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Abundant complex wave solutions for the nonautonomous Fokas–Lenells equation in presence of perturbation terms
Optik - International Journal for Light and Electron Optics 181 (2019) 503–513
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Abundant complex wave solutions for the nonautonomous Fokas–Lenells equation in presence of perturbation terms Yao Dinga, M.S. Osmanb,c, Abdul-Majid Wazwazd,
T
⁎
a
College of General Education and InternationalStudies, Chongqing College of Electronic Engineering, Chongqing 401331, China Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Department of Mathematics, Duba University College, University of Tabuk, Saudi Arabia d Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA b c
ARTICLE INFO
ABSTRACT
MSC: 35Q55 35C08 78A60
This paper mainly investigates the nonautonomous Fokas–Lenells equation (FLE) in presence of perturbation terms. Bright–dark soliton solutions, Arbitrary solutions, similarity solutions, and some exact solutions for the FLE are obtained by different techniques. Furthermore, some figures are given to explain the movement mechanism of these solutions. To our best knowledge, the discussion and results in this work are new and important in different branches of science where this equation is used to describe some physical phenomenon.
Keywords: Nonautonomous Fokas–Lenells equation Bright–dark soliton solutions Arbitrary solutions Similarity solutions Exact solutions
1. Introduction Complexity of various nonlinear evolution equations (NEE) has been arrested much attention of many researchers to devote the investigation of nonlinear phenomena in different fields of science [1–3]. The physical property of the NEE can be explained by achieving the exact and numerical solutions of these equations. In the current era of science and technology, numerous approaches and techniques such as decomposition method [4], Cole-Hopf transformation [5], the unified method [6,7], exp(−ϕ(ξ))-expansion method [8–10], Darboux transformation [11], auxiliary equation method [12,13], Lie group analysis [14,15] have been used. In this paper, we mainly focus on the following nonautonomous Fokas–Lenells equation (FLE)
i
t + 1 (t ) i [ 5 (t ) x
xx
+ 2 (t ) xt + | |2 [ 3 (t ) + i 4 (t ) x] 2 ) 2 6 (t )(| | x 7 (t )(| | )x ] = 0,
(1)
= (x , t ) represents a complex field envelope as a function of propagation distance x and retarded time t and ℓi(t), i = 1, 2, where …, 7 are arbitrary functions in t. The FLE has been introduced in optics to be a model that describes femtosecond pulse propagation through single mode optical silica fiber [16]. This equation is completely integrable when ℓi(t) are constants [17] which has been derived by utilizing the two Hamiltonian operators associated with the nonlinear Shrödinger (NLS) Equation [18,19]. Here, our work aims to search for different wave structures for the complex solutions of the FLE. To this end, we find Bright–dark
⁎
Corresponding Author. E-mail address: [email protected] (A.-M. Wazwaz).
https://doi.org/10.1016/j.ijleo.2018.12.064 Received 16 December 2018; Accepted 16 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 503–513
Y. Ding et al.
Fig. 1. Bright soliton solution | | with ℓ1(t) = ℓ 2(t) = ℓ 5(t) = υ1 = Γ3 = 1, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
soliton solutions by ansatz method (two direct test functions), new arbitrary solutions (including rogue wave solution as an example) by a direct calculations for the main equation, similarity solutions which are similar to the solutions in [20], and finally some exact solutions by using the (G′/G)-expansion method [21–23]. The remainder of this paper is organized as follows. Section 2 constructs distinguished types of complex wave solutions for the FLE such as Bright–dark soliton solutions. Section 3 recalls new arbitrary solutions which contain, for example, rogue wave solution. Section 4 presents similarity solutions. Section 5 contains some exact solutions for the FLE via the (G′/G)-expansion method. Finally, conclusions are given in Section 6. 2. Bright–dark soliton solutions Let us make the following transformation
(x , t ) = e i
(x , t )
(2)
[ (x , t )],
where ϕ(x, t) = Γ1x + Γ2(t), (x , t ) = 3x + 4 (t ) . Γ1 and Γ3 are unknown real constants. Γ2(t) and Γ4(t) are unknown real functions. Substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts, we have
504
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Fig. 2. Bright soliton solution | | with ℓ1(t) = cost, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = sint, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot. 4 (t )
+
( )+
2 (t )
4 (t )
( ) 1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 5 (t )] 3 1 (t ) ( ) + 3 (t )
3
(
)3
( )[[ 4 (t ) +
6 (t )]
( )2 (3)
= 0,
and
+ 3 6 (t ) + 2 7 (t )] ( )2 + 2 (t ) 4 (t ) 1 (t ) 5 (t )] + 2 (t )[ 3 2 (t ) + 1] = 0.
1 [[ 4 (t )
+2
3
(4)
Set the coefficients of Ξ(ϕ)2 and constant terms to zero in Eq. (4), respectively. We have 4 (t ) 4 (t )
= =
3 6 (t ) 1
+
2 7 (t ), 1 [ 5 (t )
2 3 1 (t )]
2 (t )[ 3 2 (t ) + 1]
1 2 (t )
dt.
(5)
where is an integral constant. In Eq. (3), assume that
( )=
1sech(
(6)
).
Substituting Eq. (6) into Eq. (3), we get 505
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Fig. 3. Bright soliton solution | | with ℓ1(t) = tanht, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = tanht, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot. 2 2 1 [2 3 [ 6 (t ) + 7 (t )] + 3 (t )] 2 1 1 (t ) dt, 2 1 2 (t ) 2 2 2 5 (t )]] 7 (t ) = { 1 [2 1 (t ) 2 (t )[ 1 2 (t )[2 3 6 (t ) + 3 (t )] 2[ 2 [2 ( t ) + 1] 3 2 3 6 (t ) + 3 (t )]} 1 / {2 3 12 [( 12 + 32) 2 (t ) 2 + 2 3 2 (t ) + 1]}.
2 (t ) =
2+
(7)
From what has been discussed above, we can obtain the following bright soliton solutions 1
=
1e
i
(x , t ) sech{
1 [x
2 1 (t )[2 3 + ( 1 + 2 2 ( 1 + 3 ) 2 (t ) 2
2 3 ) 2 (t )]
+2
3 2 (t )
5 (t ) dt] + +1
2}.
(8)
where υ1 is arbitrary constant, ϕ1 and ϕ2 are integral constants. The dynamical behaviors for | | in solution (8) are shown in Figs. 1–3. In the same way, suppose that
( )=
2 tanh(
(9)
).
Substituting Eq. (9) into Eq. (3), we have
506
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Fig. 4. Dark soliton solution | | with ℓ1(t) = ℓ 2(t) = ℓ 5(t) = υ2 = Γ3 = 1, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
2 (t )
=
7 (t )
=
1
2
2 2 1 (t )[2 3 + ( 3 2 1 ) 2 (t )] 5 (t ) ( 32 2 12) 2 (t ) 2+2 3 2 (t ) + 1
dt,
2 {2 12 [ 2 (t )[ 5 (t ) 2 2 (t )[2 3 6 (t ) + 3 (t )]] 2 2 + 2 [ 3 2 (t ) + 1] [2 3 6 (t ) + 3 (t )]} / {2 3 22 [( 32 2 12 ) 2 (t ) 2 + 2 3 2 (t ) + 1]}.
+
1 (t )]
(10)
From what has been discussed above, we can obtain the following bright soliton solutions 2
=
2e
i
(x , t )
tanh{ 1x
1
2 2 12 ) 2 (t )] 1 (t )[2 3 + ( 3 2 2 2 ( 3 2 1 ) 2 (t ) + 2 3 2 (t )
5 (t ) dt + +1
2}.
(11)
where υ2 is arbitrary constant, ϕ1 and ϕ2 are integral constants. The dynamical behaviors for | | in solution (11) are shown in Figs. 4–6. 3. Arbitrary solutions Suppose the coefficients of Ξ(ϕ)3, Ξ(ϕ), Ξ′(ϕ) and Ξ″(ϕ) to zero in Eq. (4), respectively. We obtain 507
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Fig. 5. Dark soliton solution | | with ℓ1(t) = cost, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = sint, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
1 (t )
=
2 (t ) 5 (t ),
3 (t )
=
2 3 [ 6 (t ) +
7 (t )],
2 (t )
=
2
+
1
5 (t ) dt.
(12)
Substituting Eqs. (5) and (12) into Eq. (2), we have 3
= ei { 3 [
5 (t ) dt + x ] + 1}
{ 1[
5 (t ) dt
+ x] +
2}.
(13)
This is a very interesting solution. Ξ does not need to satisfy any restrictions. In other words, Ξ can be replaced by arbitrary solution. As an instance, suppose that
{ 1[
5 (t ) dt
+ x] +
2}
=
xt {ln[ 1 [
5 (t ) dt
+ x] +
2]}.
(14)
This is a rogue wave-type solution. When Γ1 = ϕ2 = 1, ℓ 5(t) =− t, the dynamical behaviors for Eq. (14) are shown in Fig. 7. Similarly, we can also discuss other types of solutions. 4. Similarity solutions We suppose that Eq. (3) can reduce Eq. (1) to the following known ordinary differential equation
( )+
( )
(15)
2 ( )3 = 0. 508
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Fig. 6. Dark soliton solution | | with ℓ1(t) = Secht, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = Secht, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
By computation, we derive 2 (t )
=
7 (t )
= { 12 [ 2 (t ){
2
+
1
2 2 1 (t )[( 1 3 ) 2 (t ) 2 3] + 5 (t ) 2 2 ( 32 1 ) 2 (t ) +2 3 2 (t ) + 1
2 (t )[2 3 6 (t )
+
3 (t )]
dt,
2 5 (t )}
2 1 (t )] [
/ {2 3 [(
2 3
2 1
)
2
(t ) 2
+2
3 2 (t )
3 2 (t )
+ 1] 2 [2
3 6 (t )
+
3 (t )]}
(16)
+ 1]}.
Thus, a corresponding relation is established between Eqs. (15) and (1). It is gratifying that the elliptic function solutions for Eq. (15) have been got in Ref. [20]. Consequently, the corresponding similarity solutions for Eq. (1) can be obtained. 5. Exact solutions Following the steps of the (G′/G)-expansion method [21–23], assume that
( )=
1 (t )
G( ) + G( )
0 (t )
+
1 (t )
G( ) , G( )
(17)
with the constraint condition
G ( )+
G( )+
(18)
G ( ) = 0. 509
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Eq. (18) satisfies
G( ) = G( )
2 2
+
C1 cosh[ 1 ] + C2 sinh[ 1 ] , 1 C sinh[ 1 1 ] + C2 cosh[ 1 ]
2
+
C1 sin[ 2 ] + C2 cos[ 2 ] , 2 C cos[ 1 2 ] + C2 sin[ 2 ]
2
4
> 0,
1
=
4
< 0,
2
=
2
4
,
2 2
4 2
.
Substituting Eq. (17) with constraint (18) into Eq. (3) and equating all the coefficients of different powers of zero via symbolic computation [24–29], the results are listed below
(19) G( ) [ G ( ) ]i (i
=
3, …, 3) to
Case I: 1 (t ) 0 (t ) 2 (t )
7 (t )
= 0,
1 (t )
=
2
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
1
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2{2 3 [ 6 (t ) + 7 (t )] + 3 (t )}
1
=
=
+
{ 1 (t )[ 2 (t )[ 1 2 2 2 2 (t ) [2 3 + 1 (4 2
2 1 (t ) +[
2 (t )[ 1 (t )
1 (t ) 3 (t )
+ 4[ 6 (t ) +
+ [2 +
2
(
4 3] + 2 5 (t )}
3 2 (t ) 5 (t )] 6 (t )] 3
7 (t )]
4 ) 2 (t )[[2 1 (t ) 2 2 1 2 (t ) ] 1 (t )
4 )
2
7 (t )[(
+
1 (t ) 6 (t )] 2 (t ) 2
+ 2[ 6 (t )[(
2 32]
4 )
2)]
2 (t )]
7 (t )] 1 (t )
,
2
+ 4 3 2 (t ) + 2} dt, 7 (t )][ 2 (t )[ 1 (t ) 5 (t ) 2 ( t ) +
/{ = [4 2 (t )[[ 6 (t ) +
+
2 1(
,
2 (t ) 5 (t )]
5 (t ) 3 (t ) 2 ( t ) 4 5 (t ) 6 (t )] 2 (t ) 2 + 4[[ 6 (t ) 5 (t ) 4 1 (t )[ 6 (t ) + 7 (t )] 2 (t )] 32 2 2 + 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1
+ 2[ 5 (t )
4 ) 2 (t )[(2 1 (t ) +
2 (t )
+ 2[
+
3
2 (t ) 5 (t )]] 2 2 2 (t ) 5 (t )] 1
2 (t ) 5 (t )) 2 (t )
( 2 4 ) 12 2 (t ) 2] 1 (t ) + 2( 5 (t ) 2 (t ) + 2 (t ) 5 (t ))] + [ 1 (t ) 2 4 ) 12 2 (t ) 2 2] 6 (t ) 2 2 (t ) 3 (t )]] 3 2 (t ) 5 (t )][[( 2 2 2 [( 4 ) 1 2 (t ) 2][ 1 (t ) + 2 (t ) 5 (t )] 3 (t ) 2) 2 + 2 2] (t ) 2 + 4 3 (t )[[[(4 3 2 (t ) + 2] 1 (t ) 1 3 2 [( 2 4 ) 2 (t )(2 1 (t ) + 2 (t ) 5 (t )) 12 4 3 1 (t ) + 2 5 (t ) 2) 2 2 32 2 (t )(2 1 (t ) + 2 (t ) 5 (t ))] 2 (t ) + 2 (t )[[(4 1 2 2 2 2 2 3] 2 (t ) + 4 3 2 (t ) + 2] 5 (t )]]/[2 3 [((4 ) 1 + 2 23 ) 2 (t ) 2
+ [2 + + + +
+ +4
3 2 (t )
+ 2]( 1 (t ) +
2 (t ) 5 (t ))],
(20)
where ϵ1 = ± 1. Case II: 1 (t ) 0 (t )
= 2 =
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
2
2 (t )
=
7 (t )
= [4 2 (t )[( 6 (t ) +
+
2 1 (t ) + 2[
1 (t )
= 0,
1 [ 2 (t ) 2 (t ) + 1 1 (t )] , 2{2 3 [ 6 (t ) + 7 (t )] + 3 (t )} 2 ( 2 4 ) 2 2] 4 } + 2 5 (t ) ( t ){ ( t )[ 1 2 3 1 3 1 2)] + 4 2 2 2 2 (t ) [2 3 + 1 (4 3 2 (t ) + 2
2
2
,
2 (t )]
5 (t ) 3 (t ) 2
4 5 (t )
7 (t ))[ 2 (t )( 1 (t ) 2 (t )( 1 (t ) (t ) 3 + [
6 (t )] 2 (t )
2
+
5 (t )
+ +
1 (t ) 3 (t )
+ 4[( 6 (t ) +
+ 4( 6 (t ) +
7 (t )) 1 (t )
2 2 7 (t )) 2 (t )] 3 + 2[ 6 (t )[( 2 2 ( 2 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1 + [2 2( 5 (t ) 2 (t ) + 2 (t ) 5 (t ))] + 7 (t )[( 2 4 2 2 ( 2 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1 + [2
+ 2( 5 (t ) *
2 (t )
2
2 (t )
2]
+
2 (t ) 5 (t ))]
6 (t )
2 2 (t )
* ( 1 (t ) +
2 (t ) 5 (t )) 3 (t )
+4
3 2 (t )
+ 2]
4
3 1 (t )
+ 2 5 (t )
+
2 (t )[((4
1 (t )
+ [(
2 (t ) 5 (t ))
3 2 (t ) 5 (t )) 6 (t )] 3
4 1 (t )( 6 (t ) + +
dt,
2 (t ) +
+ ( 1 (t ) +
7 (t )) 5 (t )
1 (t ) 6 (t )] 2 (t )
4 ) 2 (t )[[2 1 (t ) 4 )
2 2 1 2 (t ) ] 1 (t )
) 2 (t )[[2 1 (t ) 4 )
2 2 1 2 (t ) ] 1 (t ) 2 4 12 (t ) 2 2]
2 (t ) 5 (t ))[[( 2 4 ) 12 2 2) 2 + 2 2] (t ) 2 + 3 (t )[[[(4 1 3 2 2 4 ) 2 (t )(2 1 (t ) + 2 (t ) 5 (t )) 12 3 (t )]] 3
+ [(
2 23 2 (t )(2 1 (t ) + 2 (t ) 5 (t ))] 2 (t ) 2) 2 + 2 2 ) (t ) 2 + 4 3 2 (t ) + 2] 5 (t )]] 1 3 2 2) 2 + 2 2] (t ) 2 1 3 2
/ [2 3 [[(4 + 4 3 2 (t ) + 2][ 1 (t ) +
(21)
2 (t ) 5 (t )]],
where ϵ2 = ± 1.
510
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Fig. 7. Rogue wave-type solution (14) (a) 3D plot, (b) contour plot.
511
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Case III: 0 (t ) 1 (t )
= 0,
1 (t )
= 2
2 (t )
=
7 (t )
= + + + + + + + + +
=
2
2
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
2
,
,
5 (t ) 1 (t )[2 3 + ( 23 4 12 ) 2 (t )] dt, 1 2 ( 32 4 12 ) 2 (t ) 2+2 3 2 (t ) + 1 2 2 [2 3 [ 6 (t )[ 1 (t )(1 4 1 2 (t ) ) + 2 (t )[4 12 2 (t )[2 1 (t ) 4 12 2 (t ) 2)] 2 (t ) 5 (t )] + 5 (t )] + 2 (t ) 5 (t )(1 2 2 4 1 2 (t ) ) + 2 (t )[4 12 2 (t )[2 1 (t ) 7 (t )[ 1 (t )(1 ( ) ( )] + ( 4 12 2 (t ) 2)] + [ 1 (t ) t t 2 5 5 t )] + 2 (t ) 5 (t )(1 2 2 1) 2 (t ) 5 (t )][ 6 (t )(4 1 2 (t ) 2 (t ) 3 (t )]] + [ 1 (t ) 2 2 1) + 3 (t )[ 1 (t )[( 32 4 12 ) 2 (t ) 2 2 (t ) 5 (t )] 3 (t )(4 1 2 (t ) 2 2 2 3 2 (t ) + 1] + 2 (t )[ 5 (t )[(4 12 3 ) 2 (t ) + 1] 2 1 (t )[ 3 + ( 32 4 12 ) 2 (t )]] + 2 (t ) 5 (t )[( 23 4 12 ) 2 (t ) 2 2 3 2 (t ) + 1]] + 2 33 2 (t )[( 6 (t ) + 7 (t ))[ 2 (t )[ 1 (t ) 5 (t ) 2 (t )
2 1 (t ) 2 (t )] 2 (t ) 5 (t )] 2 (t )( 1 (t ) + 2 (t ) 5 (t )) 6 (t )] 2 3 2 5 (t ) 2 (t ) 3 (t ) + 2 (t ) [ 1 (t ) 3 (t ) + 4( 6 (t ) + 7 (t )) 5 (t ) 3[ 4 5 (t ) 6 (t )] + 4 2 (t )[( 6 (t ) + 7 (t )) 1 (t ) 1 (t ) 6 (t )] 4 1 (t )( 6 (t ) + 7 (t )) 2 (t )]]/[2 3 ( 1 (t ) + 2 (t ) 5 (t ))
* [(
2 3
4
2 1
) 2 (t ) 2 + 2
3 2 (t )
(22)
+ 1]],
where ϵ2 = ± 1. Substituting Eqs. (20)–(22) into Eq. (2) with Eq. (19), the corresponding exact solutions for Eq. (1) can be obtained. 6. Conclusion In conclusion, new abundant complex wave solutions for the FLE was constructed by different algorithms. Among these solutions we obtained bright–dark soliton solutions via the ansatz method, arbitrary solutions, which contain rogue wave solution, by a direct calculation for the FLE, similarity solutions as in Ref. [17], and some exact solutions by the (G′/G)-expansion method. In this respect, the graphical representations of the solutions were demonstrated in Figs. 1–7, to discuss the behavior of the FLE. The results of the present study may provide useful information about the physical meaning for the solutions of the other types of complex models. References [1] J.G. Liu, Y. Tian, J.G. Hu, New non-traveling wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, Appl. Math. Lett. 79 (2018) 162–168. [2] J.G. Liu, Lump-type solutions and interaction solutions for the (2+1)-dimensional generalized fifth-order KdV equation, Appl. Math. Lett. 86 (2018) 36–41. [3] J.G. Liu, Y. Tian, Z.F. Zeng, New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in multitemperature electron plasmas, AIP Adv. 7 (10) (2017) 105013. [4] H. Zenil, S. Hernández-Orozco, N. Kiani, F. Soler-Toscano, A. Rueda-Toicen, J. Tegnér, A decomposition method for global evaluation of Shannon entropy and local estimations of algorithmic complexity, Entropy 20 (8) (2018) 605. [5] R. Vanon, K. Ohkitani, Applications of a Cole–Hopf transform to the 3D Navier–Stokes equations, J. Turbul. 19 (4) (2018) 322–333. [6] M.S. Osman, A. Korkmazc, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, The unified method for conformable time fractional Schrödinger equation with perturbation terms, Chin. J. Phys. 56 (5) (2018) 2500–2506. [7] M.S. Osman, H.I. Abdel-Gawad, M.A. El Mahdy, Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results Phys. 8 (2018) 1054–1060. [8] S. Arshed, A. Biswas, M. Ekici, S. Khan, Q. Zhou, S.P. Moshokoa, M. Alfiras, M.M. Belic, Solitons in nonlinear directional couplers with optical metamaterials by exp(−ϕ(ξ))-expansion, Optik 179 (2019) 443–462. [9] S. Arshed, A. Biswas, Q. Zhou, S. Khan, S. Adesanya, S.P. Moshokoa, M.M. Belic, Optical solitons pertutabation with Fokas–Lenells equation by exp(−ϕ(ξ))expansion method, Optik 179 (2019) 341–345. [10] F. Ferdous, M.G. Hafez, A. Biswas, M. Ekici, Q. Zhou, M. Alfiras, S.P. Moshokoa, M. Belic, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−ϕ(ξ)), Optik 178 (2019) 439–448. [11] Z. Amjad, Z.B. Haider, Darboux transformations of supersymmetric Heisenberg magnet model, J. Phys. Commun. 2 (3) (2018) 035019. [12] S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A 309 (5–6) (2003) 387–396. [13] Y. Ma, B. Li, C. Wang, A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method, Appl. Math. Comput. 211 (1) (2009) 102–107. [14] M.A.A. Hamad, M. Ferdows, Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysis, Commun. Nonlinear Sci. 17 (1) (2012) 132–140. [15] C.M. Khalique, K.R. Adem, Exact solutions of the (2+1)-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis, Math. Comput. Model. 54 (1-2) (2011) 184–189. [16] H. Triki, A.M. Wazwaz, Combined optical solitary waves of the Fokas–Lenells equation, Waves Random Complex 27 (4) (2017) 587–593. [17] M.S. Osman, B. Ghanbari, New optical solitary wave solutions of Fokas–Lenells equation in presence of perturbation terms by a novel approach, Optik 175 (2018) 328–333. [18] A.S. Fokas, On a class of physically important integrable equations, Physica D 87 (1995) 145–150. [19] J. Lenells, Exactly solvable model for nonlinear pulse propagation in optical fibers, Stud. Appl. Math. 123 (2009) 215–232. [20] P.A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A 22 (13) (1999) 2355–2367. [21] A.R. Shehata, The traveling wave solutions of the perturbed nonlinear Schrödinger equation and the cubic–quintic Ginzburg Landau equation using the modified (G′/G)-expansion method, Appl. Math. Comput. 217 (2010) 1–10. [22] S. Guo, Y. Zhou, C. Zhao, The improved (G′/G)-expansion method and its applications to the Broer–Kaup equations and approximate long water wave equations,
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Appl. Math. Comput. 216 (7) (2010) 1965–1971. [23] A. Biswas, A. Sonmezoglu, M. Ekici, M. Mirzazadeh, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Optical soliton perturbation with fractional temporal evolution by extended (G′/G)-expansion method, Optik 161 (2018) 301–320. [24] J.G. Liu, Y. He, New periodic solitary wave solutions for the (3+1)-dimensional generalized shallow water equation, Nonlinear Dyn. 90 (1) (2017) 363–369. [25] J.G. Liu, J.Q. Du, Z.F. Zeng, B. Nie, New three-wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, Nonlinear Dyn. 88 (1) (2017) 655–661. [26] J.G. Liu, J.Q. Du, Z.F. Zeng, G.P. Ai, Exact periodic cross-kink wave solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics, Chaos 26 (2016) 103114. [27] A. Biswas, Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities, Optik 176 (2019) 198–201. [28] A. Biswas, M. Ekici, A. Sonmezoglu, M. Belic, Optical solitons in birefringent fibers with quadratic–cubic nonlinearity by extended trial function scheme, Optik 176 (2019) 542–548. [29] H. Triki, A. Biswas, Q. Zhou, S.P. Moshokoa, M. Belic, Chirped envelope optical solitons for Kaup–Newell equation, Optik 177 (2019) 1–7.
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Short note
Abundant complex wave solutions for the nonautonomous Fokas–Lenells equation in presence of perturbation terms Yao Dinga, M.S. Osmanb,c, Abdul-Majid Wazwazd,
T
⁎
a
College of General Education and InternationalStudies, Chongqing College of Electronic Engineering, Chongqing 401331, China Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Department of Mathematics, Duba University College, University of Tabuk, Saudi Arabia d Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA b c
ARTICLE INFO
ABSTRACT
MSC: 35Q55 35C08 78A60
This paper mainly investigates the nonautonomous Fokas–Lenells equation (FLE) in presence of perturbation terms. Bright–dark soliton solutions, Arbitrary solutions, similarity solutions, and some exact solutions for the FLE are obtained by different techniques. Furthermore, some figures are given to explain the movement mechanism of these solutions. To our best knowledge, the discussion and results in this work are new and important in different branches of science where this equation is used to describe some physical phenomenon.
Keywords: Nonautonomous Fokas–Lenells equation Bright–dark soliton solutions Arbitrary solutions Similarity solutions Exact solutions
1. Introduction Complexity of various nonlinear evolution equations (NEE) has been arrested much attention of many researchers to devote the investigation of nonlinear phenomena in different fields of science [1–3]. The physical property of the NEE can be explained by achieving the exact and numerical solutions of these equations. In the current era of science and technology, numerous approaches and techniques such as decomposition method [4], Cole-Hopf transformation [5], the unified method [6,7], exp(−ϕ(ξ))-expansion method [8–10], Darboux transformation [11], auxiliary equation method [12,13], Lie group analysis [14,15] have been used. In this paper, we mainly focus on the following nonautonomous Fokas–Lenells equation (FLE)
i
t + 1 (t ) i [ 5 (t ) x
xx
+ 2 (t ) xt + | |2 [ 3 (t ) + i 4 (t ) x] 2 ) 2 6 (t )(| | x 7 (t )(| | )x ] = 0,
(1)
= (x , t ) represents a complex field envelope as a function of propagation distance x and retarded time t and ℓi(t), i = 1, 2, where …, 7 are arbitrary functions in t. The FLE has been introduced in optics to be a model that describes femtosecond pulse propagation through single mode optical silica fiber [16]. This equation is completely integrable when ℓi(t) are constants [17] which has been derived by utilizing the two Hamiltonian operators associated with the nonlinear Shrödinger (NLS) Equation [18,19]. Here, our work aims to search for different wave structures for the complex solutions of the FLE. To this end, we find Bright–dark
⁎
Corresponding Author. E-mail address: [email protected] (A.-M. Wazwaz).
https://doi.org/10.1016/j.ijleo.2018.12.064 Received 16 December 2018; Accepted 16 December 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 181 (2019) 503–513
Y. Ding et al.
Fig. 1. Bright soliton solution | | with ℓ1(t) = ℓ 2(t) = ℓ 5(t) = υ1 = Γ3 = 1, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
soliton solutions by ansatz method (two direct test functions), new arbitrary solutions (including rogue wave solution as an example) by a direct calculations for the main equation, similarity solutions which are similar to the solutions in [20], and finally some exact solutions by using the (G′/G)-expansion method [21–23]. The remainder of this paper is organized as follows. Section 2 constructs distinguished types of complex wave solutions for the FLE such as Bright–dark soliton solutions. Section 3 recalls new arbitrary solutions which contain, for example, rogue wave solution. Section 4 presents similarity solutions. Section 5 contains some exact solutions for the FLE via the (G′/G)-expansion method. Finally, conclusions are given in Section 6. 2. Bright–dark soliton solutions Let us make the following transformation
(x , t ) = e i
(x , t )
(2)
[ (x , t )],
where ϕ(x, t) = Γ1x + Γ2(t), (x , t ) = 3x + 4 (t ) . Γ1 and Γ3 are unknown real constants. Γ2(t) and Γ4(t) are unknown real functions. Substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts, we have
504
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Fig. 2. Bright soliton solution | | with ℓ1(t) = cost, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = sint, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot. 4 (t )
+
( )+
2 (t )
4 (t )
( ) 1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 5 (t )] 3 1 (t ) ( ) + 3 (t )
3
(
)3
( )[[ 4 (t ) +
6 (t )]
( )2 (3)
= 0,
and
+ 3 6 (t ) + 2 7 (t )] ( )2 + 2 (t ) 4 (t ) 1 (t ) 5 (t )] + 2 (t )[ 3 2 (t ) + 1] = 0.
1 [[ 4 (t )
+2
3
(4)
Set the coefficients of Ξ(ϕ)2 and constant terms to zero in Eq. (4), respectively. We have 4 (t ) 4 (t )
= =
3 6 (t ) 1
+
2 7 (t ), 1 [ 5 (t )
2 3 1 (t )]
2 (t )[ 3 2 (t ) + 1]
1 2 (t )
dt.
(5)
where is an integral constant. In Eq. (3), assume that
( )=
1sech(
(6)
).
Substituting Eq. (6) into Eq. (3), we get 505
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Fig. 3. Bright soliton solution | | with ℓ1(t) = tanht, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = tanht, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot. 2 2 1 [2 3 [ 6 (t ) + 7 (t )] + 3 (t )] 2 1 1 (t ) dt, 2 1 2 (t ) 2 2 2 5 (t )]] 7 (t ) = { 1 [2 1 (t ) 2 (t )[ 1 2 (t )[2 3 6 (t ) + 3 (t )] 2[ 2 [2 ( t ) + 1] 3 2 3 6 (t ) + 3 (t )]} 1 / {2 3 12 [( 12 + 32) 2 (t ) 2 + 2 3 2 (t ) + 1]}.
2 (t ) =
2+
(7)
From what has been discussed above, we can obtain the following bright soliton solutions 1
=
1e
i
(x , t ) sech{
1 [x
2 1 (t )[2 3 + ( 1 + 2 2 ( 1 + 3 ) 2 (t ) 2
2 3 ) 2 (t )]
+2
3 2 (t )
5 (t ) dt] + +1
2}.
(8)
where υ1 is arbitrary constant, ϕ1 and ϕ2 are integral constants. The dynamical behaviors for | | in solution (8) are shown in Figs. 1–3. In the same way, suppose that
( )=
2 tanh(
(9)
).
Substituting Eq. (9) into Eq. (3), we have
506
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Fig. 4. Dark soliton solution | | with ℓ1(t) = ℓ 2(t) = ℓ 5(t) = υ2 = Γ3 = 1, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
2 (t )
=
7 (t )
=
1
2
2 2 1 (t )[2 3 + ( 3 2 1 ) 2 (t )] 5 (t ) ( 32 2 12) 2 (t ) 2+2 3 2 (t ) + 1
dt,
2 {2 12 [ 2 (t )[ 5 (t ) 2 2 (t )[2 3 6 (t ) + 3 (t )]] 2 2 + 2 [ 3 2 (t ) + 1] [2 3 6 (t ) + 3 (t )]} / {2 3 22 [( 32 2 12 ) 2 (t ) 2 + 2 3 2 (t ) + 1]}.
+
1 (t )]
(10)
From what has been discussed above, we can obtain the following bright soliton solutions 2
=
2e
i
(x , t )
tanh{ 1x
1
2 2 12 ) 2 (t )] 1 (t )[2 3 + ( 3 2 2 2 ( 3 2 1 ) 2 (t ) + 2 3 2 (t )
5 (t ) dt + +1
2}.
(11)
where υ2 is arbitrary constant, ϕ1 and ϕ2 are integral constants. The dynamical behaviors for | | in solution (11) are shown in Figs. 4–6. 3. Arbitrary solutions Suppose the coefficients of Ξ(ϕ)3, Ξ(ϕ), Ξ′(ϕ) and Ξ″(ϕ) to zero in Eq. (4), respectively. We obtain 507
Optik - International Journal for Light and Electron Optics 181 (2019) 503–513
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Fig. 5. Dark soliton solution | | with ℓ1(t) = cost, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = sint, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
1 (t )
=
2 (t ) 5 (t ),
3 (t )
=
2 3 [ 6 (t ) +
7 (t )],
2 (t )
=
2
+
1
5 (t ) dt.
(12)
Substituting Eqs. (5) and (12) into Eq. (2), we have 3
= ei { 3 [
5 (t ) dt + x ] + 1}
{ 1[
5 (t ) dt
+ x] +
2}.
(13)
This is a very interesting solution. Ξ does not need to satisfy any restrictions. In other words, Ξ can be replaced by arbitrary solution. As an instance, suppose that
{ 1[
5 (t ) dt
+ x] +
2}
=
xt {ln[ 1 [
5 (t ) dt
+ x] +
2]}.
(14)
This is a rogue wave-type solution. When Γ1 = ϕ2 = 1, ℓ 5(t) =− t, the dynamical behaviors for Eq. (14) are shown in Fig. 7. Similarly, we can also discuss other types of solutions. 4. Similarity solutions We suppose that Eq. (3) can reduce Eq. (1) to the following known ordinary differential equation
( )+
( )
(15)
2 ( )3 = 0. 508
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Fig. 6. Dark soliton solution | | with ℓ1(t) = Secht, ℓ2(t) = υ1 = Γ3 = 1, ℓ5(t) = Secht, Γ1 =−2, ϕ2 = 0 (a) 3D plot, (b) contour plot.
By computation, we derive 2 (t )
=
7 (t )
= { 12 [ 2 (t ){
2
+
1
2 2 1 (t )[( 1 3 ) 2 (t ) 2 3] + 5 (t ) 2 2 ( 32 1 ) 2 (t ) +2 3 2 (t ) + 1
2 (t )[2 3 6 (t )
+
3 (t )]
dt,
2 5 (t )}
2 1 (t )] [
/ {2 3 [(
2 3
2 1
)
2
(t ) 2
+2
3 2 (t )
3 2 (t )
+ 1] 2 [2
3 6 (t )
+
3 (t )]}
(16)
+ 1]}.
Thus, a corresponding relation is established between Eqs. (15) and (1). It is gratifying that the elliptic function solutions for Eq. (15) have been got in Ref. [20]. Consequently, the corresponding similarity solutions for Eq. (1) can be obtained. 5. Exact solutions Following the steps of the (G′/G)-expansion method [21–23], assume that
( )=
1 (t )
G( ) + G( )
0 (t )
+
1 (t )
G( ) , G( )
(17)
with the constraint condition
G ( )+
G( )+
(18)
G ( ) = 0. 509
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Eq. (18) satisfies
G( ) = G( )
2 2
+
C1 cosh[ 1 ] + C2 sinh[ 1 ] , 1 C sinh[ 1 1 ] + C2 cosh[ 1 ]
2
+
C1 sin[ 2 ] + C2 cos[ 2 ] , 2 C cos[ 1 2 ] + C2 sin[ 2 ]
2
4
> 0,
1
=
4
< 0,
2
=
2
4
,
2 2
4 2
.
Substituting Eq. (17) with constraint (18) into Eq. (3) and equating all the coefficients of different powers of zero via symbolic computation [24–29], the results are listed below
(19) G( ) [ G ( ) ]i (i
=
3, …, 3) to
Case I: 1 (t ) 0 (t ) 2 (t )
7 (t )
= 0,
1 (t )
=
2
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
1
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2{2 3 [ 6 (t ) + 7 (t )] + 3 (t )}
1
=
=
+
{ 1 (t )[ 2 (t )[ 1 2 2 2 2 (t ) [2 3 + 1 (4 2
2 1 (t ) +[
2 (t )[ 1 (t )
1 (t ) 3 (t )
+ 4[ 6 (t ) +
+ [2 +
2
(
4 3] + 2 5 (t )}
3 2 (t ) 5 (t )] 6 (t )] 3
7 (t )]
4 ) 2 (t )[[2 1 (t ) 2 2 1 2 (t ) ] 1 (t )
4 )
2
7 (t )[(
+
1 (t ) 6 (t )] 2 (t ) 2
+ 2[ 6 (t )[(
2 32]
4 )
2)]
2 (t )]
7 (t )] 1 (t )
,
2
+ 4 3 2 (t ) + 2} dt, 7 (t )][ 2 (t )[ 1 (t ) 5 (t ) 2 ( t ) +
/{ = [4 2 (t )[[ 6 (t ) +
+
2 1(
,
2 (t ) 5 (t )]
5 (t ) 3 (t ) 2 ( t ) 4 5 (t ) 6 (t )] 2 (t ) 2 + 4[[ 6 (t ) 5 (t ) 4 1 (t )[ 6 (t ) + 7 (t )] 2 (t )] 32 2 2 + 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1
+ 2[ 5 (t )
4 ) 2 (t )[(2 1 (t ) +
2 (t )
+ 2[
+
3
2 (t ) 5 (t )]] 2 2 2 (t ) 5 (t )] 1
2 (t ) 5 (t )) 2 (t )
( 2 4 ) 12 2 (t ) 2] 1 (t ) + 2( 5 (t ) 2 (t ) + 2 (t ) 5 (t ))] + [ 1 (t ) 2 4 ) 12 2 (t ) 2 2] 6 (t ) 2 2 (t ) 3 (t )]] 3 2 (t ) 5 (t )][[( 2 2 2 [( 4 ) 1 2 (t ) 2][ 1 (t ) + 2 (t ) 5 (t )] 3 (t ) 2) 2 + 2 2] (t ) 2 + 4 3 (t )[[[(4 3 2 (t ) + 2] 1 (t ) 1 3 2 [( 2 4 ) 2 (t )(2 1 (t ) + 2 (t ) 5 (t )) 12 4 3 1 (t ) + 2 5 (t ) 2) 2 2 32 2 (t )(2 1 (t ) + 2 (t ) 5 (t ))] 2 (t ) + 2 (t )[[(4 1 2 2 2 2 2 3] 2 (t ) + 4 3 2 (t ) + 2] 5 (t )]]/[2 3 [((4 ) 1 + 2 23 ) 2 (t ) 2
+ [2 + + + +
+ +4
3 2 (t )
+ 2]( 1 (t ) +
2 (t ) 5 (t ))],
(20)
where ϵ1 = ± 1. Case II: 1 (t ) 0 (t )
= 2 =
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
2
2 (t )
=
7 (t )
= [4 2 (t )[( 6 (t ) +
+
2 1 (t ) + 2[
1 (t )
= 0,
1 [ 2 (t ) 2 (t ) + 1 1 (t )] , 2{2 3 [ 6 (t ) + 7 (t )] + 3 (t )} 2 ( 2 4 ) 2 2] 4 } + 2 5 (t ) ( t ){ ( t )[ 1 2 3 1 3 1 2)] + 4 2 2 2 2 (t ) [2 3 + 1 (4 3 2 (t ) + 2
2
2
,
2 (t )]
5 (t ) 3 (t ) 2
4 5 (t )
7 (t ))[ 2 (t )( 1 (t ) 2 (t )( 1 (t ) (t ) 3 + [
6 (t )] 2 (t )
2
+
5 (t )
+ +
1 (t ) 3 (t )
+ 4[( 6 (t ) +
+ 4( 6 (t ) +
7 (t )) 1 (t )
2 2 7 (t )) 2 (t )] 3 + 2[ 6 (t )[( 2 2 ( 2 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1 + [2 2( 5 (t ) 2 (t ) + 2 (t ) 5 (t ))] + 7 (t )[( 2 4 2 2 ( 2 2 (t ) 5 (t )] 2 (t ) 2 (t ) 5 (t )] 1 + [2
+ 2( 5 (t ) *
2 (t )
2
2 (t )
2]
+
2 (t ) 5 (t ))]
6 (t )
2 2 (t )
* ( 1 (t ) +
2 (t ) 5 (t )) 3 (t )
+4
3 2 (t )
+ 2]
4
3 1 (t )
+ 2 5 (t )
+
2 (t )[((4
1 (t )
+ [(
2 (t ) 5 (t ))
3 2 (t ) 5 (t )) 6 (t )] 3
4 1 (t )( 6 (t ) + +
dt,
2 (t ) +
+ ( 1 (t ) +
7 (t )) 5 (t )
1 (t ) 6 (t )] 2 (t )
4 ) 2 (t )[[2 1 (t ) 4 )
2 2 1 2 (t ) ] 1 (t )
) 2 (t )[[2 1 (t ) 4 )
2 2 1 2 (t ) ] 1 (t ) 2 4 12 (t ) 2 2]
2 (t ) 5 (t ))[[( 2 4 ) 12 2 2) 2 + 2 2] (t ) 2 + 3 (t )[[[(4 1 3 2 2 4 ) 2 (t )(2 1 (t ) + 2 (t ) 5 (t )) 12 3 (t )]] 3
+ [(
2 23 2 (t )(2 1 (t ) + 2 (t ) 5 (t ))] 2 (t ) 2) 2 + 2 2 ) (t ) 2 + 4 3 2 (t ) + 2] 5 (t )]] 1 3 2 2) 2 + 2 2] (t ) 2 1 3 2
/ [2 3 [[(4 + 4 3 2 (t ) + 2][ 1 (t ) +
(21)
2 (t ) 5 (t )]],
where ϵ2 = ± 1.
510
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Y. Ding et al.
Fig. 7. Rogue wave-type solution (14) (a) 3D plot, (b) contour plot.
511
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Case III: 0 (t ) 1 (t )
= 0,
1 (t )
= 2
2 (t )
=
7 (t )
= + + + + + + + + +
=
2
2
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
1 [ 2 (t ) 2 (t ) + 1 1 (t )] 2 3 [ 6 (t ) + 7 (t )] + 3 (t )
2
,
,
5 (t ) 1 (t )[2 3 + ( 23 4 12 ) 2 (t )] dt, 1 2 ( 32 4 12 ) 2 (t ) 2+2 3 2 (t ) + 1 2 2 [2 3 [ 6 (t )[ 1 (t )(1 4 1 2 (t ) ) + 2 (t )[4 12 2 (t )[2 1 (t ) 4 12 2 (t ) 2)] 2 (t ) 5 (t )] + 5 (t )] + 2 (t ) 5 (t )(1 2 2 4 1 2 (t ) ) + 2 (t )[4 12 2 (t )[2 1 (t ) 7 (t )[ 1 (t )(1 ( ) ( )] + ( 4 12 2 (t ) 2)] + [ 1 (t ) t t 2 5 5 t )] + 2 (t ) 5 (t )(1 2 2 1) 2 (t ) 5 (t )][ 6 (t )(4 1 2 (t ) 2 (t ) 3 (t )]] + [ 1 (t ) 2 2 1) + 3 (t )[ 1 (t )[( 32 4 12 ) 2 (t ) 2 2 (t ) 5 (t )] 3 (t )(4 1 2 (t ) 2 2 2 3 2 (t ) + 1] + 2 (t )[ 5 (t )[(4 12 3 ) 2 (t ) + 1] 2 1 (t )[ 3 + ( 32 4 12 ) 2 (t )]] + 2 (t ) 5 (t )[( 23 4 12 ) 2 (t ) 2 2 3 2 (t ) + 1]] + 2 33 2 (t )[( 6 (t ) + 7 (t ))[ 2 (t )[ 1 (t ) 5 (t ) 2 (t )
2 1 (t ) 2 (t )] 2 (t ) 5 (t )] 2 (t )( 1 (t ) + 2 (t ) 5 (t )) 6 (t )] 2 3 2 5 (t ) 2 (t ) 3 (t ) + 2 (t ) [ 1 (t ) 3 (t ) + 4( 6 (t ) + 7 (t )) 5 (t ) 3[ 4 5 (t ) 6 (t )] + 4 2 (t )[( 6 (t ) + 7 (t )) 1 (t ) 1 (t ) 6 (t )] 4 1 (t )( 6 (t ) + 7 (t )) 2 (t )]]/[2 3 ( 1 (t ) + 2 (t ) 5 (t ))
* [(
2 3
4
2 1
) 2 (t ) 2 + 2
3 2 (t )
(22)
+ 1]],
where ϵ2 = ± 1. Substituting Eqs. (20)–(22) into Eq. (2) with Eq. (19), the corresponding exact solutions for Eq. (1) can be obtained. 6. Conclusion In conclusion, new abundant complex wave solutions for the FLE was constructed by different algorithms. Among these solutions we obtained bright–dark soliton solutions via the ansatz method, arbitrary solutions, which contain rogue wave solution, by a direct calculation for the FLE, similarity solutions as in Ref. [17], and some exact solutions by the (G′/G)-expansion method. In this respect, the graphical representations of the solutions were demonstrated in Figs. 1–7, to discuss the behavior of the FLE. The results of the present study may provide useful information about the physical meaning for the solutions of the other types of complex models. References [1] J.G. Liu, Y. Tian, J.G. Hu, New non-traveling wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, Appl. Math. Lett. 79 (2018) 162–168. [2] J.G. Liu, Lump-type solutions and interaction solutions for the (2+1)-dimensional generalized fifth-order KdV equation, Appl. Math. Lett. 86 (2018) 36–41. [3] J.G. Liu, Y. Tian, Z.F. Zeng, New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in multitemperature electron plasmas, AIP Adv. 7 (10) (2017) 105013. [4] H. Zenil, S. Hernández-Orozco, N. Kiani, F. Soler-Toscano, A. Rueda-Toicen, J. Tegnér, A decomposition method for global evaluation of Shannon entropy and local estimations of algorithmic complexity, Entropy 20 (8) (2018) 605. [5] R. Vanon, K. Ohkitani, Applications of a Cole–Hopf transform to the 3D Navier–Stokes equations, J. Turbul. 19 (4) (2018) 322–333. [6] M.S. Osman, A. Korkmazc, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, The unified method for conformable time fractional Schrödinger equation with perturbation terms, Chin. J. Phys. 56 (5) (2018) 2500–2506. [7] M.S. Osman, H.I. Abdel-Gawad, M.A. El Mahdy, Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results Phys. 8 (2018) 1054–1060. [8] S. Arshed, A. Biswas, M. Ekici, S. Khan, Q. Zhou, S.P. Moshokoa, M. Alfiras, M.M. Belic, Solitons in nonlinear directional couplers with optical metamaterials by exp(−ϕ(ξ))-expansion, Optik 179 (2019) 443–462. [9] S. Arshed, A. Biswas, Q. Zhou, S. Khan, S. Adesanya, S.P. Moshokoa, M.M. Belic, Optical solitons pertutabation with Fokas–Lenells equation by exp(−ϕ(ξ))expansion method, Optik 179 (2019) 341–345. [10] F. Ferdous, M.G. Hafez, A. Biswas, M. Ekici, Q. Zhou, M. Alfiras, S.P. Moshokoa, M. Belic, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−ϕ(ξ)), Optik 178 (2019) 439–448. [11] Z. Amjad, Z.B. Haider, Darboux transformations of supersymmetric Heisenberg magnet model, J. Phys. Commun. 2 (3) (2018) 035019. [12] S. Jiong, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A 309 (5–6) (2003) 387–396. [13] Y. Ma, B. Li, C. Wang, A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method, Appl. Math. Comput. 211 (1) (2009) 102–107. [14] M.A.A. Hamad, M. Ferdows, Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: a Lie group analysis, Commun. Nonlinear Sci. 17 (1) (2012) 132–140. [15] C.M. Khalique, K.R. Adem, Exact solutions of the (2+1)-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis, Math. Comput. Model. 54 (1-2) (2011) 184–189. [16] H. Triki, A.M. Wazwaz, Combined optical solitary waves of the Fokas–Lenells equation, Waves Random Complex 27 (4) (2017) 587–593. [17] M.S. Osman, B. Ghanbari, New optical solitary wave solutions of Fokas–Lenells equation in presence of perturbation terms by a novel approach, Optik 175 (2018) 328–333. [18] A.S. Fokas, On a class of physically important integrable equations, Physica D 87 (1995) 145–150. [19] J. Lenells, Exactly solvable model for nonlinear pulse propagation in optical fibers, Stud. Appl. Math. 123 (2009) 215–232. [20] P.A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A 22 (13) (1999) 2355–2367. [21] A.R. Shehata, The traveling wave solutions of the perturbed nonlinear Schrödinger equation and the cubic–quintic Ginzburg Landau equation using the modified (G′/G)-expansion method, Appl. Math. Comput. 217 (2010) 1–10. [22] S. Guo, Y. Zhou, C. Zhao, The improved (G′/G)-expansion method and its applications to the Broer–Kaup equations and approximate long water wave equations,
512
Optik - International Journal for Light and Electron Optics 181 (2019) 503–513
Y. Ding et al.
Appl. Math. Comput. 216 (7) (2010) 1965–1971. [23] A. Biswas, A. Sonmezoglu, M. Ekici, M. Mirzazadeh, Q. Zhou, A.S. Alshomrani, S.P. Moshokoa, M. Belic, Optical soliton perturbation with fractional temporal evolution by extended (G′/G)-expansion method, Optik 161 (2018) 301–320. [24] J.G. Liu, Y. He, New periodic solitary wave solutions for the (3+1)-dimensional generalized shallow water equation, Nonlinear Dyn. 90 (1) (2017) 363–369. [25] J.G. Liu, J.Q. Du, Z.F. Zeng, B. Nie, New three-wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, Nonlinear Dyn. 88 (1) (2017) 655–661. [26] J.G. Liu, J.Q. Du, Z.F. Zeng, G.P. Ai, Exact periodic cross-kink wave solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics, Chaos 26 (2016) 103114. [27] A. Biswas, Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities, Optik 176 (2019) 198–201. [28] A. Biswas, M. Ekici, A. Sonmezoglu, M. Belic, Optical solitons in birefringent fibers with quadratic–cubic nonlinearity by extended trial function scheme, Optik 176 (2019) 542–548. [29] H. Triki, A. Biswas, Q. Zhou, S.P. Moshokoa, M. Belic, Chirped envelope optical solitons for Kaup–Newell equation, Optik 177 (2019) 1–7.
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