nonlinearity gain (or loss ). γ (t ) is refer to the strength of fluid dispersion, γ (t ) < 0 . The Eq. (1) is have been used as the models to describe physical phenomena in portraying in nonuniform media with time dependent density gradients. Now, we consider the coupled evolution equations:
where Fk are polynomials in their argument. When x, y and t are missing in the Eqs. (2), then it has traveling wave solutions (TWS), we have
Gk (Ui1, Ui2, …Ui1 ′, Ui2 ′, …Ui1 ″, Ui2 ″…) = 0, k , i j = 1, 2, …sU ′ = η = κx +
dUi , dη
∫ ω (t ) dt.
(3)
Here, we use the unified method UM [49], to find the (TWS). By this method are classified to be polynomials or rational functions in an auxiliary function (namely φ (η)) that satisfies an appropriate auxiliary equation. By the notion of exact solubility of real valued partial differential equations (PDEs) of order k here mean that there exists of least one solution with k-free parameters. 2. The generalizing unified method In the present paper, we use the UM [49]. By this method are classified to be polynomials or rational functions solutions, which follow as: (i) Polynomial Solutions . To search for solutions of Eq. (1), the unified method suggests that the single (TWS) is given by nj
u j (η , t ) =
Corresponding author. E-mail address: [email protected] (M. Tantawy).
http://dx.doi.org/10.1016/j.optcom.2016.09.064 Received 8 August 2016; Received in revised form 28 September 2016; Accepted 29 September 2016 0030-4018/ © 2016 Elsevier B.V. All rights reserved.
j = pk
∑ ai j (t ) φi (η), j = 1, 2, …s (φ′(η, t )) pk = ∑ i =0
p = 1, 2,
⁎
(2)
cj (t ) φ j (η),
j =0
(4)
Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
3. Solutions of the(BBEs)
where φ (η) is the auxiliary function, and ai j (t ), cj (t ) are unknown parameters. In the Eq. (4) for nj and k is detrermined from the leading analysis and in this case the balance condition in the first and second Eq. (1) is reads n1 = (k − 1), and n2 = 2(k − 1) respectively. The consistency condition relates the number of equations obtained by substituting the Eq. (4) into (3), (namely n 2j ), the number of free parameters in polynomial and auxiliary functions (namely n1j ) and the integrability property of Eq. (3). As the Eq. (1) is completely integrable, then consistency condition reads n 2j − n1j ≤ mj . We mention that, when p=1, the solution of the auxiliary equation gives rise to (explicit or implicit) solutions in elementary functions. While when p=2 they give rise to explicit solutions in Jacobi-elliptic or periodic. (ii) Rational Solutions . To get the rational solutions of Eq. (3) we assume that n
u j (η , t ) =
The purpose of this section is to investigates the different geometrical structures of multi-soliton via rational solutions of the Eq. (1), and to the aim that the balance condition between the leading terms to exist, we use the transformation v (x, t ) = −ux (x, t ). When substituting by this condition into Eq. (1), then the first Eq. (1) (or in the second equation in (1) and integration with neglecting the constant of integration ) [50,51], reduce to:
3.1. Single-soliton
r
∑ pi j (t ) φi (η)/ ∑ qi j (t ) φi (η), j = 1, 2, i =0
Here, we find the exact (TWS) of Eq. (1) by using the UM. The propagation on (TWS) under more than varying dispersion or nonlinearity coefficients. Let us present, we use the transformations u (x, t ) = u (η1), and η1 = κ1 x + ∫ ω1 (t ) dt , where κ1−1 and ω1 (t ) are designate the characteristic wave lengths and frequency. Thus Eq. (8) becomes BBEs is becomes:
i =0 i = pk
∑ ci (t ) φi (η), p = 1, 2,
…s (φ′(η, t )) pk =
(5)
i =0
where pi j (t ), qi j (t ) are arbitrary parameters, n , r and k are detrermined from the leading analysis. It is worth to mention that the balance conditions in this case be obtained as in the case of polynomial solutions but n is replaced by n − r . Here a gain, the condition for the existence of the solutions in Eq. (3) is determined from the consistency equation. Indeed, when k=1 in the solution of the second equation in (5) was suggested to describe “a jet stream” or (wave pattern). Multi-waves solutions are found bythe generalizing unified method (GUM) that is by accounting for multi-auxiliary functions (with multiauxiliary equations ). In this case, solution are also given polynomial solutions or rational solutions. We bear in mind that, multi-wave polynomial solutions describe direct nonlinear interactions (DNLI) of basic waves, which are the solutions of the auxiliary equations. While multi-waves rational solutions describe indirect nonlinear interactions (IDNLI) of multi-waves (or wave patterns). The (GUM) suggests to get N-nonautonomous solitons (or N-waves) solutions in the auxiliary functions φi;i = 1, ⋯N + q − 1, where φi satisfy appropriate auxiliary equations. Here, we focus our attention to find N-nonautonomous solitons via rational solutions, namely.
u (φ1, φ2, …, φ) = PN (φi )/ QN (φi ),
ω1 (t ) u η1 − κ1 δ (t ) uu η1 − κ12 γ (t ) u η1η1 = 0,
∑
PN (φi ) = ai0 (t ) +
i1=1
u (η1, t ) = ( p1 (t ) φ (η1) + p0 (t ))/(q1 (t ) φ (η1) + q0 (t )), φη1 = c0 (t ) + c1 (t ) φ (η1),
u (x, t ) = (2κ1 c1 (t ) ρ (t )exp[λ (x, t )])/ a (c1 (t )(q1 (t )exp[λ (x, t )] + q0 (t )) − c0 (t ) q1 (t )), v (x, t ) = −ux (x, t ), v (x, t ) = 2κ12 c13 (t ) q1 (t ) ρ (t )exp[λ (x, t )]/ a (c1 (t )(c1 (t )(q1 (t )exp[λ (x, t )] + q0 (t )) − c0 (t ) q1 (t ))2 , λ (x, t ) = κ1 c1 (t )(x + κ
(6)
∑
ai2 (t ) φi1 φi2 N
∑
ai3 (t ) φi1 φi2 φi3+⋯+
i1≠ i2 ≠ i3
∑
∫ c1 (t ) γ (t ) dt, ρ (t ) = c0 (t ) q1 (t ) − c1 (t ) q0 (t ),
(11)
where qi (t ), and cj (t ), i , j = 0, 1 are arbitrary parameters. Under this condition, which is always supposed to be true in the similar circumstances thereafter, we have δ (t ) = a γ (t ), where a is arbitrary constant and ωj (t ) = κ j2 cj (t ) γ (t ), j = 1, 2, 3, for single, two and there solutions respectively.
i1≠ i2
N + q −1
+
(10)
by substituting from Eq. (10) into (9), we find that the solutions are
N + q −1
ai1 (t ) φi1 +
(9)
when p=1. By taking n=r (when k=1) and by using Eq. (5), the solution of Eq. (9) has the form
where PN is given by: N + q −1
(8)
ut + δ (t ) uux − γ (t ) uxx = 0.
ai(N −1) (t ) φi1 φi2 φi3 ⋯
i1≠ i2 ≠ i3≠⋯ i (N −1) N
+ aiN (t )
∏ φi , φi;′ = c0j (t ) + cj (t ) φi i =1
(7)
3.2. Two-soliton
By the same way the equation for QN (φi ) is found. In the case of the (1+1)-dimensional in the Eq. (7) , we haveq=1. Steps of computation: When substituting from Eq. (4) (or (5)) into the Eq. (3), we get the principle equations and the following steps are done.
For two-soliton solutions, we write transformation u (x, t ) = u (η1, η2 ), ηj = κj x + ∫ ωj (t ) dt and j = 1, 2 . By taking N = 2, q = 1, into the Eq. (6) , the solution of Eq. (8) with auxiliary equations is 2
1. 2. 3. 4.
Solving the principle equations. Solving the auxiliary equations. Finding the exact solution. We check that the solutions obtained satisfies the Eq. (3).
u (η1, η2 ) = PN (φi )/ QN (φi ), PN (φi ) = a 0 (t ) +
∑ ai (t ) φi + a3 (t ) φ1 φ2 φ1 ′(η1, i =1
t ) = c1 (t ) φ1 (η1) + c01 (t ), φ2 (η2 , t ) = c2 (t ) φ2 (η2 ) + c02 (t ),
(12)
and by same way QN (φi ) is given as in PN (φi ), but ai (t ) → qi (t ). By using a computer algebra system or otherwise, we find the solution Eq. (8) are
For convenience, we confine ourselves to find the solutions for one and multi-solution in the form of rational functions. 8
Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
In this case, we find that the we find the solution Eq. (8) are
u (x, t ) = P2 (φ1,2 )/ Q2 (φ1,2 ), v (x, t ) = ψ1i (x , t )/Q22 (φ1,2 ), i = 1, 2v (x, t ) = −ux (x, t ), P2 (φ1,2 ) = −2c1 (t ) c2 (t )(κ1 ρ1 (t )exp[λ2 (x,
u (x, t ) = P3 (φ1,2,3)/ Q3 (φ1,2,3),
t )] + κ2 ρ2 (t )exp[λ1 (x, t )]) Q2 (φ1,2 ) = ac1 (t ) ρ2 (t )exp[λ1 (x,
P3 (φ1,2,3) = −2c1 (t ) c2 (t ) c3 (t )(κ1 ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + exp[λ1 (x,
t )] + ac2 (t )exp[λ2 (x, t )](c1 (t )(q2 (t ) + q3 (t )exp[λ1 (x,
t )](κ2 ρ4 (t )exp[λ3 (x, t )] − κ3 ρ5 (t )exp[λ2 (x, t )]),
t )]) − c01 (t ) q3 (t )) ψ11 (x, t ) = −2c1 (t ) c2 (t )exp[λ1 (x, t ) + λ2 (x,
Q3 (φ1,2,3) = ac2 (t ) c3 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + ac1 (t )exp[λ1 (x,
t )](−κ22 c01 (t ) c2 (t )2 q3 (t ) + κ2 ρ2 (t ) c1 (t ) c2 (t )(2κ1 c01 (t ) q3 (t )
t )](c2 (t )exp[λ2 (x, t )](ρ5 (t ) + c3 (t ) q7 (t )exp[λ3 (x, t )]) c3 (t ) ρ4 (t )exp[λ3 (x,
+ κ2 c2 (t )(q2 (t ) q3 (t )exp[λ1 (x,
(16)
t )]),
t )])) + κ12 c13 (t ) q2 (t )(− c02 (t ) q3 (t ) + c2 (t )(q1 (t ) + q3 (t )exp[λ2 (x,
and
t )] )) + κ1 c12 (t )(−κ1 c01 (t ) c2 (t ) q3 (t )2 exp[λ2 (x, t )]) + (2κ2 c2 (t ) q2 (t ) + κ1 c01 (t ) q3 (t ))(−ρ2 (t )))) ψ12 (x ,
v = − u x , v (x , t ) =
t ) = a (c10 (t ) c2 (t ) q3 (t )exp[λ2 (x , t )]) + c1 (t )(c02 (t ) q3 (t )
ψ11 (x, t ) ψ12 (x, t ) + ψ21 (x, t ) ψ22 (x, t ) Q32 (φ1,2,3)
,
ψ11 (x, t ) = (ac2 (t ) c3 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + ac1 (t )exp[λ1 (x,
exp[λ1 (x , t )] q1 (t ) + exp[λ2 (x , t )](q2 (t ) + q3 (t )exp[λ1 (x , t )]))))2 ,
t )](c2 (t )exp[λ2 (x, t )](c3 (t ) q4 (t ) + q7 (t )(c3 (t )exp[λ3 (x,
(13)
t )] − c30 (t ))) + c3 (t ) ρ4 (t )exp[λ3 (x, t )])),
where
ψ12 (x, t ) = (−κ1 σ23 (t ) ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] + exp[λ1 (x,
ρ1 (t ) = c2 (t ) q2 (t ) − c01 (t ) q3 (t ), ρ2 (t ) = c2 (t ) q1 (t ) − c02 (t ) q3 (t ),
t )](κ3 σ23 (t ) ρ5 (t )exp[λ2 (x, t )] − κ2 σ13 (t ) ρ4 (t )exp[λ2 (x, t ) + λ3 (x, t )] ))
λj (x, t ) = κj cj (t )(x + κj
∫ cj (t ) dt, ), j = 1, 2.
ψ21 (x, t ) = exp[λ1 (x , t )](κ3 ρ5 (t )exp[λ2 (x , t )] − κ 4 ρ4 (t )exp[λ3 (x,
(14)
t )]) − κ1 ρ3 (t )exp[λ2 (x, t ) + λ3 (x, t )] ψ22 (x, t ) = c3 (t )exp[λ3 (x,
And qi (t ), cj (t ) and c0j (t ), i , =0, 1, 2, 3, j = 1, 2 are arbitrary parameters. In this figures c0j (t ) = 1, j = 0, 1, 2 and qi (t ) = 1, i = 0, 1, 2, 3. These (Fig. 1 a and b) shows the graded-index for two-solitons to interconnect optical fibers and the long periodic zigzag (or antenna with a wide band of frequencies), for soliton similariton pairs [52,53] for the height wave v (x, t ).
t )](c2 (t ) σ23 (t ) ρ3 (t )exp[λ2 (x, t )] + c1 (t )exp[λ1 (x, t )](c2 (t ) q7 (t ) σ13 (t )exp[λ2 (x, t )] + σ12 (t ) ρ4 (t ))) − c1 (t ) c2 (t ) σ13 (t ) ρ4 (t )exp[λ1 (x, t ) + λ2 (x, t )],
(17)
where
ωj (t ) = κ j2 cj (t ) γ (t ), j = 1, 2, 3.ρ3 (t ) = c1 (t ) q5 (t ) − c01 (t ) q7 (t ),
3.3. Three-soliton
ρ4 (t ) = c2 (t ) q6 (t ) − c02 (t ) q7 (t ) ρ3 (t ) = c03 (t ) q7 (t ) − c3 (t ) q4 (t ),
In this case, we take N = 3, q = 1, into the Eq. (6), the solution of Eq. (8) with auxiliary equations is
λj (x, t ) = κj cj (t )(x + κj i ≠ j,
∫ cj (t ) dt ) σij (t ) = κi ci (t ) + κj cj (t ), i, j = 1, 2, 3, (18)
u (η1, η2 , η3) = PN (φi )/ QN (φi ), 3
PN (φi ) = a 0 (t ) +
and we take qi (t ) = 1, i = 4, 6, 7, but a7 and κi, i = 1, 2, 3 are arbitrary parameters. In this figures qi (t ) = 1, i = 4, 5, 6, 7 and c0j (t ) = 1, j = 1, 2, 3. These (Fig. 2 a and b) shows multi-graded-index and periodic zigzag propagations for the three- soliton collisions at v (x, t ).
∑ ai (t ) φi + a4 (t ) φ1 φ2 + a5 (t ) φ2 φ3 + a6 (t ) φ1 φ3 i =1
+ +a7 (t ) φ1 φ2 φ3, φηj = c0j (t ) + cj (t ) φi , i , j = 1, 2, 3. (15)
Fig. 1. a and b, the solutions of Eq. (13) are displayed against x and t. (a) c1 (t ) = 2, c2 (t ) = 5.6, γ (t ) = sn (t , 0.5) + tanh(t ), κ1 = 0.5, κ 2 = 0.4 and a = −1 and (b) c1 (t ) = 3.5, c2 (t ) = 4.5, γ (t ) = cos(t ) + sech (t ), κ1 = −0.5, κ 2 = 0.4 and a = −1.
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Optics Communications 384 (2017) 7–10
H.I. Abdel-Gawad, M. Tantawy
Fig. 2. a and b the solutions of Eq. (17) are displayed against x and t. (a) c1 (t ) = −10, c2 (t ) = 5, c3 (t ) = 4, γ (t ) = sn (t , 0.5) + tanh(t ), κ1 = −.35, κ 2 = 0.4, κ3 = 0.25 , and a = −1. (b) c1 (t ) = 8.5, c2 (t ) = 5, c3 (t ) = 3.2γ (t ) = cos(t ) + sech (t ), κ1 = −0.35, κ 2 = 0.3, κ 2 = 0.48 and a = −1. Optoelectron. Adv. Mater. 9 (2015) 384. [23] M. Savescu, A.A. Alshaery, E.M. Hilal, A.H. Bhrawy, Q. Zhou, A. Biswas, Optoelectron. Adv. Mater. 9 (2015) 14. [24] Q. Zhou, L. Liu, H. Zhang, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad, S. Moshokoa, A. Biswas, Opt. Appl. 46 (1) (2016) 79. [25] M. Mirzazadeh, M. Ekici, A. Sonomezoglu, M. Eslami, Q. Zhou, A.H. Kara, D. Milovic, F.B. Majid, A. Biswas, M. Belic, Nonlinear Dyn. 85 (3) (2016) 1979. [26] J. Wang, L. Li, S. Jia, Opt. Commun. 274 (2007) 223. [27] H. Kumar, F. Chand, Optik 125 (2014) 2949. [28] R. Murali, K. Senthilnathan, K. Porsezian, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 025401. [29] C.-Q. Dai, Y.-J. Xu, Opt. Commun. 311 (2013) 216. [30] C.-Q. Dai, R. Chen, J. Zhang, Chaos Solitons Fractals 44 (2011) 862. [31] M.S.M. Rajan, A. Mahalingam, A. Uthayakumar, K. Porsezian, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1410. [32] G. Yang, R. Hao, L. Li, Z. Li, G. Zhou, Opt. Commun. 260 (2006) 282. [33] J.-D. He, J. Zhang, Y. Zhang, C.-Q. Dai, Opt. Commun. 285 (2012) 755. [34] Y. Zheng, Y. Meng, Y. Liu, Opt. Commun. 315 (2014) 63. [35] J. Yang, D.J. Kaup, SIAM J. Appl. Math. 60 (2000) 967. [36] H.M. Baskonus, Nonlinear Dyn. 86 (1) (2016) 77. [37] H. Bulut, Abstr. Appl. Anal. 2013 (2013) 742643. [38] H.M. Baskonus, H. Bulut, Waves Random Complex 25 (4) (2015) 720. [39] X. Li, A. Chen, Phys. Lett. A 342 (2005) 413. [40] A. Chen, X. Li, Chaos Solitons Fractals 27 (2006) 43. [41] J. Mei, M. Zhangyun, Appl. Math. Comput. 219 (2013) 6163. [42] A.S. Abdel Rady, E.S. Osman, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1172. [43] P. Wang, W.-J. Bo Tian, X.L. Liu, Y. Jiang, Appl. Math. Comput. 218 (2011) 1726. [44] H.-Yi Wang, Appl. Math. Lett. 38 (2014) 100. [45] C.-C. Zhang, A.-H. Chen, Appl. Math. Lett. 58 (2016) 133. [46] G. Ebadi, N. Yousefzadeh, H. Triki, A. Yildirim, A. Biswas, Rom. Rep. Phys. 64 (4) (2012) 915. [47] D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450 (2007) 1054. [48] S. Birkholz, C. Bre, A. Demircan, G. Steinmeyer, Phys. Rev. Lett. 114 (2015) 213901. [49] H.I. Abdel-Gawad, J. Stat. Phys. 147 (2012) 506. [50] N.A. Kudryashov, M.B. Soukharev, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1765. [51] A.S. Abdel Rady, E.S. Osman, M. Khalfallah, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1172. [52] C.-Q. Dai, Y.J. Xu, R. Chen, S.Q. Zhu, Eur. Phys. J. D 59 (2010) 457. [53] C.-Q. Dai, Y. Wang, J. Zhang, Opt. Lett. 35 (2010) 1437.
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