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Optical solitons with generalized anti-cubic nonlinearity by Lie symmetry
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
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Original research article
Optical solitons with generalized anti-cubic nonlinearity by Lie symmetry Sachin Kumara, Sandeep Malika, Anjan Biswasb,c,d,e, Yakup Yıldırımf,*, Ali Saleh Alshomranic, Milivoj R. Belicg a
Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151001, Punjab, India Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA c Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Applied Mathematics, National Research Nuclear University, 31 Kashirskoe Shosse, Moscow 115409, Russian Federation e Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa f Department of Mathematics, Faculty of Arts and Sciences, Near East University, 99138 Nicosia, Cyprus g Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar b
A R T IC LE I N F O
ABS TRA CT
Keywords: Solitons Generalized anti-cubic nonlinearity Lie group
Lie symmetry analysis handled the nonlinear Schrödinger's equation that is studied with generalized anti-cubic nonlinear form. These yielded bright, dark and singular optical soliton solutions as well as a bright-singular combo optical soliton solution to the model. The restrictions on the parameters for the existence of such solitons are also presented. OCIS Codes: 060.2310; 060.4510; 060.5530; 190.3270; 190.4370
1. Introduction The concept of anti-cubic (AC) nonlinearity for nonlinear Schrödinger's equation (NLSE) was first proposed by Fedelle et al. during 2003 to study solitons in optical fibers [1]. Later, this concept of AC nonlinearity was generalized by Biswas et al. during 2019 [2,3]. These lead to a plethora of results that has flourished across the board [4–17]. There are various features of these two forms of refractive indices that has been addressed with NLSE as visible in a variety of reported results. These include polarization preserving fibers, birefringent fibers, meta-optics, conservation laws, perturbation theory, soliton cooling and many others. It is time to move on with additional studies. This paper therefore is taking up the task of addressing soliton dynamics with generalized AC nonlinearity, that was introduced during 2019, by the aid of the classic Lie symmetry analysis [18–20]. Subsequently, soliton solutions are retrieved with the implementation of two subsidiary integration norms. These are Kudryashov's approach and the method undetermined coefficients. They yield bright, dark, singular and bright-singular combo soliton solutions. The details are enumerated in the rest of the paper after the governing model is put forth. 1.1. Governing model The governing NLSE with generalized AC nonlinearity is of the form [2,3]:
iq t + aq xx + bq xt + {c1 |q|−(2n + 2) + c2 |q|2n + c3 |q|(2n + 2) } q = 0, ⁎
Corresponding author. E-mail address: [email protected] (Y. Yıldırım).
https://doi.org/10.1016/j.ijleo.2019.163638 Received 2 August 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: Sachin Kumar, et al., Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163638
(1)
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where a , b , c1, c2 and c3 are arbitrary real-valued constants. Here x and t are independent variables, which represents spatial and temporal coordinates, respectively. q is a complex-valued function which depends upon x and t . If c1 = 0, (1) collapses to NLSE with parabolic law or cubic-quintic law of nonlinearity that has been extensively studied [21–25]. In this case, c1 introduces the AC nonlinearity term. In this work, first we generated infinitesimals and Lie symmetries of NLSE with generalized anti-cubic nonlinearity. Then we obtained two vector fields. Eq. (1) is reduced into system of ordinary differential equations (ODE) by using these vector fields, and exact solutions of system of ODE are carried out using various methods. Subsequently, the soliton solutions and other explicit solutions of equation are carried out. 2. Lie symmetry analysis of NLSE In this section, to obtain the infinitesimals of Eq. (1), Lie classical method [18–20] is applied by separating its real and imaginary parts. Firstly, let us consider
q (x , t ) = u (x , t ) e ιv (x , t ),
(2)
where u , v are real valued functions. Using (2) in (1), and separating real and imaginary parts, we have
ut + 2au x vx + a uvxx + bu x vt + but vx + b uvxt = 0 au(2n + 1) u xx − u(2n + 2) vt − au(2n + 2) vx2 + bu(2n + 1) u xt − bu(2n + 2) vx vt + c1 + c2 u(4n + 2) + c3 u(4n + 4) = 0.
(3)
Now let us consider one-parameter (ϵ ) transformation for the system of equations (3) as
x * = x + ϵξ (x , t , u, v ) + O (ξ 2) t * = t + ϵτ (x , t , u, v ) + O (ξ 2) u* = u + ϵη (x , t , u, v ) + O (ξ 2) v * = v + ϵϕ (x , t , u, v ) + O (ξ 2)
(4)
where ξ , τ , η and ϕ are infinitesimals, depending upon x , t , u and v , have to be determined. Associated vector field for these transformation is
V = ξ ∂x + τ ∂t + η∂u + ϕ∂v .
(5)
The second prolongations formula [19,20] for system of equation (3), are
pr (2) V = V + η x
∂ ∂ ∂ ∂ + ϕx + ηt + ϕ xx ∂u x ∂vx ∂ut ∂vxx
pr (2) V = V + ϕ x
∂ ∂ ∂ + ϕt + η xx , ∂vx ∂vt ∂u xx
(6)
where η x , ηt , ϕ x , ϕt , η xx and ϕ xx represent extended infinitesimal. Making use of invariance condition pr (2) V (Δ) = 0 whenever Δ = 0 , the invariance surface condition of (3) became
0 = a η vxx + b η vxt + a u ϕ xx + 2 a (vx η x + u x ϕ x ) + b (vt η x + ut ϕ x + vx ηt ) + ηt 0 = a u2n + 1 η xx − (2 a vx ϕ x + b vt ϕ x + ϕt + b vx ϕt ) u(2n + 2) + ((2n + 1) (a u xx + b u xt ) u2n − (2 n + 2)(vt − a vx2) u(2n + 1) + (4 n + 2) c2 u4 n + 2 + (4 n + 4) c3 u4 n + 3) η .
(7)
Substituting the value of extended infinitesimal and equating the coefficient of various partial derivativs of u and v equal to zero, we get the system of partial differential equations (PDEs). The general solution of desired system of PDEs in ξ , τ , η and ϕ gives the following result
2a ξ = C2 + ⎛−x + t ⎞ C4 b ⎠ ⎝ τ = C1 + t C4 η=0 ϕ = C3 +
x C4, b
(8)
where C1, C2, C3 and C4 are arbitrary constants. Hence we have the corresponding Lie algebra of the symmetry group of system (3) is defined by the infinitesimal generators of the form 2
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V1 =
∂ ∂t
V2 =
∂ ∂x
V3 =
∂ ∂v
∂ ∂ 2a x ∂ +t + V4 = ⎛−x + t⎞ . ∂t b ⎠ ∂x b ∂v ⎝
(9)
The corresponding commutation relations of V1, V2, V3 and V4 are given by
[V1, V2] = 0 [V1, V3] = 0 [V1, V4] =
2a V2 + V1 b
[V2, V3] = 0 [V2, V4] = −V2 +
1 V3 b
[V3, V4] = 0.
(10)
3. Symmetry reduction and invariant solutions In order to find invariant solutions of system (3), we have to solve the corresponding characteristic equation given as
dx dt du dv = = = , ξ τ η ϕ
(11)
where ξ , τ , η and ϕ are given by (8). To solve characteristic equation (11), we will consider two cases of vector fields:
• V + βV + μV •V 3
2
1
4
where β and μ are arbitrary non-zero real numbers. Case (i) V3 + βV2 + μV1 In this case, after solving the characteristic equation (11), we obtain following similarity variables
s =μx − βt u (x , t ) = P (s ) x v (x , t ) = + Q (s ) β
(12)
where s is new independent variable and P , Q are new dependent variable, which depend on s . Substituting (12) in system (3), we have following similarity reduction
0 = (a μ2 β − b μ β 2) P Q″ + (2 a μ − β 2 − b β ) P′ + (2 a μ2 β − 2 b μ β 2) P′ Q′
(13)
0= (a μ2 β 2 − b μ β 3) P (2n + 1) P″ − (2 a μ β − β 3 − b β 2) P (2n + 2) Q′ − (a μ2 β 2 − b μ β 3) P (2n + 2) Q′2 − a P (2n + 2) + c3 β 2 P (4n + 4) + c2 β 2 P (4n + 2) + c1 β 2,
(14)
where (′) denotes derivative with respct to s . Integrating (13) twice with respct to s , we obtained
Q=
k1 μ β (aμ − bβ )
2
aμ − β − bβ ) s + k2 ∫ P12 ds − 12 (2μβ (aμ − bβ )
(15)
where k1 and k2 are arbitrary constant. Substituting the Q from (15) into (14), we have
P″= − +
c3 c2 c1 P (2n + 3) − P (2n + 1) − P −(2n + 1) μ (aμ − bβ ) μ (aμ − bβ ) μ (aμ − bβ ) k2 1 ( − 2 b β − b2 + 4 a μ − β 2 ) P −3. P+ 2 2 1 4 μ β (aμ − bβ )2 μ2 (aμ − bβ )2 3
(16)
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Integrating (16), by taking integrable constant equal to zero, we have
(P ′)2 c3 c2 c1 = − P (2n + 4) − P (2n + 2) + P −2n 2 (2n + 4) μ (aμ − bβ ) (2n + 2) μ (aμ − bβ ) 2 n μ (aμ − bβ ) +
k12 ( − 2 b β − b2 + 4 a μ − β 2 ) 2 P −2. P − 2 2 2 2 8 μ (aμ − bβ ) 2 μ β (aμ − bβ )2
(17)
Let us consider
P 2 (s ) = A (s ),
(18)
we have
(P′)2 =
(A′)2 , 4A
(19)
where (′) denotes derivative with respct to s . Using (19) along with (18) in (17), we have
(A′)2= − +
4c3 4c2 4c1 A(n + 3) − A(n + 2) + A(−n + 1) (n + 2) μ (aμ − bβ ) (n + 1) μ (aμ − bβ ) nμ (aμ − bβ ) 4 k12 (−2bβ − b2 + 4aμ − β 2) 2 . A − 2 2 2 2 μ β (aμ − bβ )2 μ (aμ − bβ )
(20)
4. Kudryashov method Here, the algorithm for Kudryashov method [1] is presented. This method will be used to obtain exact solutions of (1). Let us consider the nonlinear PDE as below:
T (u, ut , u x , utt , u xx , u xt , …) = 0.
(21)
Use the traveling wave solution
u (x , t ) = y (η),
η = kx − wt
(22)
Eq. (21) will be reduced into nonlinear ODE as below:
T (y, wy′, ky′, w y ′′, k y ′′, wky ′′, …) = 0.
(23)
Now to wind the dominant term, substitute
y (η) = η−p
(24)
where p > 0 , into (23). Now compare the degree of all the terms and choose two or more smallest degree terms. The maximum value of p is pole of (23) and denote it as N . Now let us consider
y (η) = a0 + a1 R (η) + ⋯+aN R (η) N
(25)
where a 0 , a1, …, aN are unknown to be determined and
R (η) =
1 . 1 + eη
(26)
The function R (η) satisfy the following relation (27)
R′ = R2 − R
where (′) denotes the derivative with respect to η . Using (27), we can calculate the derivatives of y (η) in form of power of R . Differentiating (25) with respect to η and using (27), we have N
y′ =
∑ ai i (R − 1) Ri i=1 N
y″ =
∑ ai i ((i + 1) R2 − (2i + 1) R + i) Ri .
(28)
i=1
Now substitute (25), along with (27) and (28), into (23), and equating the coefficients of same power of R to zero, we obtain algebraic system of equations. Then by solving this system, we obtain the value of various parameters w, k , a0 , a1, …, aN . Now, to apply the Kudryashov's method to (20), first we substitute
A (s ) = s−p, p > 0 into (20), we obtain the pole N =
(29) 2 . n+1
Then we move for exact solution of (20) in following form 4
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A (s ) = a1 R (s ) n + 1
(30)
with a1 is unknown parameters to be determined and R (s ) satisfying the relation (31)
R′ = R2 − R
where (′) denotes derivative with respct to s . Substituting (30) into (20) and by comparing coefficients of same power of R equal to zero, we have following system of equations:
0 = μ β 2 n (c3 a1n + 3 (n + 1)2 + μ a12 (2 + n)(μ a − β b))(μ a − β b) 0 = μ β 2 c2 a12 + n n (n + 1)(2 + n)(μ a − β b) 0 = a12β 2μ2 n (2 + n)(μ a − β b)2 0 = a12 β 2 n (2 + n)(n2b2 + 2 n2β b + n2β 2 − 4 n2μ a + 2 nβ 2 − 8 nμ a + 2 nb2 + 4 nβ b + β 2 + 4 β 2μ2 b2 + 2 β b − 8 β bμ3a + 4 μ4 a2 − 4 μ a + b2) 0 = k12 n (n + 1)2 (2 + n) 0 = μ β 2 c1 a1−n + 1 (n + 1)2 (2 + n)(μ a − β b).
(32)
After solving the system (32), we have
μ=
b2 a
k1 = 0 β=b
(33)
and a1 is arbitrary. Corresponding soliton solution of (1), is therefore given by
a1
q (x , t ) =
2
e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(1 + e (μ x − β t ) ) n + 1 a1
=
1
e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
{1 + cosh(μx − βt ) + sinh(μx − βt )} n + 1
(34)
where a1, μ , β are given by (33). Thus, (34) represents a bright-singular combo optical soliton solution to the model and it exists provided a1 > 0 . 5. Undetermined coefficients In this section, soliton solutions of the main equation (1) will be found from the corresponding reduction (20). For this hyperbolic function method will be used to obtained the bright, dark and singular 1-soliton solutions of (1). 5.1. Bright soliton solution To obtain the solution by sech function method [13], in Eq. (20), use the following
A (s ) = a1 sechn1 (m1 s ) A′ (s ) = −a1 n1 m1 sechn1 (m1 s ) tanh(m1 s ).
(35)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of sech function equal to zero, we have following system of equations:
2 . n+1
Now by comparing coefficients of same
0 = c2 a1n + 2 μ β 2 n (n + 1)(n + 2)(aμ − β b) 0 = n μ β 2 a12 (c3 a1 (n + 1)2a1n − (n + 2) m12 μ (aμ − β b))(aμ − β b) 0 = a1−n + 1 c1 μ β 2 (n + 1)2 (n + 2)(aμ − β b) 0 = β 2 a12 n (n + 2)(n2 b2 + 2 β b n2 + β 2 n2 − 4 a μ n2 + 2 β 2 n − 8 a μ n + 2 nb2 + 4 β b n + 4 β 2 m12 μ2 b2 + β 2 − 8 β m12 μ3 a b + 2 β b + 4 m12 μ4 a2 − 4 a μ + b2) 0 = k12 n (n + 1)2 (n + 2).
(36)
After solving the system (36), we have 5
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μ=
b2 a
k1 = 0 β=b
(37)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 sechn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ ι⎜ x − 1 (μ x − β t ) + k2⎟ β 2 μβ (aμ − bβ ) ⎝ ⎠
(38)
where a1, μ, β , m1 are given by (37).
5.2. Dark soliton solution To obtain the solution by tanh function method [11], in Eq. (20), use the following
A (s ) = a1 tanhn1 (m1 s ),
|s| ≤
π 2 m1
A′ (s ) = a1 n1 m1tanhn1− 1 (m1 s ) − a1 n1 m1tanhn1+ 1 (m1 s ).
(39)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of tanh function equal to zero, we have following system of equations:
2 . n+1
Now by comparing coefficients of same
0 = μ β 2 (n + 2)(c1 (n + 1)2a1−n + 1 + n μ a12 m12 (aμ − bβ ))(aμ − bβ ) 0 = μ β 2 n (c3 (n + 1)2a1n + 3 + μ a12 m12 (aμ − bβ )(n + 2))(aμ − bβ ) 0 = a1n + 2 c2 μ β 2 n (n + 1)(n + 2)(aμ − bβ ) 0 = a12 β 2 n (n + 2)(−2 n2 b β − n2 β 2 − n2 b2 + 4 n2 a μ − 4 b β n − 2 nb2 + 8 a μ n − 2 n β 2 + 8 β 2 m12 μ2 b2 − β 2 − 16 β m12 μ3 a b − 2 b β − b2 + 8 m12 μ4 a2 + 4 a μ) 0 = k12 n (n + 1)2 (n + 2).
(40)
After solving the system (40), we have
μ=
b2 a
k1 = 0 β=b
(41)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 tanhn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(42)
where a1, μ, β , m1 are given by (41).
5.3. Singular soliton solution To obtain the solution by csch function method [9], in Eq. (20), use the following
A (s ) = a1 cschn1 (m1 s ) A′ (s ) = −a1 n1 m1 cschn1 (m1 s ) coth(m1 s ).
(43)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of csch function equal to zero, we have following system of equations: 6
2 . n+1
Now by comparing coefficients of same
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0 = μ β 2 a1−n + 1 c1 (n + 2)(n + 1)2 (aμ − bβ ) 0 = a1n + 2 c2 μ β 2 n (n + 1)(n + 2)(aμ − bβ ) 0 = μ β 2 n (c3 (n + 1)2a1n + 3 + μ m12 a12 (n + 2)(aμ − bβ ))(aμ − bβ ) 0 = a12 β 2 n (n + 2)(−4 n2 a μ + 2 n2 b β + n2 b2 + n2 β 2 + 4 n b β + 2 n β 2 − 8 n a μ + 2 n b2 + 4 β 2 m12 μ2 b2 + β 2 + 2 b β − 8 β m12 μ3 a b + 4 m12 μ4 a2 − 4 a μ + b2) 0 = k12 n (n + 1)2 (n + 2).
(44)
After solving the system (44), we have
μ=
b2 a
k1 = 0 β=b
(45)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 cschn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(46)
where a1, μ, β , m1 are given by (45). Again, these bright, dark and singular soliton solutions are guaranteed to exist provided a1 > 0 . Case (ii) V4 In this case, after solving the characteristic equation (11), we obtain following similarity variables
w =xt −
a 2 t b
u (x , t ) = F (w ) w a v (x , t ) = − + 2 t + G (w ), bt b
(47)
where w is new independent variable and F , G are new dependent variable, which depend upon w . Substituting (47) in (3), we have following similarity reduction (48)
w F G″ + 2 w F ′ G′ + F G′ = 0
b3 w F 2n + 1 F ″ − a F 2n + 2 − b3 w F 2n + 2 (G′)2 + b3 F 2n + 1 F ′ + c3 b2 F 4n + 4 + c2 b2 F 4n + 2 + c1 b2 = 0,
(49)
where (′) denotes derivative with respct to w . Integrating (48) twice with respct to w , we have
G = l1 + l2
∫ w1F 2 dw
(50)
where l1 and l2 are arbitrary constant. Substituting the value of G from (50) into (49), we have
F″ = −
l2 1 c3 2n + 3 c c a F − 2 F 2n + 1 − 1 F −2n − 1 + 3 F − F ′ + 22 F −3. w bw bw bw b w w
(51)
Corresponding solution of (1), is given by
q (x , t ) = F (w ) e
ι ⎛⎜− w + a2 t + l1+ l2 bt b ⎝
∫ w F1(w)2 dw⎞ ⎟
(52)
⎠
where w is given by (47) and F satisfies Eq. (51). 6. Conclusions The paper extracted bright, dark and singular solitons as well as bright-singular combo soliton solutions to the governing NLSE that is considered with generalized AC nonlinearity. The integration algorithm is Lie symmetry analysis, powerful as it is and one of the most classic forms. The enlightening results are thus very encouraging to venture further along in this direction. In order to fill in the gap, one needs to address soliton cooling, as well as extend the study to perturbed NLSE with generalized AC nonlinearity. In this context, both deterministic as well as stochastic perturbation terms need to be addressed. This would mean studying this extended model by additional norms such as semi-inverse variational principle, extended trial function scheme, F -expansion and extended Jacobi's elliptic function method, just to name a few. Generalized AC nonlinearity being a new model, a lot is on the table. 7
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Consequently, possibilities with the model are endless. Conflict of interest The authors declare that there is no conflict of interest. Acknowledgements The research work of the sixth author (MRB) was supported by the grant NPRP 11S-1126-170033 from QNRF and he is thankful for it. References [1] R. Fedele, H. Schamel, V.I. Karpman, P.K. Shukla, Envelope solitons of nonlinear Schrödinger equation with an anti-cubic nonlinearity, J. Phys. A 36 (2003) 1169–1173. [2] A. Biswas, Conservation laws for optical solitons with anti-cubic and generalized anti-cubic nonlinearities, Optik 176 (2019) 198–201. [3] A. Biswas, M. Ekici, A. Sonmezoglu, M. Belic, Chirped and chirp-free optical solitons with generalized anti-cubic nonlinearity by extended trial function scheme, Optik 178 (2019) 636–644. [4] K.S. Al-Ghafri, E.V. Krishnan, A. Biswas, M. Ekici, Optical solitons having anti-cubic nonlinearity with a couple of exotic integration schemes, Optik 172 (2018) 794–800. [5] A.H. Arnous, A. Biswas, M. Asma, M. Belic, Dark and singular solitons in optical metamaterials with anti-cubic nonlinearity by modified simple equation approach, Optoelectron. Adv. Mater. Rapid Commun. 12 (5–6) (2018) 332–336. [6] A. Biswas, Q. Zhou, M.Z. Ullah, M. Asma, S.P. Moshokoa, M. Belic, Perturbation theory and optical soliton cooling with anti-cubic nonlinearity, Optik 142 (2017) 73–76. [7] A. Biswas, Q. Zhou, M.Z. Ullah, H. Triki, S.P. Moshokoa, M. Belic, Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik 143 (2017) 131–134. [8] A. Biswas, A.J.M. Jawad, Q. Zhou, Resonant optical solitons with anti-cubic nonlinearity, Optik 157 (2018) 525–531. [9] M. Ekici, M. Mirzazadeh, A. Sonmezoglu, M.Z. Ullah, Q. Zhou, H. Triki, S.P. Moshokoa, A. Biswas, Optical solitons with anti-cubic nonlinearity by extended trial equation method, Optik 136 (2017) 368–373. [10] M. Foroutan, J. Manafian, A. Ranjbaran, Solitons in optical metamaterials with anti-cubic law of nonlinearity by generalized G′/ G -expansion method, Optik 162 (2018) 86–94. [11] M. Foroutan, J. Manafian, A. Ranjbaran, Solitons in optical metamaterials with anti-cubic law of nonlinearity by ETEM and IGEM, J. Eur. Opt. Soc. Rapid Publ. 14 (2018) Article 16. [12] A.J.M. Jawad, M. Mirzazadeh, Q. Zhou, A. Biswas, Optical solitons with anti-cubic nonlinearity using three integration schemes, Superlattices Microstruct. 105 (2017) 1–10. [13] S. Kumar, A. Biswas, M. Ekici, Q. Zhou, S.P. Moshokoa, M.R. Belic, Optical solitons and other solutions with anti-cubic nonlinearity by Lie symmetry analysis and additional integration architecture, Optik 185 (2019) 30–38. [14] E.V. Krishnan, A. Biswas, Q. Zhou, M.M. Babatin, Optical solitons with anti-cubic nonlinearity by mapping methods, Optik 170 (2018) 520–526. [15] H. Triki, A.H. Kara, A. Biswas, S.P. Moshokoa, M. Belic, Optical solitons and conservation laws with anti-cubic nonlinearity, Optik 127 (24) (2016) 12056–12062. [16] E.M.E. Zayed, M.E.M. Alngar, A.-G. Al-Nowehy, On solving the nonlinear Schrödinger equation with an anti-cubic nonlinearity in presence of Hamiltonian perturbation terms, Optik 178 (2019) 488–508. [17] E.M.E. Zayed, M.E.M. Alngar, M. El-Horbaty, A. Biswas, M. Ekici, H. Triki, Q. Zhou, S.P. Moshokoa, M.R. Belic, Optical solitons having anti-cubic nonlinearity with strategically sound integration architectures, Optik 185 (2019) 57–70. [18] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. [19] G. Bluman, A. Stephen, Symmetry and Integration Methods for Differential Equations, vol. 154, Springer Science & Business Media, 2008. [20] P.J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer Science & Business Media, 2000. [21] B. Li, J. Zhao, H. Triki, M. Ekici, M. Mirzazadeh, Q. Zhou, W. Liu, Soliton interactions for optical switching systems with symbolic computation, Optik 175 (2018) 177–180. [22] A. Zubair, N. Raza, M. Mirzazadeh, W. Liu, Q. Zhou, Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities, Optik 173 (2018) 249–262. [23] W. Yu, M. Ekici, M. Mirzazadeh, Q. Zhou, W. Liu, Periodic oscillations of dark solitons in nonlinear optics, Optik 165 (2018) 341–344. [24] W. Liu, Y. Zhang, A. Wazwaz, Q. Zhou, Analytic study on triple-S, triple-triangle structure interactions for solitons in inhomogeneous multi-mode fiber, Appl. Math. Comput. 361 (2019) 325–331. [25] W. Liu, Y. Zhang, Z. Luan, Q. Zhou, M. Mirzazadeh, M. Ekici, A. Biswas, Dromion-like soliton interactions for nonlinear Schrodinger equation with variable coefficients in inhomogeneous optical fibers, Nonlinear Dyn. 96 (2019) 729–736.
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons with generalized anti-cubic nonlinearity by Lie symmetry Sachin Kumara, Sandeep Malika, Anjan Biswasb,c,d,e, Yakup Yıldırımf,*, Ali Saleh Alshomranic, Milivoj R. Belicg a
Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151001, Punjab, India Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500, USA c Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Applied Mathematics, National Research Nuclear University, 31 Kashirskoe Shosse, Moscow 115409, Russian Federation e Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa f Department of Mathematics, Faculty of Arts and Sciences, Near East University, 99138 Nicosia, Cyprus g Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar b
A R T IC LE I N F O
ABS TRA CT
Keywords: Solitons Generalized anti-cubic nonlinearity Lie group
Lie symmetry analysis handled the nonlinear Schrödinger's equation that is studied with generalized anti-cubic nonlinear form. These yielded bright, dark and singular optical soliton solutions as well as a bright-singular combo optical soliton solution to the model. The restrictions on the parameters for the existence of such solitons are also presented. OCIS Codes: 060.2310; 060.4510; 060.5530; 190.3270; 190.4370
1. Introduction The concept of anti-cubic (AC) nonlinearity for nonlinear Schrödinger's equation (NLSE) was first proposed by Fedelle et al. during 2003 to study solitons in optical fibers [1]. Later, this concept of AC nonlinearity was generalized by Biswas et al. during 2019 [2,3]. These lead to a plethora of results that has flourished across the board [4–17]. There are various features of these two forms of refractive indices that has been addressed with NLSE as visible in a variety of reported results. These include polarization preserving fibers, birefringent fibers, meta-optics, conservation laws, perturbation theory, soliton cooling and many others. It is time to move on with additional studies. This paper therefore is taking up the task of addressing soliton dynamics with generalized AC nonlinearity, that was introduced during 2019, by the aid of the classic Lie symmetry analysis [18–20]. Subsequently, soliton solutions are retrieved with the implementation of two subsidiary integration norms. These are Kudryashov's approach and the method undetermined coefficients. They yield bright, dark, singular and bright-singular combo soliton solutions. The details are enumerated in the rest of the paper after the governing model is put forth. 1.1. Governing model The governing NLSE with generalized AC nonlinearity is of the form [2,3]:
iq t + aq xx + bq xt + {c1 |q|−(2n + 2) + c2 |q|2n + c3 |q|(2n + 2) } q = 0, ⁎
Corresponding author. E-mail address: [email protected] (Y. Yıldırım).
https://doi.org/10.1016/j.ijleo.2019.163638 Received 2 August 2019; Accepted 13 October 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: Sachin Kumar, et al., Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163638
(1)
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where a , b , c1, c2 and c3 are arbitrary real-valued constants. Here x and t are independent variables, which represents spatial and temporal coordinates, respectively. q is a complex-valued function which depends upon x and t . If c1 = 0, (1) collapses to NLSE with parabolic law or cubic-quintic law of nonlinearity that has been extensively studied [21–25]. In this case, c1 introduces the AC nonlinearity term. In this work, first we generated infinitesimals and Lie symmetries of NLSE with generalized anti-cubic nonlinearity. Then we obtained two vector fields. Eq. (1) is reduced into system of ordinary differential equations (ODE) by using these vector fields, and exact solutions of system of ODE are carried out using various methods. Subsequently, the soliton solutions and other explicit solutions of equation are carried out. 2. Lie symmetry analysis of NLSE In this section, to obtain the infinitesimals of Eq. (1), Lie classical method [18–20] is applied by separating its real and imaginary parts. Firstly, let us consider
q (x , t ) = u (x , t ) e ιv (x , t ),
(2)
where u , v are real valued functions. Using (2) in (1), and separating real and imaginary parts, we have
ut + 2au x vx + a uvxx + bu x vt + but vx + b uvxt = 0 au(2n + 1) u xx − u(2n + 2) vt − au(2n + 2) vx2 + bu(2n + 1) u xt − bu(2n + 2) vx vt + c1 + c2 u(4n + 2) + c3 u(4n + 4) = 0.
(3)
Now let us consider one-parameter (ϵ ) transformation for the system of equations (3) as
x * = x + ϵξ (x , t , u, v ) + O (ξ 2) t * = t + ϵτ (x , t , u, v ) + O (ξ 2) u* = u + ϵη (x , t , u, v ) + O (ξ 2) v * = v + ϵϕ (x , t , u, v ) + O (ξ 2)
(4)
where ξ , τ , η and ϕ are infinitesimals, depending upon x , t , u and v , have to be determined. Associated vector field for these transformation is
V = ξ ∂x + τ ∂t + η∂u + ϕ∂v .
(5)
The second prolongations formula [19,20] for system of equation (3), are
pr (2) V = V + η x
∂ ∂ ∂ ∂ + ϕx + ηt + ϕ xx ∂u x ∂vx ∂ut ∂vxx
pr (2) V = V + ϕ x
∂ ∂ ∂ + ϕt + η xx , ∂vx ∂vt ∂u xx
(6)
where η x , ηt , ϕ x , ϕt , η xx and ϕ xx represent extended infinitesimal. Making use of invariance condition pr (2) V (Δ) = 0 whenever Δ = 0 , the invariance surface condition of (3) became
0 = a η vxx + b η vxt + a u ϕ xx + 2 a (vx η x + u x ϕ x ) + b (vt η x + ut ϕ x + vx ηt ) + ηt 0 = a u2n + 1 η xx − (2 a vx ϕ x + b vt ϕ x + ϕt + b vx ϕt ) u(2n + 2) + ((2n + 1) (a u xx + b u xt ) u2n − (2 n + 2)(vt − a vx2) u(2n + 1) + (4 n + 2) c2 u4 n + 2 + (4 n + 4) c3 u4 n + 3) η .
(7)
Substituting the value of extended infinitesimal and equating the coefficient of various partial derivativs of u and v equal to zero, we get the system of partial differential equations (PDEs). The general solution of desired system of PDEs in ξ , τ , η and ϕ gives the following result
2a ξ = C2 + ⎛−x + t ⎞ C4 b ⎠ ⎝ τ = C1 + t C4 η=0 ϕ = C3 +
x C4, b
(8)
where C1, C2, C3 and C4 are arbitrary constants. Hence we have the corresponding Lie algebra of the symmetry group of system (3) is defined by the infinitesimal generators of the form 2
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V1 =
∂ ∂t
V2 =
∂ ∂x
V3 =
∂ ∂v
∂ ∂ 2a x ∂ +t + V4 = ⎛−x + t⎞ . ∂t b ⎠ ∂x b ∂v ⎝
(9)
The corresponding commutation relations of V1, V2, V3 and V4 are given by
[V1, V2] = 0 [V1, V3] = 0 [V1, V4] =
2a V2 + V1 b
[V2, V3] = 0 [V2, V4] = −V2 +
1 V3 b
[V3, V4] = 0.
(10)
3. Symmetry reduction and invariant solutions In order to find invariant solutions of system (3), we have to solve the corresponding characteristic equation given as
dx dt du dv = = = , ξ τ η ϕ
(11)
where ξ , τ , η and ϕ are given by (8). To solve characteristic equation (11), we will consider two cases of vector fields:
• V + βV + μV •V 3
2
1
4
where β and μ are arbitrary non-zero real numbers. Case (i) V3 + βV2 + μV1 In this case, after solving the characteristic equation (11), we obtain following similarity variables
s =μx − βt u (x , t ) = P (s ) x v (x , t ) = + Q (s ) β
(12)
where s is new independent variable and P , Q are new dependent variable, which depend on s . Substituting (12) in system (3), we have following similarity reduction
0 = (a μ2 β − b μ β 2) P Q″ + (2 a μ − β 2 − b β ) P′ + (2 a μ2 β − 2 b μ β 2) P′ Q′
(13)
0= (a μ2 β 2 − b μ β 3) P (2n + 1) P″ − (2 a μ β − β 3 − b β 2) P (2n + 2) Q′ − (a μ2 β 2 − b μ β 3) P (2n + 2) Q′2 − a P (2n + 2) + c3 β 2 P (4n + 4) + c2 β 2 P (4n + 2) + c1 β 2,
(14)
where (′) denotes derivative with respct to s . Integrating (13) twice with respct to s , we obtained
Q=
k1 μ β (aμ − bβ )
2
aμ − β − bβ ) s + k2 ∫ P12 ds − 12 (2μβ (aμ − bβ )
(15)
where k1 and k2 are arbitrary constant. Substituting the Q from (15) into (14), we have
P″= − +
c3 c2 c1 P (2n + 3) − P (2n + 1) − P −(2n + 1) μ (aμ − bβ ) μ (aμ − bβ ) μ (aμ − bβ ) k2 1 ( − 2 b β − b2 + 4 a μ − β 2 ) P −3. P+ 2 2 1 4 μ β (aμ − bβ )2 μ2 (aμ − bβ )2 3
(16)
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Integrating (16), by taking integrable constant equal to zero, we have
(P ′)2 c3 c2 c1 = − P (2n + 4) − P (2n + 2) + P −2n 2 (2n + 4) μ (aμ − bβ ) (2n + 2) μ (aμ − bβ ) 2 n μ (aμ − bβ ) +
k12 ( − 2 b β − b2 + 4 a μ − β 2 ) 2 P −2. P − 2 2 2 2 8 μ (aμ − bβ ) 2 μ β (aμ − bβ )2
(17)
Let us consider
P 2 (s ) = A (s ),
(18)
we have
(P′)2 =
(A′)2 , 4A
(19)
where (′) denotes derivative with respct to s . Using (19) along with (18) in (17), we have
(A′)2= − +
4c3 4c2 4c1 A(n + 3) − A(n + 2) + A(−n + 1) (n + 2) μ (aμ − bβ ) (n + 1) μ (aμ − bβ ) nμ (aμ − bβ ) 4 k12 (−2bβ − b2 + 4aμ − β 2) 2 . A − 2 2 2 2 μ β (aμ − bβ )2 μ (aμ − bβ )
(20)
4. Kudryashov method Here, the algorithm for Kudryashov method [1] is presented. This method will be used to obtain exact solutions of (1). Let us consider the nonlinear PDE as below:
T (u, ut , u x , utt , u xx , u xt , …) = 0.
(21)
Use the traveling wave solution
u (x , t ) = y (η),
η = kx − wt
(22)
Eq. (21) will be reduced into nonlinear ODE as below:
T (y, wy′, ky′, w y ′′, k y ′′, wky ′′, …) = 0.
(23)
Now to wind the dominant term, substitute
y (η) = η−p
(24)
where p > 0 , into (23). Now compare the degree of all the terms and choose two or more smallest degree terms. The maximum value of p is pole of (23) and denote it as N . Now let us consider
y (η) = a0 + a1 R (η) + ⋯+aN R (η) N
(25)
where a 0 , a1, …, aN are unknown to be determined and
R (η) =
1 . 1 + eη
(26)
The function R (η) satisfy the following relation (27)
R′ = R2 − R
where (′) denotes the derivative with respect to η . Using (27), we can calculate the derivatives of y (η) in form of power of R . Differentiating (25) with respect to η and using (27), we have N
y′ =
∑ ai i (R − 1) Ri i=1 N
y″ =
∑ ai i ((i + 1) R2 − (2i + 1) R + i) Ri .
(28)
i=1
Now substitute (25), along with (27) and (28), into (23), and equating the coefficients of same power of R to zero, we obtain algebraic system of equations. Then by solving this system, we obtain the value of various parameters w, k , a0 , a1, …, aN . Now, to apply the Kudryashov's method to (20), first we substitute
A (s ) = s−p, p > 0 into (20), we obtain the pole N =
(29) 2 . n+1
Then we move for exact solution of (20) in following form 4
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A (s ) = a1 R (s ) n + 1
(30)
with a1 is unknown parameters to be determined and R (s ) satisfying the relation (31)
R′ = R2 − R
where (′) denotes derivative with respct to s . Substituting (30) into (20) and by comparing coefficients of same power of R equal to zero, we have following system of equations:
0 = μ β 2 n (c3 a1n + 3 (n + 1)2 + μ a12 (2 + n)(μ a − β b))(μ a − β b) 0 = μ β 2 c2 a12 + n n (n + 1)(2 + n)(μ a − β b) 0 = a12β 2μ2 n (2 + n)(μ a − β b)2 0 = a12 β 2 n (2 + n)(n2b2 + 2 n2β b + n2β 2 − 4 n2μ a + 2 nβ 2 − 8 nμ a + 2 nb2 + 4 nβ b + β 2 + 4 β 2μ2 b2 + 2 β b − 8 β bμ3a + 4 μ4 a2 − 4 μ a + b2) 0 = k12 n (n + 1)2 (2 + n) 0 = μ β 2 c1 a1−n + 1 (n + 1)2 (2 + n)(μ a − β b).
(32)
After solving the system (32), we have
μ=
b2 a
k1 = 0 β=b
(33)
and a1 is arbitrary. Corresponding soliton solution of (1), is therefore given by
a1
q (x , t ) =
2
e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(1 + e (μ x − β t ) ) n + 1 a1
=
1
e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
{1 + cosh(μx − βt ) + sinh(μx − βt )} n + 1
(34)
where a1, μ , β are given by (33). Thus, (34) represents a bright-singular combo optical soliton solution to the model and it exists provided a1 > 0 . 5. Undetermined coefficients In this section, soliton solutions of the main equation (1) will be found from the corresponding reduction (20). For this hyperbolic function method will be used to obtained the bright, dark and singular 1-soliton solutions of (1). 5.1. Bright soliton solution To obtain the solution by sech function method [13], in Eq. (20), use the following
A (s ) = a1 sechn1 (m1 s ) A′ (s ) = −a1 n1 m1 sechn1 (m1 s ) tanh(m1 s ).
(35)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of sech function equal to zero, we have following system of equations:
2 . n+1
Now by comparing coefficients of same
0 = c2 a1n + 2 μ β 2 n (n + 1)(n + 2)(aμ − β b) 0 = n μ β 2 a12 (c3 a1 (n + 1)2a1n − (n + 2) m12 μ (aμ − β b))(aμ − β b) 0 = a1−n + 1 c1 μ β 2 (n + 1)2 (n + 2)(aμ − β b) 0 = β 2 a12 n (n + 2)(n2 b2 + 2 β b n2 + β 2 n2 − 4 a μ n2 + 2 β 2 n − 8 a μ n + 2 nb2 + 4 β b n + 4 β 2 m12 μ2 b2 + β 2 − 8 β m12 μ3 a b + 2 β b + 4 m12 μ4 a2 − 4 a μ + b2) 0 = k12 n (n + 1)2 (n + 2).
(36)
After solving the system (36), we have 5
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μ=
b2 a
k1 = 0 β=b
(37)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 sechn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ ι⎜ x − 1 (μ x − β t ) + k2⎟ β 2 μβ (aμ − bβ ) ⎝ ⎠
(38)
where a1, μ, β , m1 are given by (37).
5.2. Dark soliton solution To obtain the solution by tanh function method [11], in Eq. (20), use the following
A (s ) = a1 tanhn1 (m1 s ),
|s| ≤
π 2 m1
A′ (s ) = a1 n1 m1tanhn1− 1 (m1 s ) − a1 n1 m1tanhn1+ 1 (m1 s ).
(39)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of tanh function equal to zero, we have following system of equations:
2 . n+1
Now by comparing coefficients of same
0 = μ β 2 (n + 2)(c1 (n + 1)2a1−n + 1 + n μ a12 m12 (aμ − bβ ))(aμ − bβ ) 0 = μ β 2 n (c3 (n + 1)2a1n + 3 + μ a12 m12 (aμ − bβ )(n + 2))(aμ − bβ ) 0 = a1n + 2 c2 μ β 2 n (n + 1)(n + 2)(aμ − bβ ) 0 = a12 β 2 n (n + 2)(−2 n2 b β − n2 β 2 − n2 b2 + 4 n2 a μ − 4 b β n − 2 nb2 + 8 a μ n − 2 n β 2 + 8 β 2 m12 μ2 b2 − β 2 − 16 β m12 μ3 a b − 2 b β − b2 + 8 m12 μ4 a2 + 4 a μ) 0 = k12 n (n + 1)2 (n + 2).
(40)
After solving the system (40), we have
μ=
b2 a
k1 = 0 β=b
(41)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 tanhn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(42)
where a1, μ, β , m1 are given by (41).
5.3. Singular soliton solution To obtain the solution by csch function method [9], in Eq. (20), use the following
A (s ) = a1 cschn1 (m1 s ) A′ (s ) = −a1 n1 m1 cschn1 (m1 s ) coth(m1 s ).
(43)
By balancing the highest nonlinear term and highest derivative term, we have n1 = power of csch function equal to zero, we have following system of equations: 6
2 . n+1
Now by comparing coefficients of same
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0 = μ β 2 a1−n + 1 c1 (n + 2)(n + 1)2 (aμ − bβ ) 0 = a1n + 2 c2 μ β 2 n (n + 1)(n + 2)(aμ − bβ ) 0 = μ β 2 n (c3 (n + 1)2a1n + 3 + μ m12 a12 (n + 2)(aμ − bβ ))(aμ − bβ ) 0 = a12 β 2 n (n + 2)(−4 n2 a μ + 2 n2 b β + n2 b2 + n2 β 2 + 4 n b β + 2 n β 2 − 8 n a μ + 2 n b2 + 4 β 2 m12 μ2 b2 + β 2 + 2 b β − 8 β m12 μ3 a b + 4 m12 μ4 a2 − 4 a μ + b2) 0 = k12 n (n + 1)2 (n + 2).
(44)
After solving the system (44), we have
μ=
b2 a
k1 = 0 β=b
(45)
and a1, m1 is arbitrary. Corresponding bright soliton solution of (1), is given by
q (x , t ) =
1
a1 cschn + 1 [m1 (μ x − β t )] e
(2aμ − β 2 − bβ ) ⎛ ⎞ (μ x − β t ) + k2⎟ ι⎜ x − 1 β 2 μβ (aμ − bβ ) ⎝ ⎠
(46)
where a1, μ, β , m1 are given by (45). Again, these bright, dark and singular soliton solutions are guaranteed to exist provided a1 > 0 . Case (ii) V4 In this case, after solving the characteristic equation (11), we obtain following similarity variables
w =xt −
a 2 t b
u (x , t ) = F (w ) w a v (x , t ) = − + 2 t + G (w ), bt b
(47)
where w is new independent variable and F , G are new dependent variable, which depend upon w . Substituting (47) in (3), we have following similarity reduction (48)
w F G″ + 2 w F ′ G′ + F G′ = 0
b3 w F 2n + 1 F ″ − a F 2n + 2 − b3 w F 2n + 2 (G′)2 + b3 F 2n + 1 F ′ + c3 b2 F 4n + 4 + c2 b2 F 4n + 2 + c1 b2 = 0,
(49)
where (′) denotes derivative with respct to w . Integrating (48) twice with respct to w , we have
G = l1 + l2
∫ w1F 2 dw
(50)
where l1 and l2 are arbitrary constant. Substituting the value of G from (50) into (49), we have
F″ = −
l2 1 c3 2n + 3 c c a F − 2 F 2n + 1 − 1 F −2n − 1 + 3 F − F ′ + 22 F −3. w bw bw bw b w w
(51)
Corresponding solution of (1), is given by
q (x , t ) = F (w ) e
ι ⎛⎜− w + a2 t + l1+ l2 bt b ⎝
∫ w F1(w)2 dw⎞ ⎟
(52)
⎠
where w is given by (47) and F satisfies Eq. (51). 6. Conclusions The paper extracted bright, dark and singular solitons as well as bright-singular combo soliton solutions to the governing NLSE that is considered with generalized AC nonlinearity. The integration algorithm is Lie symmetry analysis, powerful as it is and one of the most classic forms. The enlightening results are thus very encouraging to venture further along in this direction. In order to fill in the gap, one needs to address soliton cooling, as well as extend the study to perturbed NLSE with generalized AC nonlinearity. In this context, both deterministic as well as stochastic perturbation terms need to be addressed. This would mean studying this extended model by additional norms such as semi-inverse variational principle, extended trial function scheme, F -expansion and extended Jacobi's elliptic function method, just to name a few. Generalized AC nonlinearity being a new model, a lot is on the table. 7
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