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CUBIC–QUARTIC OPTICAL SOLITONS IN BIREFRINGENT FIBERS WITHFOUR FORMS OF NONLINEAR REFRACTIVE INDEX
Journal Pre-proof CUBIC–QUARTIC OPTICAL SOLITONS IN BIREFRINGENT FIBERS WITHFOUR FORMS OF NONLINEAR REFRACTIVE INDEX Yakup Yildirim, Anjan Biswas, Padmaja Guggilla, Fouad Mallawi, Milivoj R. Belic
PII:
S0030-4026(19)31783-8
DOI:
https://doi.org/10.1016/j.ijleo.2019.163885
Reference:
IJLEO 163885
To appear in:
Optik
Received Date:
16 September 2019
Accepted Date:
23 November 2019
Please cite this article as: { doi: https://doi.org/ This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
*Manuscript
CUBIC–QUARTIC OPTICAL SOLITONS IN BIREFRINGENT FIBERS WITH FOUR FORMS OF NONLINEAR REFRACTIVE INDEX Yakup Yıldırım 1 , Anjan Biswas 1
2, 3, 4, 5
, Padmaja Guggilla 2 , Fouad Mallawi
3
& Milivoj R. Belic
Department of Mathematics, Faculty of Arts and Sciences, Near East University, 99138 Nicosia, Cyprus 2
Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762–7500, USA
4
of
Department of Mathematics, King Abdulaziz University, Jeddah–21589, Saudi Arabia Department of Applied Mathematics, National Research Nuclear University 31 Kashirskoe Shosse, Moscow–115409, Russian Federation
ro
3
5
Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar
re
6
-p
Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria–0008, South Africa
Abstract
lP
This paper studies cubic–quartic solitons in birefringent fibers having Kerr, parabolic, quadratic–cubic and non–local laws of refractive index. Bright, dark and singular solitons are recovered by Riccati function approach, sine-Gordon function technique and F-expansion scheme.
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OCIS Codes: 060.2310; 060.4510; 060.5530; 190.3270; 190.4370 Key words: solitons; riccati function; sine-Gordon function; F-expansion principle.
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1
6
1
One of the most appreciated models that govern the dynamics of pulse transfer through optical fibers and other forms of waveguides is the cubic–quartic nonlinear Schr¨odinger’s equation (CQ-NLSE) where third order dispersion (3OD) and fourth order dispersion (4OD) terms are included when group velocity dispersion (GVD) is negligible and consequently discarded. This model has been extensively studied for polarization–preserving fibers all across [1–8]. Therefore, our next stop is with birefringent fibers. This paper will thus address the dynamics of soliton propagation with differential group delay for coupled CQ-NLSE with and without fourwave mixing (4WM) that depends on the type of nonlinearity. These four nonlinear forms that will be handled in this paper are with Kerr, parabolic, quadratic-cubic and non-local laws. The three integration schemes adopted in the paper are the Riccati function, sine-Gordon function and F-expansion principles. These three approaches collectively reveal bright, dark and singular solitons to the models in birefringent fibers. Additional soliton solutions that emerges from the scheme are also listed as byproducts of the algorithms. The details are presented in the subsequent couple of upcoming sections along with the parameter restrictions that make the soliton solutions viable. These restrictions are listed as constraints.
2
F-EXPANSION: A RAPID SKIM THROUGH
of
The steps of applying the method to any governing model are as follows: Step-1: A model can be taken as The important equation emerged from Eq.(1) is as G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
-p
by using of
ro
S (u, ut , ux , utt , uxt , uxx , ...) = 0.
u(x, t) = U (ϑ), ϑ = x − pt. Step-2: The solution structure of Eq.(2) is considered as N X
re
U (ϑ) =
µi F i (ϑ)
(1)
(2) (3)
(4)
i=0
lP
and also the number N comes from Eq.(2) in consequence of using balancing condition. The function F holds p F ′ (ϑ) = P F 2 (ϑ) + QF (ϑ) + R (5) where P , Q and R are constants. The strategic solutions of Eq.(5) are as
= sn (ϑ) = tanh (ϑ) , P = m2 , Q = −(1 + m2 ), R = 1, m → 1,
F (ϑ)
= ns (ϑ) = coth (ϑ) , P = 1, Q = −(1 + m2 ), R = m2 , m → 1,
F (ϑ)
= sc (ϑ) = tan (ϑ) , P = 1 − m2 , Q = 2 − m2 , R = 1, m → 0,
F (ϑ)
= cs (ϑ) = cot (ϑ) , P = 1, Q = 2 − m2 , R = 1 − m2 , m → 0,
ur na
F (ϑ)
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INTRODUCTION
F (ϑ)
= cn (ϑ) = sech (ϑ) , P = −m2 , Q = 2m2 − 1, R = 1 − m2 , m → 1.
F (ϑ)
= ds (ϑ) = csch (ϑ) , P = 1, Q = 2m2 − 1, R = −m2 1 − m2 , m → 1
F (ϑ)
= nc (ϑ) = sec (ϑ) , P = 1 − m2 , Q = 2m2 − 1, R = −m2 , m → 0
F (ϑ)
= ns (ϑ) = csc (ϑ) , P = 1, Q = −(1 + m2 ), R = m2 , m → 0
(6)
Step-3: The strategic equations are easily obtained as long as the solution structure (4) along with Eq.(5) is put in Eq.(2). Thus, the important results µi emerged from these strategic equations are obtained. If we use the results µi in Eq.(4), solitons in an optical fiber communication system are recovered by Eq.(6). 2
2.1
KERR LAW
The CQ-NLSE equation in polarization-preserving fibers with the kerr law is: iqt + iaqxxx + bqxxxx + c |q|2 q = 0.
(7)
In the model (7), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficient c provides the Kerr media effect which is known as the basic case of fiber nonlinearity. Most optical fibers which has been quite popular recently comply with this law nonlinearity. The CQ-NLSE equation in coupled vector form without FWM is: iut + ia1 uxxx + b1 uxxxx + c1 |u|2 + d1 |v|2 u = 0, ivt + ia2 vxxx + b2 vxxxx + c2 |v|2 + d2 |u|2 v = 0
(8)
of
with the constants cj , dj which assure the existence of the self–phase modulation (SPM) and cross-phase modulation (XPM) sequentially.
u(x, t) = U1 (ϑ)eiϕ1 (x,t) v(x, t) = U2 (ϑ)eiϕ2 (x,t) .
ro
Solitons with the CQ-NLSE in coupled vector form without FWM along with the F -expansion principle are recovered in this section. Firstly, the strategic equations are easily employed as
(9)
ϑ = x − pt
re
while the amplitude component is as
-p
In Eq.(9), the phase component is shown as Uj (ϑ) and the velocity is given by
ϕj (x, t) = −kj x + wj t + ζj
(10) (11)
The real and imaginary equations are (4)
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + cj Uj3 + dj Uj U˜j2 = 0,
ur na
bj Uj
lP
where the parameters kj , ζj , wj sequentially correspond to the frequency, phase and wave number.
(aj − 4 bj kj ) Uj′′ − p + 3 aj kj 2 − 4 bj kj 3 Uj = 0
(12)
(13)
respectively as long as Eq.(9) is put in Eqs.(8)-(13). In order to use the balancing rule, the important results emerged from Eq.(12) by using of U˜j = Uj are as (4)
bj Uj
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + (cj + dj ) Uj3 = 0,
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(14)
aj = 4 bj kj ,
p = −3 aj kj 2 + 4 bj kj 3 .
(15)
Uj = µ0 + µ1 Fj + µ2 Fj2
(16)
Eq.(4) can be given as which comes from Eq.(14) in consequence of using balancing rule.
3
The strategic equations are easily obtained as long as Eq.(16) along with Eq.(5) is put in Eq.(14). Thus, the important results are as (27 cj 3 kj 6 + 81 cj 2 dj kj 6 + 81 cj dj 2 kj 6 + 27 dj 3 kj 6 + 540 P Rcj 3 kj 2 +1620 P Rcj 2 dj kj 2 + 1620 P Rcj dj 2 kj 2 + 540 P Rdj 3 kj 2 − 180 Q2 cj 3 kj 2 −540 Q2cj 2 dj kj 2 − 540 Q2cj dj 2 kj 2 − 180 Q2dj 3 kj 2 + 1800 P QRcj 3 +5400 P QRcj 2 dj + 5400 P QRcj dj 2 + 1800 P QRdj 3 − 400 Q3cj 3 −1200 Q3cj 2 dj − 1200 Q3cj dj 2 − 400 Q3dj 3 )f 6 + (4860 aj cj 2 kj 5
+9720 aj cj dj kj 5 + 4860 aj dj 2 kj 5 + 32400 P Raj cj 2 kj + 64800 P Raj cj dj kj +32400 P Raj dj 2 kj − 10800 Q2aj cj 2 kj − 21600 Q2aj cj dj kj − 10800 Q2aj dj 2 kj )f 4 + 291600 aj 2 cj kj 4 + 291600 aj 2 dj kj 4 f 2 + 5832000 aj 3 kj 3 = 0,
wj = −
−3 f 2cj kj 2 − 3 f 2 dj kj 2 + 10 f 2Qcj + 10 f 2 Qdj − 180 aj kj , 30f (cj + dj )
ro
µ0 =
f 2 (cj + dj ) , µ1 = 0, µ2 = f P, 120
of
bj = −
1 (−13 f 4 cj 2 kj 4 − 26 f 4 cj dj kj 4 − 13 f 4dj 2 kj 4 (600 cj + 600 dj ) f 2
-p
+360 f 4P Rcj 2 + 720 f 4 P Rcj dj + 360 f 4P Rdj 2 − 120 f 4Q2 cj 2 − 240 f 4Q2 cj dj −120 f 4Q2 dj 2 − 1560 f 2aj cj kj 3 − 1560 f 2aj dj kj 3 − 64800 aj 2 kj 2 ).
lP
re
If we use the results (17) along with Eq.(6) in Eq.(16), dark solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 − 20 f 2c1 − 20 f 2 d1 − 180 a1k1 i(−k x+w t+ζ ) 30f (c1 + d1 ) 1 1 1 e u(x, t) = +f tanh2 x + 3 a1 k1 2 − 4 b1 k1 3 t −3 f 2c2 k2 2 − 3 f 2 d2 k2 2 − 20 f 2c2 − 20 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f tanh2 x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
Similarly, singular solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 − 20 f 2c1 − 20 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = +f coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t
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i(−k x+w t+ζ ) 1 1 1 e
−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 − 20 f 2c2 − 20 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f coth2 x + 3 a2 k2 2 − 4 b2 k2 3 t
Similarly, bright solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 + 10 f 2c1 + 10 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = −f sech2 x + 3 a1 k1 2 − 4 b1 k1 3 t 4
i(−k x+w t+ζ ) 1 1 1 e
(17)
(18)
(19)
(20)
(21)
(22)
Similarly, singular solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 + 10 f 2c1 + 10 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = +f csch2 x + 3 a1 k1 2 − 4 b1 k1 3 t
i(−k x+w t+ζ ) 1 1 1 e
−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 + 10 f 2c2 + 10 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f csch2 x + 3 a2 k2 2 − 4 b2 k2 3 t
2.2
(23)
(24)
(25)
of
PARABOLIC LAW
ro
The CQ-NLSE equation in polarization-preserving fibers with the parabolic law is given by 2 4 iqt + iaqxxx + bqxxxx + c |q| + d |q| q = 0.
(26)
-p
In the model (26), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficients c, d provide the parabolic nonlinearity effect which has been quite popular recently and is known as a generalized state of kerr fiber nonlinearity.
re
The CQ-NLSE equation in coupled vector form without FWM is: iut + ia1 uxxx + b1 uxxxx + c1 |u|2 + d1 |v|2 u + e1 |u|4 + g1 |u|2 |v|2 + h1 |v|4 u = 0,
lP
2 2 4 2 2 4 ivt + ia2 vxxx + b2 vxxxx + c2 |v| + d2 |u| v + e2 |v| + g2 |v| |u| + h2 |u| v = 0
(27)
with the constants cj ,ej and dj , gj , hj which assure SPM and XPM sequentially.
ur na
The important result is as
(4)
bj Uj
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + (cj + dj ) Uj3 + (ej + gj + hj ) Uj5 = 0
(28)
as long as Eq.(9) is put in Eq.(27) along with U˜j = Uj .
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−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 + 10 f 2c2 + 10 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = −f sech2 x + 3 a2 k2 2 − 4 b2 k2 3 t
Eq.(4) can be given as
Uj = µ0 + µ1 Fj
(29)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(29) along with Eq.(5) is put in Eq.(28). Thus, the important results are as bj = −
6 P aj kj + cj µ21 + dj µ1 2 , µ0 = 0, 4P −3 kj 2 + 5 Q
−3 ej kj 2 − 3 gj kj 2 − 3 hj kj 2 + 5 Qej + 5 Qgj + 5 Qhj µ41 5
+ (−6 P cj − 6 P dj ) µ21 − 36 P 2 aj kj = 0,
wj = −
1 (−6 P aj kj 5 + µ1 2 cj kj 4 + µ1 2 dj kj 4 2 4P −3 kj + 5 Q
+20 P Qaj kj 3 − 6µ1 2 Qcj kj 2 − 6 µ1 2 Qdj kj 2 + 72 P 2 Raj kj
−54 P Q2aj kj + 12 µ1 2 P Rcj + 12 µ1 2 P Rdj + µ1 2 Q2 cj + µ1 2 Q2 dj ).
(30)
If we use the results (30) along with Eq.(6) in Eq.(29), dark solitons are obtained as µ1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t ×e
=
f 2 (c1 + d1 )(k14 + 12k12 + 16) − 2a1 k1 (3k14 + 20k12 + 72) t+ζ1 4 (3k12 + 10)
µ1 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
i−k2 x+
f 2 (c2 + d2 )(k24 + 12k22 + 16) − 2a2 k2 (3k24 + 20k22 + 72) t+ζ2 4 (3k22 + 10)
Similarly, singular solitons are obtained as
×e
=
f 2 (c1 + d1 )(k14 + 12k12 + 16) − 2a1 k1 (3k14 + 20k12 + 72) t+ζ1 4 (3k12 + 10)
µ1 coth x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
v(x, t)
i−k1 x+
re
µ1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
i−k2 x+
(31)
(32)
lP
u(x, t) =
ro
v(x, t)
i−k1 x+
of
-p
u(x, t) =
f 2 (c2 + d2 )(k24 + 12k22 + 16) − 2a2 k2 (3k24 + 20k22 + 72) t+ζ2 4 (3k22 + 10)
(33)
(34)
Similarly, bright solitons are obtained as
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u(x, t) =
v(x, t)
=
µ1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
i−k1 x−
f 2 (c1 + d1 )(k14 − 6k12 + 1) + 2a1 k1 (3k14 − 10k12 + 27) t+ζ1 4 (3k12 − 5)
µ1 sech x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
i−k2 x−
f 2 (c2 + d2 )(k24 − 6k22 + 1) + 2a2 k2 (3k24 − 10k22 + 27 t+ζ2 4 (3k22 − 5) 6
(35)
(36)
Similarly, singular solitons are obtained as = µ1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t
u(x, t)
×e
i−k1 x+
×e
i−k2 x+
f 2 (c2 + d2 )(k24 − 6k22 + 1) − 2a2 k2 (3k24 − 10k22 + 27 t+ζ2 4 (3k22 − 5)
(37)
(38)
QUADRATIC-CUBIC LAW
of
2.3
f 2 (c1 + d1 )(k14 − 6k12 + 1) − 2a1 k1 (3k14 − 10k12 + 27) t+ζ1 4 (3k12 − 5)
= µ1 csch x + 3 a2 k2 2 − 4 b2 k2 3 t
v(x, t)
(39)
ro
The CQ-NLSE equation in polarization-preserving fibers with the quadratic-cubic law is given by 2 iqt + iaqxxx + bqxxxx + c |q| + d |q| q = 0.
-p
In the model (39), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficients c, d provide the quadratic-cubic law effect.
q 2 2 2 2 |v| + |u| + vu∗ + v ∗ u + d2 |v| + e2 |u| v = 0
(40)
lP
ivt + ia2 vxxx + b2 vxxxx + c2 v
re
The CQ-NLSE in coupled vector form without FWM is: q iut + ia1 uxxx + b1 uxxxx + c1 u |u|2 + |v|2 + uv ∗ + u∗ v + d1 |u|2 + e1 |v|2 u = 0,
with the constants dj and ej which assure SPM and XPM sequentially. Also, the constant cj assures the existence of the quadratic media in birefringent fibers where the first two terms correspond to SPM and XPM sequentially while the last two terms refer to 4WM.
ur na
The important result is as (4)
bj Uj
+ 3aj k − 6bj k 2 Uj′′ − w + aj k 3 − bj k 4 Uj + 2cj Uj2 + (dj + ej ) Uj3 = 0
(41)
as long as Eq.(9) is put in Eq.(41) along with
U˜j = Uj ,
ϕ(x, t) = −kx + wt + ζ.
(42)
Uj = µ0 + µ1 Fj + µ2 Fj2
(43)
Eq.(4) can be given as
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which comes from Eq.(41) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(43) along with Eq.(5) is put in Eq.(41). Thus, the important results are as (27 k 6 dj 3 + 81 k 6dj 2 ej + 81 k 6 dj ej 2 + 27 k 6ej 3 + 540 P Rk 2dj 3 +1620 P Rk 2dj 2 ej + 1620 P Rk 2dj ej 2 + 540 P Rk 2ej 3 − 180 Q2k 2 dj 3 −540 Q2k 2 dj 2 ej − 540 Q2k 2 dj ej 2 − 180 Q2 k 2 ej 3 + 1800 P QRdj 3
+5400 P QRdj 2 ej + 5400 P QRdj ej 2 + 1800 P QRej 3 − 400 Q3dj 3 −1200 Q3dj 2 ej − 1200 Q3dj ej 2 − 400 Q3ej 3 )f 6 + (270 k 4 cj dj 2 7
+540 k 4cj dj ej + 270 k 4cj ej 2 − 5400 P Rcj dj 2 − 10800 P Rcj dj ej
+(4860 k 5aj dj 2 + 9720 k 5aj dj ej + 4860 k 5aj ej 2 + 32400 P Rkaj dj 2
+64800 P Rkaj dj ej + 32400 P Rkaj ej 2 − 10800 Q2kaj dj 2 − 21600 Q2kaj dj ej −10800 Q2kaj ej 2 )f 4 + 291600 k 4aj 2 dj + 291600 k 4aj 2 ej f 2 + (32400 k 3aj cj dj +32400 k 3aj cj ej − 4000 cj 3 )f 3 + 972000 k 2 aj 2 cj f + 5832000 k 3aj 3 = 0
w=−
1 (−13 f 4 k 4 dj 2 − 26 f 4k 4 dj ej − 13 f 4 k 4 ej 2 (600 dj + 600 ej ) f 2
+360 f 4P Rdj 2 + 720 f 4P Rdj ej + 360 f 4P Rej 2 − 120 f 4Q2 dj 2 − 240 f 4Q2 dj ej −120 f 4Q2 ej 2 − 1560 f 2k 3 aj dj − 1560 f 2k 3 aj ej + 800 f 2 cj 2 − 64800 k 2aj 2 ),
bj = −
f 2 (dj + ej ) , µ1 = 0, µ2 = f P. 120
of
−3 f 2 k 2 dj − 3 f 2 k 2 ej + 10 f 2 Qdj + 10 f 2 Qej − 20 f cj − 180 aj k , 30f (dj + ej )
ro
µ0 =
re
-p
If we use the results (44) along with Eq.(6) in Eq.(43), dark solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 − 20 f 2 d1 − 20 f 2e1 − 20f c1 − 180 a1k i(−kx+wt+ζ) 30f (d1 + e1 ) e u(x, t) = +f tanh2 x + 3 a1 k 2 − 4 b1 k 3 t
lP
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 − 20 f 2 d2 − 20 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = +f tanh2 x + 3 a2 k 2 − 4 b2 k 3 t
ur na
Similarly, singular solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 − 20 f 2 d1 − 20 f 2e1 − 20f c1 − 180 a1k 30f (d1 + e1 ) u(x, t) = +f coth2 x + 3 a1 k 2 − 4 b1 k 3 t
i(−kx+wt+ζ) e
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 − 20 f 2 d2 − 20 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = 2 2 3 +f coth x + 3 a2 k − 4 b2 k t
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−5400 P Rcj ej 2 + 1800 Q2cj dj 2 + 3600 Q2cj dj ej + 1800 Q2cj ej 2 )f 5
Similarly, bright solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 + 10 f 2 d1 + 10 f 2e1 − 20f c1 − 180 a1k 30f (d1 + e1 ) u(x, t) = −f sech2 x + 3 a1 k 2 − 4 b1 k 3 t
i(−kx+wt+ζ) e
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 + 10 f 2 d2 + 10 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = 2 2 3 −f sech x + 3 a2 k − 4 b2 k t
8
(44)
(45)
(46)
(47)
(48)
(49)
(50)
× ei(−kx+wt+ζ)
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 + 10 f 2 d2 + 10 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = +f csch2 x + 3 a2 k 2 − 4 b2 k 3 t
2.4
(51)
(52)
NON-LOCAL LAW
The CQ-NLSE equation in polarization-preserving fibers with the non-local law is given by 2 iqt + iaqxxx + bqxxxx + c |q| q = 0.
(53)
of
xx
In the model (53), the constant a is coefficient of the 3OD while the constants b assure the existence of the 4OD. Also, the coefficient c provides the non-local law effect.
ro
The CQ-NLSE equation in coupled vector form without FWM is: 2 2 u = 0, + d1 |v| iut + ia1 uxxx + b1 uxxxx + c1 |u| xx
2 ivt + ia2 vxxx + b2 vxxxx + c2 |v|
-p
xx
xx
2 v=0 + d2 |u| xx
(54)
re
with the constants cj and dj which assure SPM and XPM sequentially. The important result is as
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj
lP
(4)
bj Uj
ur na
+2 (cj + dj ) Uj Uj′
2
+ 2 (cj + dj ) Uj2 Uj′′ = 0
(55)
as long as Eq.(9) is put in Eq.(55) along with U˜j = Uj . Eq.(4) can be given as
Uj = µ0 + µ1 Fj
(56)
which comes from Eq.(55) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(56) along with Eq.(5) is put in Eq.(55). Thus, the important results are as 3aj kj , µ0 = 0, bj = − 2 −3 kj 2 + Q
Jo
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Similarly, singular solitons are obtained as −3 f 2 d1 k 2 − 3 f 2 e1 k 2 + 10 f 2 d1 + 10 f 2 e1 − 20f c1 − 180 a1 k 30f (d1 + e1 ) u(x, t) = +f csch2 x + 3 a1 k 2 − 4 b1 k 3 t
µ1 = ±
wj = −
s
6P aj kj , −3 cj kj 2 − 3 dj kj 2 + Qcj + Qdj
−3 kj 4 + 2 Qkj 2 + 12 RP − 3 Q2 aj kj 2 −3 kj 2 + Q
9
(57)
If we use the results (57) along with Eq.(6) in Eq.(56), dark solitons are obtained as
± −
2
3 c1 k1
3 k1 4 + 4 k1 2 a1 k1 t+ζ1 2 3 k1 2 + 2
×e
v(x, t)
= ±
i−k1 x−
s
−
3 c2 k2
2
i−k2 x−
6a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 + 2 c2 + 2 d2
Similarly, singular solitons are obtained as
= ±
−
3 c1 k1
±
−
re
3 c2 k2
×e
(60)
lP
=
2
6a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 + 2 c2 + 2 d2
ur na
v(x, t)
i−k1 x−
s
6a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 + 2 c1 + 2 d1
3 k1 4 + 4 k1 2 a1 k1 t+ζ1 2 3 k1 2 + 2
×e
2
(59)
-p
u(x, t)
s
(58)
3 k2 4 + 4 k2 2 a2 k2 t+ζ2 2 3 k2 2 + 2
×e
6a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 + 2 c1 + 2 d1
of
s
ro
u(x, t) =
i−k2 x−
3 k2 4 + 4 k2 2 a2 k2 t+ζ2 2 3 k2 2 + 2
(61)
Similarly, bright solitons are obtained as
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
u(x, t) =
±
s
3 c1 k1
×e
2
i−k1 x−
6a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 − c1 − d1 3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
10
(62)
v(x, t)
=
±
s
3 c2 k2
2
6a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 − c2 − d2 3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
×e
i−k2 x−
(63)
Similarly, singular solitons are obtained as
3 c1 k1
= ±
−
×e
i−k2 x−
(64)
6a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 − c2 − d2
3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
(65)
RICCATI EQUATION TECHNIQUE: A QUICK OVERVIEW
lP
3
3 c2 k2
2
re
v(x, t)
i−k1 x−
s
6a1 k1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 − c1 − d1
3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
×e
2
of
± −
ro
s
-p
u(x, t) =
The steps of applying the method to any governing model are as follows : Step-1: A model can be taken as
ur na
S (u, ut , ux , utt , uxt , uxx , ...) = 0.
(66)
The important equation emerged from Eq.(66) is as
by using of
G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
(67)
u(x, t) = U (ϑ), ϑ = x − pt.
(68)
Step-2: The solution structure of Eq.(67) is considered as
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
U (ϑ) =
N X
Ai V i (ϑ)
(69)
i=0
and also the number N comes from Eq.(67) in consequence of using balancing condition. The function V (ϑ) holds the Riccati equation as V ′ (ϑ) = S2 V 2 (ϑ) + S1 V (ϑ) + S0 , S2 6= 0 (70)
11
V (ϑ) =
−
S1 1 − , µ=0 2S2 S2 ϑ + ϑ0
(71)
where µ = S12 − 4S0 S2 and ϑ0 is an arbitrary real constant.
3.1
ro
of
Step-3: The strategic equations are easily obtained as long as the solution structure (69) along with Eq.(70) is put in Eq.(67). Thus, the important results Ai ,S2 , S1 , S0 emerged from these strategic equations are obtained. If we use the results Ai ,S2 , S1 , S0 in Eq.(69), solitons in an optical fiber communication system are recovered by Eq.(71).
KERR LAW
-p
Solitons with the CQ-NLSE in coupled vector form without FWM along with the riccati function principle are recovered in this section. Eq.(69) can be given as
Uj = A0 + A1 Vj + A2 Vj2
(72)
re
which comes from Eq.(14) in consequence of using balancing rule.
Result-1:
A2 5 bj S1 2 − 6 bj kj 2 + 3 aj kj 2
20S2 bj
ur na
A0 =
lP
The strategic equations are easily obtained as long as Eq.(72) along with Eq.(70) is put in Eq.(14). Thus, the important results are as
S0 =
, A1 =
A2 S1 , S2
120 S24 bj + A2 2 dj 5 bj S1 2 − 6 bj kj 2 + 3 aj kj , cj = − , 20S2 bj A2 2 kj 2 119 bj 2 kj 2 − 119 aj bj kj + 36 aj 2 wj = − 25bj
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where S2 , S1 and S0 are constants. The strategic solutions of Eq.(70) are as √ √ µ µ S1 − tanh ϑ + ϑ0 , µ > 0 V (ϑ) = − 2S2 2S2 2 √ √ µ µ S1 V (ϑ) = − − coth ϑ + ϑ0 , µ > 0 2S2 2S2 2 √ √ S1 −µ −µ + tan ϑ + ϑ0 , µ < 0 V (ϑ) = − 2S2 2S2 2 √ √ S1 −µ −µ − cot ϑ + ϑ0 , µ < 0 V (ϑ) = − 2S2 2S2 2
and
µ = S12 − 4S0 S2 = −
3kj (−2 kj bj + aj ) . 5bj
(73)
If we use the results (73) along with Eq.(71) in Eq.(72), dark solitons are obtained as
u(x, t) =
3 A2 k1 (2 b1 k1 − a1 + (a1 − 2b1 k1 )) − 20 S2 2 b1 r 15k1 (−2 b1 k1 + a1 ) 2 − x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh 100b1 ×ei(−k1 x+w1 t+ζ1 )
12
(74)
v(x, t)
=
3 A2 k2 (2 b2 k2 − a2 + (a2 − 2b2 k2 )) − 20 S2 2 b2 r 15k2 (−2 b2 k2 + a2 ) 2 3 2 x + 3 a2 k2 − 4 b2 k2 t + ϑ0 − ×tanh 100b2
×ei(−k2 x+w2 t+ζ2 )
(75)
with bj kj (−2 bj kj + aj ) < 0. Similarly, singular solitons are obtained as
-p
×ei(−k1 x+w1 t+ζ1 )
3 A2 k2 (2 b2 k2 − a2 + (a2 − 2b2 k2 )) − 20 S2 2 b2 = r 15k2 (−2 b2 k2 + a2 ) 2 3 2 x + 3 a2 k2 − 4 b2 k2 t + ϑ0 − ×coth 100b2
re
v(x, t)
of
3 A2 k1 (2 b1 k1 − a1 + (a1 − 2b1 k1 )) − 20 S2 2 b1 r 15k1 (−2 b1k1 + a1 ) 2 ×coth − x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 100b1
ro
u(x, t) =
lP
×ei(−k2 x+w2 t+ζ2 )
with
(76)
(77)
Result-2:
ur na
bj kj (−2 bj kj + aj ) < 0.
A0 =
A1 =
Jo
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A2 5 bj S1 2 − 6 bj kj 2 + 3 aj kj + 40 f 60 S2 2 bj
,
f 120 S24 bj + A2 2 dj A2 S1 , S0 = , cj = − , S2 b j S2 A2 2
160 f 2 + −80 bj S1 2 − 120 bj kj 2 + 60 aj kj f + 10 S1 4 bj 2 + 30 S1 2 bj 2 kj 2 +36 bj 2 kj 4 − 15 S1 2 aj kj bj − 36 aj bj kj 3 + 9 aj 2 kj 2 = 0,
wj =
kj (90 S1 2 bj 2 kj + 56 bj 2 kj 3 − 45 S1 2 aj bj 20bj
−56 aj bj kj 2 − 360 f bj kj + 9 aj 2 kj + 180 f aj ) and µ = S12 − 4S0 S2 = − 13
−bj S1 2 + 4 f . bj
(78)
If we use the results (78) along with Eq.(71) in Eq.(72), dark solitons are obtained as
u(x, t) =
A2 10 S1 2 b1 + 6 b1 k1 2 − 3 a1 k1 − 40 f + 60f − 15S1 2 b1 − 60 S2 2 b1 s 2 b + 4 f −S 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh2 − 4b1
A2 10 S1 2 b2 + 6 b2 k2 2 − 3 a2 k2 − 40 f + 60f − 15S1 2 b2 − 60 S2 2 b2 s 2 b + 4 f −S 2 1 2 3 2 ×tanh − x + 3 a2 k2 − 4 b2 k2 t + ϑ0 4b2
×ei(−k1 x+w1 t+ζ1 )
×ei(−k2 x+w2 t+ζ2 )
with
re
Similarly, singular solitons are obtained as
lP
A2 10 S1 2 b1 + 6 b1 k1 2 − 3 a1 k1 − 40 f + 60f − 15S1 2 b1 − 60 S2 2 b1 s = 2 b + 4 f −S 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×coth2 − 4b1
ur na
ei(−k1 x+w1 t+ζ1 )
v(x, t)
=
A2 10 S1 2 b2 + 6 b2 k2 2 − 3 a2 k2 − 40 f + 60f − 15S1 2 b2 − 60 S2 2 b2 q 2 ×coth2 − −S1 4bb22+4 f x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 ×ei(−k2 x+w2 t+ζ2 )
with
3.2
PARABOLIC LAW
(80)
-p
bj kj −S1 2 bj + 4 f < 0.
u(x, t)
of
=
(79)
ro
v(x, t)
Jo
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(81)
(82)
bj kj −S1 2 bj + 4 f < 0.
Eq.(69) can be given as Uj = A0 + A1 Vj
(83)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(83) along with Eq.(70) is put in Eq.(28). Thus, 14
the important results are as S1 = 2
A1 4 gj + A1 4 hj + 24 S2 4 bj A0 S2 , , ej = − A1 A1 4
cj =
1 (40 A0 2 S2 4 bj − 40 A1 2 S0 S2 3 bj A1 4
+12 A1 2 S2 2 bj kj 2 − 6 A1 2 S2 2 aj kj − A1 4 dj ),
wj =
1 (16 A0 4 S2 4 bj − 32 A0 2 A1 2 S0 S2 3 bj A1 4
+12 A0 2 A1 2 S2 2 bj kj 2 + 16 A1 4 S0 2 S2 2 bj − 12 A1 4 S0 S2 bj kj 2
and µ=
S12
− 4S0 S2 =
4S2 A0 2 S2 − A1 2 S0 A1 2
.
of
+A1 4 bj kj 4 − 6 A0 2 A1 2 S2 2 aj kj + 6 A1 4 S0 S2 aj kj − A1 4 aj kj 3 )
(84)
A0 2 S2 − A1 2 S0 S2 s
×tanh
S2 A0 2 S2 − A1 2 S0 A1 2
=
−
s
×tanh
S2 A0 2 S2 − A1 2 S0 A1 2
×ei(−k2 x+w2 t+ζ2 )
with
Similarly, singular solitons are obtained as
u(x, t)
= −
s
x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0
(85) (86) (87)
A0 2 S2 − A1 2 S0 S2
ur na
v(x, t)
lP
×ei(−k1 x+w1 t+ζ1 )
s
-p
−
re
u(x, t) =
s
ro
If we use the results (84) along with Eq.(71) in Eq.(83), dark solitons are obtained as
Jo
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(88)
x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0
(89) (90)
S2 A0 2 S2 − A1 2 S0 > 0.
A0 2 S2 − A1 2 S0 S2 (91) s
×coth
S2 A0 2 S2 − A1 2 S0 A1
2
×ei(−k1 x+w1 t+ζ1 ) 15
x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0
(92) (93)
= −
v(x, t)
s
A0 2 S2 − A1 2 S0 S2 (94) s
×coth
S2 A0 2 S2 − A1 2 S0 A1
2
×ei(−k2 x+w2 t+ζ2 ) with
3.3
x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0
(95) (96)
S2 A0 2 S2 − A1 2 S0 > 0.
QUADRATIC-CUBIC LAW
Eq.(69) can be given as
of
Uj = A0 + A1 Vj + A2 Vj2 which comes from Eq.(41) in consequence of using balancing rule.
(97)
ro
The strategic equations are easily obtained as long as Eq.(97) along with Eq.(70) is put in Eq.(41). Thus, the important results are as
w=
-p
Result-1:
1 (k 4 A2 4 bj − 6 k 2 A1 2 A2 2 S2 2 bj + 24 k 2 A2 4 S0 S2 bj A2 4
re
+A1 4 S2 4 bj − 8 A1 2 A2 2 S0 S2 3 bj + 16 A2 4 S0 2 S2 2 bj
3S2 2 6 k 2 A2 2 bj − 5 A1 2 S2 2 bj + 20 A2 2 S0 S2 bj − 3 kA2 2 aj A2 3
A0 =
and
,
A2 S0 A1 S2 120 S24 bj + A2 2 ej , , S1 = , dj = − S2 A2 A2 2
ur na
cj =
lP
−k 3 A2 4 aj + 3 kA1 2 A2 2 S2 2 aj − 12 kA2 4 S0 S2 aj )
µ=
S12
− 4S0 S2 =
S2 A1 2 S2 − 4 A2 2 S0 A2 2
.
(98)
If we use the results (98) along with Eq.(71) in Eq.(97), dark solitons are obtained as
Jo
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u(x, t) =
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S S − 4 A A S 0 2 2 1 2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 ×tanh2 4A2 2 ×ei(−kx+wt+ζ)
16
(99)
v(x, t)
=
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S S − 4 A S A 0 2 2 2 1 ×tanh2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 4A2 2
×ei(−kx+wt+ζ) with
(100)
S2 A1 2 S2 − 4 A2 2 S0 > 0.
Similarly, singular solitons are obtained as
of
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S2 A1 S2 − 4 A2 S0 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 ×coth2 4A2 2
ro
u(x, t) =
re
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 = 4A2 S2 s 2 2 S S − 4 A A S 0 2 2 1 2 ×coth2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 4A2 2
lP
v(x, t)
3.4
ur na
×ei(−kx+wt+ζ)
with
(101)
-p
×ei(−kx+wt+ζ)
(102)
S2 A1 2 S2 − 4 A2 2 S0 > 0.
NON-LOCAL LAW
Eq.(69) can be given as
Uj = A0 + A1 Vj
(103)
which comes from Eq.(55) in consequence of using balancing rule.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The strategic equations are easily obtained as long as Eq.(103) along with Eq.(70) is put in Eq.(55). Thus, the important results are as A0 =
cj = −
S1 2 bj + 6 bj kj 2 − 3 aj kj A1 S1 , S0 = , 2S2 4S2 bj
A1 2 dj + 4 S2 2 bj , wj = −kj 3 (−bj kj + aj ) A1 2
and µ = S12 − 4S0 S2 =
3kj (−2 bj kj + aj ) . bj
17
(104)
If we use the results (104) along with Eq.(71) in Eq.(103), dark solitons are obtained as
u(x, t)
s
3A21 k1 (−2 b1 k1 + a1 ) 4S22 b1 s 3k (−2 b k + a ) 1 1 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh 4b1
= −
×ei(−k1 x+w1 t+ζ1 )
of
3A21 k2 (−2 b2 k2 + a2 ) 4S22 b2 s 3k2 (−2 b2 k2 + a2 ) ×tanh x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 4b2
= −
×ei(−k2 x+w2 t+ζ2 ) with
re
Similarly, singular solitons are obtained as s
3A21 k1 (−2 b1 k1 + a1 ) 4S22 b1 s 3k (−2 b k + a ) 1 1 1 1 ×coth x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 4b1
lP
−
(106)
-p
bj kj (−2 bj kj + aj ) > 0.
u(x, t) =
(105)
s
ro
v(x, t)
(107)
ur na
×ei(−k1 x+w1 t+ζ1 )
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
=
s
3A21 k2 (−2 b2 k2 + a2 ) 4S22 b2 s 3k2 (−2 b2k2 + a2 ) ×coth x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 4b2 −
×ei(−k2 x+w2 t+ζ2 )
with bj kj (−2 bj kj + aj ) > 0.
4
SINE-GORDON EQUATION: A QUICK BRUSH-UP
The steps of applying the method to any governing model are as follows:
18
(108)
Step-1: A model can be taken as S (u, ut , ux , utt , uxt , uxx , ...) = 0.
(109)
The important equation emerged from Eq.(109) is as G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
(110)
u(x, t) = U (ϑ), ϑ = x − pt.
(111)
by using of
Step-2: The solution structure of Eq.(110) is considered as U (ϑ) =
N X
cosi−1 (V (ϑ))[Bi sin(V (ϑ)) + Ai cos(V (ϑ))] + A0
(112)
i=1
of
and also the number N comes from Eq.(110) in consequence of using balancing condition. The function V (ϑ) holds V ′ (ϑ) = sin (V (ϑ)) .
ro
(113)
The elementary solutions of Eq.(113) are given as
-p
sin (V (ϑ)) = sech (ϑ) or sin (V (ϑ)) = icsh (ϑ) ,
re
cos (V (ϑ)) = tanh (ϑ) or cos (V (ϑ)) = coth (ϑ)
(114)
4.1
lP
Step-3: The strategic equations are easily obtained as long as the solution structure (112) along with Eq.(113) is put in Eq.(110). Thus, the important results Ai ,Bi , A0 emerged from these strategic equations are obtained. If we use the results Ai ,Bi , A0 in Eq.(112), solitons in an optical fiber communication system are recovered by Eq.(114).
KERR LAW
ur na
Solitons with the CQ-NLSE in coupled vector form without FWM along with sine-Gordon function principle are recovered in this section. Eq.(112) can be given as
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + cos(Vj (ϑ)) [B2 sin(Vj (ϑ)) + A2 cos(Vj (ϑ))] + A0
(115)
which comes from Eq.(14) in consequence of using balancing rule.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The strategic equations are easily obtained as long as Eq.(115) along with Eq.(113) is put in Eq.(14). Thus, the important results are as Result-1:
A0 = 0, A1 = 0, A2 = 0, B1 = 0,
B2 = ±
s
180aj kj 3aj kj , , bj = 2 3 kj 2 + 5 3 kj + 5 (cj + dj ) 2
19
(116)
If we put Eq.(116) along with Eq.(114) in Eq.(115), one can get combo dark-bright solitons
u(x, t)
= ±
s
180a1k1 3 k1 + 5 (c1 + d1 ) 2
× tanh x + 3 a1 k1 2 − 4 b1 k1 3 t sech x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t)
=
±
a1 k1 i−k1 x−
s
3 k1 4 + 10 k1 2 − 33 2 3 k1 2 + 5
t+ζ1
(117)
180a2 k2j 3 k2 + 5 (c2 + d2 ) 2
of
×e
×e
a2 k2 i−k2 x−
3 k2 4 + 10 k2 2 − 33 2 3 k2 2 + 5
u(x, t) =
± −
t+ζ2
(118)
re
Similarly, singular solitons are obtained as s
-p
ro
× tanh x + 3 a2 k2 2 − 4 b2 k2 3 t sech x + 3 a2 k2 2 − 4 b2 k2 3 t
180a1k1 3 k1 + 5 (c1 + d1 ) 2
a1 k1 i−k1 x−
3 k1 4 + 10 k1 2 − 33 2 3 k1 2 + 5
t+ζ1
(119)
ur na
×e
lP
× coth x + 3 a1 k1 2 − 4 b1 k1 3 t csch x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
aj kj 3 kj 4 + 10 kj 2 − 33 wj = − . 2 3 kj 2 + 5
=
s
± −
180a2 k2j 3 k2 + 5 (c2 + d2 ) 2
× coth x + 3 a2 k2 2 − 4 b2 k2 3 t csch x + 3 a2 k2 2 − 4 b2 k2 3 t a2 k2 3 k2 4 + 10 k2 2 − 33 i−k2 x− 2 3 k2 2 + 5
×e
t+ζ2
(120)
Result-2: A1 = 0, B1 = 0, B2 = 0,
aj kj 3 kj 4 − 20 kj 2 + 192 3aj kj bj = , w = − j 2 3 kj 2 − 10 2 3 kj 2 − 10 20
A0 = ±
180aj kj − , A2 = ± 3 kj 2 − 10 (cj + dj )
s
−
180aj kj . 3 kj 2 − 10 (cj + dj )
(121)
If we put Eq.(121) along with Eq.(114) in Eq.(115), one can get dark solitons
u(x, t) =
s s 180a1 k1 180a1 k1 ± − ± − 3 k1 2 − 10 (c1 + d1 ) 3 k1 2 − 10 (c1 + d1 )
× tanh2
x + 3 a1 k1 2 − 4 b1 k1 3 t
a1 k1 3 k1 4 − 20 k1 2 + 192 i−k1 x− 2 3 k1 2 − 10
t+ζ1
(122)
of
×e
× tanh2
x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
a2 k2 i−k2 x−
3 k2 4 − 20 k2 2 + 192 2 3 k2 2 − 10
× coth2
x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
v(x, t)
=
t+ζ2
s s 180a1 k1 180a1 k1 ± − ± − 2 2 3 k1 − 10 (c1 + d1 ) 3 k1 − 10 (c1 + d1 )
ur na
u(x, t) =
a1 k1 i−k1 x−
3 k1 4 − 20 k1 2 + 192 2 3 k1 2 − 10
× coth2
×e
3 k2 4 − 20 k2 2 + 192 2 3 k2 2 − 10 21
t+ζ1
x + 3 a2 k2 2 − 4 b2 k2 3 t
a2 k2 i−k2 x−
(123)
s s 180a2 k2j 180a2 k2j ± − ± − 2 2 3 k2 − 10 (c2 + d2 ) 3 k2 − 10 (c2 + d2 )
lP
Similarly, combo singular solitons are obtained as
ro
-p
=
s s 180a2 k2j 180a2 k2j ± − ± − 3 k2 2 − 10 (c2 + d2 ) 3 k2 2 − 10 (c2 + d2 )
re
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
(124)
t+ζ2
(125)
Result-3: A1 = 0, B1 = 0, bj =
wj = −
3 kj 4 − 5 kj 2 + 12 aj kj 6 kj 2 − 5
3aj kj , 6 kj 2 − 5 s
, A0 = ± −
s
90aj kj A2 = ± − , B2 = ± 2 6 kj − 5 (cj + dj )
s
90aj kj , 6 kj 2 − 5 (cj + dj ) 6 kj
2
90aj kj , − 5 (cj + dj )
(126)
If we put Eq.(126) along with Eq.(114) in Eq.(115), one can get combo dark-bright solitons
6 k1 2 − 5
-p
3 k1 4 − 5 k1 2 + 12
t+ζ1
lP
×e
a1 k1 i−k1 x−
s s 90a2 k2j 90a2 k2j ± − ± 2 2 6 k2 − 5 (c2 + d2 ) 6 k2 − 5 (c2 + d2 ) ×sech x + 3 a2 k2 2 − 4 b2 k2 3 t tanh x + 3 a2 k2 2 − 4 b2 k2 3 t s 90a2 k2j 3 2 2 ± − 6 k 2 − 5 (c + d ) tanh x + 3 a2 k2 − 4 b2 k2 t 2 2 2
ur na =
×e
a2 k2 i−k2 x−
re
v(x, t)
of
s s 90a1 k1 90a1 k1 ± − ± 2 2 6 k1 − 5 (c1 + d1 ) 6 k1 − 5 (c1 + d1 ) ×sech x + 3 a1 k1 2 − 4 b1 k1 3 t tanh x + 3 a1 k1 2 − 4 b1 k1 3 t s 90a1 k1 3 2 2 ± − 6 k 2 − 5 (c + d ) tanh x + 3 a1 k1 − 4 b1 k1 t 1 1 1
ro
u(x, t) =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
3 k2 4 − 5 k2 2 + 12 6 k2 2 − 5
Similarly, combo singular solitons are obtained as
22
(127)
t+ζ2
(128)
u(x, t)
i−k1 x−
a1 k1 3 k1 4 − 5 k1 2 + 12 6 k1 2 − 5
t+ζ1
(129)
of
×e
ro
4.2
PARABOLIC LAW
6 k2 2 − 5
t+ζ2
ur na
Eq.(112) can be given as
3 k2 4 − 5 k2 2 + 12
lP
×e
a2 k2 i−k2 x−
re
-p
v(x, t)
s s 90a2 k2j 90a2 k2j ± − ± − 2 2 − 5 (c + d ) 6 k 6 k 2 2 2 2 − 5 (c2 + d2 ) = ×csch x + 3 a2 k2 2 − 4 b2 k2 3 t coth x + 3 a2 k2 2 − 4 b2 k2 3 t s 90a2 k2j 3 2 2 ± − 6 k 2 − 5 (c + d ) coth x + 3 a2 k2 − 4 b2 k2 t 2 2 2
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + A0
(130)
(131)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(131) along with Eq.(113) is put in Eq.(28). Thus, the important results are as Result-1:
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s s 90a1 k1 90a1 k1 ± − ± − 2 2 6 k1 − 5 (c1 + d1 ) 6 k1 − 5 (c1 + d1 ) = ×csch x + 3 a1 k1 2 − 4 b1 k1 3 t coth x + 3 a1 k1 2 − 4 b1 k1 3 t s 90a1 k1 3 2 2 ± − 6 k 2 − 5 (c + d ) coth x + 3 a1 k1 − 4 b1 k1 t 1 1 1
wj = bj kj 4 − aj kj 3 + 12 bj kj 2 − 6 aj kj + 16 bj
A0 = 0, B1 = 0, A1 = ±
ej = −
s
12 bj kj 2 − 6 aj kj + 40 bj , cj + dj
1 (36 bj 2 gj kj 4 + 36 bj 2 hj kj 4 (−6bj kj2 + 3aj kj − 20bj )2
−36 aj bj gj kj 3 − 36 aj bj hj kj 3 + 9 aj 2 gj kj 2 + 9 aj 2 hj kj 2 23
+240 bj 2 gj kj 2 + 240 bj 2 hj kj 2 − 120 aj bj gj kj − 120 aj bj hj kj +400 bj 2 gj + 400 bj 2 hj + 6 bj cj 2 + 12 bj cj dj + 6 bj dj 2 )
(132)
If we put Eq.(132) along with Eq.(114) in Eq.(131), one can get dark solitons
= ±
s
12 b1 k1 2 − 6 a1 k1 + 40 b1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t c1 + d1
×ei(−k1 x+(b1 k1
v(x, t)
= ±
s
−a1 k1 3 +12 b1 k1 2 −6 a1 k1 +16 b1 )t+ζ1 )
4
12 b2 k2 2 − 6 a2 k2 + 40 b2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t c2 + d2
×ei(−k2 x+(b2 k2
4
−a2 k2 3 +12 b2 k2 2 −6 a2 k2 +16 b2 )t+ζ2 )
= ±
12 b1k1 2 − 6 a1 k1 + 40 b1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t c1 + d1
=
±
−a1 k1 3 +12 b1 k1 2 −6 a1 k1 +16 b1 )t+ζ1 )
×ei(−k2 x+(b2 k2
4
−a2 k2 3 +12 b2 k2 2 −6 a2 k2 +16 b2 )t+ζ2 )
ur na
Result-2: A1 = ±
(135)
12 b2k2 2 − 6 a2 k2 + 40 b2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t c2 + d2
lP
v(x, t)
s
4
re
×ei(−k1 x+(b1 k1
-p
u(x, t)
(134)
ro
Similarly, singular solitons are obtained as s
(133)
of
u(x, t)
s
45aj kj , B1 = ± − 2 6 kj + 5 (cj + dj )
A0 = 0, wj = −
Jo
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ej = −
s
−
aj kj 3 kj 4 + 5 kj 2 + 27 2
6 kj + 5
6 kj
2
, bj =
(136)
45aj kj , + 5 (cj + dj ) 3aj kj 6 kj 2 + 5
1 (−12 cj 2 kj 2 − 24 cj dj kj 2 − 12 dj 2 kj 2 225 aj kj
+225 aj gj kj + 225 aj hj kj − 10 cj 2 − 20 cj dj − 10 dj 2 ).
If we put Eq.(137) along with Eq.(114) in Eq.(131), one can get combo dark-bright solitons
24
(137)
u(x, t) =
s 45a1 k1 3 2 ± − 6 k 2 + 5 (c + d ) sech x + 3 a1 k1 − 4 b1 k1 t 1 1 1
×e
=
a1 k1 i−k1 x−
3 k1 4 + 5 k1 2 + 27 6 k1 2 + 5
t+ζ1
(138)
s 45a2 k2 ± − tanh x + 3 a2 k2 2 − 4 b2 k2 3 t 2 6 k2 + 5 (c2 + d2 )
6 k2 2 + 5
t+ζ2
lP
s 45a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t ± − 2 6 k1 + 5 (c1 + d1 )
ur na
s 45a1 k1 2 3 ± csch x + 3 a k − 4 b k t 1 1 1 1 2 6 k1 + 5 (c1 + d1 )
×e
=
a1 k1 i−k1 x−
3 k1 4 + 5 k1 2 + 27 6 k1 2 + 5
×e
3 k2 4 + 5 k2 2 + 27 6 k2 2 + 5
25
t+ζ1
s 45a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t 2 ± 6 k2 + 5 (c2 + d2 ) a2 k2 i−k2 x−
(139)
s 45a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t ± − 2 6 k2 + 5 (c2 + d2 )
ro
3 k2 4 + 5 k2 2 + 27
Similarly, combo singular solitons are obtained as
v(x, t)
-p
×e
a2 k2 i−k2 x−
re
=
s 45a2 k2 3 2 ± − 6 k 2 + 5 (c + d ) sech x + 3 a2 k2 − 4 b2 k2 t 2 2 2
u(x, t)
of
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s 45a1 k1 ± − tanh x + 3 a1 k1 2 − 4 b1 k1 3 t 2 6 k1 + 5 (c1 + d1 )
(140)
t+ζ2
(141)
4.3
QUADRATIC-CUBIC LAW
Eq.(112) can be given as Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + cos(Vj (ϑ)) [B2 sin(Vj (ϑ)) + A2 cos(Vj (ϑ))] + A0
(142)
which comes from Eq.(41) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(142) along with Eq.(113) is put in Eq.(41). Thus, the important results are as Result-1: k 3 aj − 12 kaj + w , A2 = ± bj = k 4 − 24 k 2 + 16
−
120 (k 3 aj − 12 kaj + w) (dj + ej ) (k 4 − 24 k 2 + 16)
s
3 (3 k 5 aj − 20 k 3 aj + 6 k 2 w + 192 kaj − 20 w) (dj + ej ) 40 (k 4 − 24 k 2 + 16) ( k 3 aj − 12 kaj + w)
2
re
-p
If we put Eq.(143) along with Eq.(114) in Eq.(142), one can get dark solitons s s 3 120 k a − 12 ka + w 120 k 3 a1 − 12 ka1 + w 1 1 ± − ± − (d1 + e1 ) (k 4 − 24 k 2 + 16) (d1 + e1 ) (k 4 − 24 k 2 + 16) u(x, t) = × tanh2 x + 3 a1 k1 2 − 4 b1 k1 3 t s s 3 120 k a − 12 ka + w 120 k 3 a2 − 12 ka2 + w 2 2 ± − ± − (d2 + e2 ) (k 4 − 24 k 2 + 16) (d2 + e2 ) (k 4 − 24 k 2 + 16) × tanh2 x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
lP
v(x, t) =
Similarly, combo singular solitons are obtained as s s 120 k 3 a1 − 12 ka1 + w 120 k 3 a1 − 12 ka1 + w ± − ± − (d1 + e1 ) (k 4 − 24 k 2 + 16) (d1 + e1 ) (k 4 − 24 k 2 + 16) u(x, t) = × coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t) =
Result-2:
s s 3 120 k a − 12 ka + w 120 k 3 a2 − 12 ka2 + w 2 2 ± − ± − (d2 + e2 ) (k 4 − 24 k 2 + 16) (d2 + e2 ) (k 4 − 24 k 2 + 16)
(143)
ro
−
of
120 ( k 3 aj − 12 kaj + w) , A1 = 0, B1 = 0, B2 = 0, (dj + ej ) (k 4 − 24 k 2 + 16)
A0 = ± −
cj = ±
s
s
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
× coth2 x + 3 a2 k2 2 − 4 b2 k2 3 t s
ei(−kx+wt+ζ)
(144)
ei(−kx+wt+ζ)
(145)
ei(−kx+wt+ζ)
(146)
ei(−kx+wt+ζ)
(147)
s 30 k 3 aj − 90 kaj + 30 w 30 k 3 aj − 90 kaj + 30 w A0 = ± − , A2 = ± − 4 2 (dj + ej ) (k − 6 k + 1) (dj + ej ) (k 4 − 6 k 2 + 1)
26
A1 = 0, B1 = 0, B2 = ±
cj = ±
s
−
k 3 aj − 3 kaj + w 30 (k 3 aj − 3 kaj + w) , b = j (dj + ej ) (k 4 − 6 k 2 + 1) k4 − 6 k2 + 1 2
3 (dj + ej ) (3 k 5 aj − 5 k 3 aj + 6 k 2 w + 12 kaj − 5 w) . 40 (k 4 − 6 k 2 + 1) ( k 3 aj − 3 kaj + w)
(148)
If we put Eq.(148) along with Eq.(114) in Eq.(142), one can get combo dark-bright solitons
ro
of
u(x, t) =
s s 30 k 3 a1 − 3ka1 + w 30 k 3 a1 − 3ka1 + w ± ± − (d1 + e1 ) (k 4 − 6 k 2 + 1) (d1 + e1 ) (k 4 − 6 k 2 + 1) ×sech x + 3 a1 k1 2 − 4 b1 k1 3 t tanh x + 3 a1 k1 2 − 4 b1 k1 3 t s 3 ± − 30 k a1 − 3ka1 + w tanh2 x + 3 a k 2 − 4 b k 3 t 1 1 1 1 (d1 + e1 ) (k 4 − 6 k 2 + 1)
-p
×ei(−kx+wt+ζ)
re
=
ur na
lP
v(x, t)
s s 3 30 k 3 a2 − 3ka2 + w 30 k a − 3ka + w 2 2 ± − ± (d2 + e2 ) (k 4 − 6 k 2 + 1) (d2 + e2 ) (k 4 − 6 k 2 + 1) ×sech x + 3 a2 k2 2 − 4 b2 k2 3 t tanh x + 3 a2 k2 2 − 4 b2 k2 3 t s 3 ± − 30 k a2 − 3ka2 + w tanh2 x + 3 a k 2 − 4 b k 3 t 2 2 2 2 (d2 + e2 ) (k 4 − 6 k 2 + 1)
(149)
×ei(−kx+wt+ζ)
(150)
Similarly, combo singular solitons are obtained as s s 3 30 k a − 3ka + w 30 k 3 a1 − 3ka1 + w 1 1 ± − ± − (d1 + e1 ) (k 4 − 6 k 2 + 1) (d1 + e1 ) (k 4 − 6 k 2 + 1) = ×csch x + 3 a1 k1 2 − 4 b1 k1 3 t coth x + 3 a1 k1 2 − 4 b1 k1 3 t s 30 k 3 a1 − 3ka1 + w coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t ± − (d1 + e1 ) (k 4 − 6 k 2 + 1)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
u(x, t)
×ei(−kx+wt+ζ)
(151)
27
v(x, t)
s s 30 k 3 a2 − 3ka2 + w 30 k 3 a2 − 3ka2 + w ± − ± − (d2 + e2 ) (k 4 − 6 k 2 + 1) (d2 + e2 ) (k 4 − 6 k 2 + 1) = ×csch x + 3 a2 k2 2 − 4 b2 k2 3 t coth x + 3 a2 k2 2 − 4 b2 k2 3 t s 3 ± − 30 k a2 − 3ka2 + w coth2 x + 3 a k 2 − 4 b k 3 t 2 2 2 2 (d2 + e2 ) (k 4 − 6 k 2 + 1) ×ei(−kx+wt+ζ)
(152)
NON-LOCAL LAW
Eq.(112) can be given as
which comes from Eq.(55) in consequence of using balancing rule.
(153)
ro
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + A0
of
4.4
-p
The strategic equations are easily obtained as long as Eq.(153) along with Eq.(113) is put in Eq.(55). Thus, the important results are as Result-1: s
re
3 kj 2 + 4 aj kj 3 6aj kj , wj = − − 3 kj 2 + 2 (cj + dj ) 2 3 kj 2 + 2
lP
A1 = ±
A0 = 0, B1 = 0, bj =
3aj kj . 2 3 kj 2 + 2
(154)
ur na
If we put Eq.(154) along with Eq.(114) in Eq.(153), one can get dark solitons
u(x, t) =
±
s
−
3 k1
2
×e
v(x, t)
=
i−k1 x−
s
± −
3 k2
×e
i−k2 x−
6a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t + 2 (c1 + d1 )
3 k1 2 + 4 a1 k1 3 t+ζ1 2 3 k1 2 + 2
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2
(155)
6a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t + 2 (c2 + d2 ) 3 k2 2 + 4 a2 k2 3 t+ζ2 2 3 k2 2 + 2
Similarly, singular solitons are obtained as
28
(156)
=
± −
6a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t + 2 (c1 + d1 )
2
3 k1
3 k1 2 + 4 a1 k1 3 t+ζ1 2 3 k1 2 + 2
×e
v(x, t)
= ±
i−k1 x−
s
−
3 k2
6a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t + 2 (c2 + d2 )
2
3 k2 2 + 4 a2 k2 3 t+ζ2 2 3 k2 2 + 2
×e
(157)
i−k2 x−
(158)
of
u(x, t)
Result-2:
3 kj 4 − 2 kj 2 + 3 aj kj 6aj kj , wj = − 3 kj 2 − 1 (cj + dj ) 2 3 kj 2 − 1
ro
s
A0 = 0, A1 = 0,
bj =
3aj kj . 2 3 kj 2 − 1
-p
B1 = ±
(159)
±
6a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t − 1 (c1 + d1 )
i−k1 x−
3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
(160)
ur na
×e
3 k1
2
lP
u(x, t) =
s
re
If we put Eq.(159) along with Eq.(114) in Eq.(153), one can get bright solitons
u(x, t) =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
±
s
3 k2
2
6a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t − 1 (c2 + d2 )
×e
i−k2 x−
3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
(161)
Similarly, singular solitons are obtained as
u(x, t) =
s
± −
×e
3 k1
i−k1 x−
2
6a1 k1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t − 1 (c1 + d1 ) 3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
29
(162)
s
u(x, t) =
± −
3 k2
×e
i−k2 x−
Result-3:
2
6a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t − 1 (c2 + d2 ) 3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
s
3aj kj A1 = ± − , B1 = ± 6 kj 2 + 1 (cj + dj )
s
(163)
3aj kj , 6 kj 2 + 1 (cj + dj )
3 kj 2 + 1 aj kj 3 3aj kj A0 = 0, bj = , wj = − . 6 kj 2 + 1 6 kj 2 + 1
of
(164)
If we put Eq.(164) along with Eq.(114) in Eq.(153), one can get combo dark-bright solitons
3 k1 2 + 1 a1 k1 3 6 k1 2 + 1
t+ζ1
s 3a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t ± − 2 6 k2 + 1 (c2 + d2 )
ur na =
s 3a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t 2 ± 6 k2 + 1 (c2 + d2 )
×e
i−k2 x−
3 k2 2 + 1 a2 k2 3 6 k2 2 + 1
Similarly, combo singular solitons are obtained as
30
lP
×e
i−k1 x−
re
s 3a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t 2 ± 6 k1 + 1 (c1 + d1 )
v(x, t)
ro
s 3a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t ± − 2 6 k1 + 1 (c1 + d1 )
-p
u(x, t) =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(165)
t+ζ2
(166)
u(x, t)
=
s 3a1 k1 3 2 ± − 6 k 2 + 1 (c + d ) csch x + 3 a1 k1 − 4 b1 k1 t 1 1 1
×e
=
3 k1 2 + 1 a1 k1 3 6 k1 2 + 1
t+ζ1
(167)
s 3a2 k2 ± − coth x + 3 a2 k2 2 − 4 b2 k2 3 t 2 6 k2 + 1 (c2 + d2 )
×e
i−k2 x−
3 k2 2 + 1 a2 k2 3 6 k2 2 + 1
t+ζ2
(168)
re
CONCLUSIONS
-p
ro
s 3a2 k2 3 2 ± − 6 k 2 + 1 (c + d ) csch x + 3 a2 k2 − 4 b2 k2 t 2 2 2
5
of
v(x, t)
i−k1 x−
ur na
lP
This paper, for the first time, introduced CQ-NLSEs in birefringent fibers with Kerr, parabolic, quadratic-cubic and non-local nonlinear laws. The three integration schemes implemented in this paper together revealed bright, dark and singular optical solitons to the models. The existence criteria for these solitons are also enumerated. Thus, the results laid the foundation for further future activities with the models. Later, these models will be extended to study soliton dynamics for DWDM networks as the fundamental governing models. Furthermore, studies with fractional temporal evolution in birefringent fibers for the models will also be considered. Later, Langevin equation will be derived for stochastic perturbation of optical solitons in birefringent fibers. These studies and many more advanced research activities will be conducted in future and the results will be gradually disseminated. Those knowledge–hungry readers must stay tuned... ACKNOWLEDGEMENTS
The research work of the fifth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. The authors also declare that there is no conflict of interest.
Jo
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s 3a1 k1 ± − coth x + 3 a1 k1 2 − 4 b1 k1 3 t 2 6 k1 + 1 (c1 + d1 )
31
References [1] A. Bansal, A. Biswas, Q. Zhou & M. M. Babatin. “Lie symmetry analysis for cubic–quartic nonlinear Schr¨odinger’s equation”. Optik. Volume 169, 12–15. (2018). [2] A. Biswas, H. Triki, Q. Zhou, S. P. Moshokoa, M. Z. Ullah & M. Belic. “Cubic–quartic optical solitons in Kerr and power–law media”. Optik. Volume 144, 357–362. (2017). [3] A. Biswas, A. H. Kara, M. Z. Ullah, Q. Zhou, H. Triki & M. Belic. “Conservation laws for cubic–quartic optical solitons in Kerr and power–law media”. Optik. Volume 145, 650–654. (2017). [4] A. Biswas & S. Arshed. “Application of semi–inverse variational principle to cubic–quartic optical solitons having Kerr and power law nonlinearity”. Optik. Volume 172, 847–850. (2018). [5] A. Blanco–Redondo, C. M. D. Sterke, J. E. Sipe, T. F. Krauss, B. J. Eggleton & C. Husko. “Pure-quartic solitons”. Nature Communications. Volume 7, 10427. (2016). [6] A. Blanco–Redondo, C. M. D. Sterke, J. E. Sipe, T. F. Krauss, B. J. Eggleton, C. Husko. “Erratum: Pure–quartic solitons”. Nature Communications. Volume 7, 11048. (2016).
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[7] A. Das, A. Biswas, M. Ekici, S. Khan, Q. Zhou & S. P. Moshokoa. “Suppressing internet bottleneck with fractional temporal evolution of cubic–quartic optical solitons”. Optik. Volume 182, 303–307. (2019).
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[8] R. W. Kohl, A. Biswas, M. Ekici, Q. Zhou, S. P. Moshokoa & M. R. Belic. “Cubic–quartic optical soliton perturbation by semi–inverse variational principle”. Optik. Volume 185, 45–49. (2019).
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32
PII:
S0030-4026(19)31783-8
DOI:
https://doi.org/10.1016/j.ijleo.2019.163885
Reference:
IJLEO 163885
To appear in:
Optik
Received Date:
16 September 2019
Accepted Date:
23 November 2019
Please cite this article as: { doi: https://doi.org/ This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
*Manuscript
CUBIC–QUARTIC OPTICAL SOLITONS IN BIREFRINGENT FIBERS WITH FOUR FORMS OF NONLINEAR REFRACTIVE INDEX Yakup Yıldırım 1 , Anjan Biswas 1
2, 3, 4, 5
, Padmaja Guggilla 2 , Fouad Mallawi
3
& Milivoj R. Belic
Department of Mathematics, Faculty of Arts and Sciences, Near East University, 99138 Nicosia, Cyprus 2
Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762–7500, USA
4
of
Department of Mathematics, King Abdulaziz University, Jeddah–21589, Saudi Arabia Department of Applied Mathematics, National Research Nuclear University 31 Kashirskoe Shosse, Moscow–115409, Russian Federation
ro
3
5
Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar
re
6
-p
Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria–0008, South Africa
Abstract
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This paper studies cubic–quartic solitons in birefringent fibers having Kerr, parabolic, quadratic–cubic and non–local laws of refractive index. Bright, dark and singular solitons are recovered by Riccati function approach, sine-Gordon function technique and F-expansion scheme.
ur na
OCIS Codes: 060.2310; 060.4510; 060.5530; 190.3270; 190.4370 Key words: solitons; riccati function; sine-Gordon function; F-expansion principle.
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1
6
1
One of the most appreciated models that govern the dynamics of pulse transfer through optical fibers and other forms of waveguides is the cubic–quartic nonlinear Schr¨odinger’s equation (CQ-NLSE) where third order dispersion (3OD) and fourth order dispersion (4OD) terms are included when group velocity dispersion (GVD) is negligible and consequently discarded. This model has been extensively studied for polarization–preserving fibers all across [1–8]. Therefore, our next stop is with birefringent fibers. This paper will thus address the dynamics of soliton propagation with differential group delay for coupled CQ-NLSE with and without fourwave mixing (4WM) that depends on the type of nonlinearity. These four nonlinear forms that will be handled in this paper are with Kerr, parabolic, quadratic-cubic and non-local laws. The three integration schemes adopted in the paper are the Riccati function, sine-Gordon function and F-expansion principles. These three approaches collectively reveal bright, dark and singular solitons to the models in birefringent fibers. Additional soliton solutions that emerges from the scheme are also listed as byproducts of the algorithms. The details are presented in the subsequent couple of upcoming sections along with the parameter restrictions that make the soliton solutions viable. These restrictions are listed as constraints.
2
F-EXPANSION: A RAPID SKIM THROUGH
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The steps of applying the method to any governing model are as follows: Step-1: A model can be taken as The important equation emerged from Eq.(1) is as G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
-p
by using of
ro
S (u, ut , ux , utt , uxt , uxx , ...) = 0.
u(x, t) = U (ϑ), ϑ = x − pt. Step-2: The solution structure of Eq.(2) is considered as N X
re
U (ϑ) =
µi F i (ϑ)
(1)
(2) (3)
(4)
i=0
lP
and also the number N comes from Eq.(2) in consequence of using balancing condition. The function F holds p F ′ (ϑ) = P F 2 (ϑ) + QF (ϑ) + R (5) where P , Q and R are constants. The strategic solutions of Eq.(5) are as
= sn (ϑ) = tanh (ϑ) , P = m2 , Q = −(1 + m2 ), R = 1, m → 1,
F (ϑ)
= ns (ϑ) = coth (ϑ) , P = 1, Q = −(1 + m2 ), R = m2 , m → 1,
F (ϑ)
= sc (ϑ) = tan (ϑ) , P = 1 − m2 , Q = 2 − m2 , R = 1, m → 0,
F (ϑ)
= cs (ϑ) = cot (ϑ) , P = 1, Q = 2 − m2 , R = 1 − m2 , m → 0,
ur na
F (ϑ)
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INTRODUCTION
F (ϑ)
= cn (ϑ) = sech (ϑ) , P = −m2 , Q = 2m2 − 1, R = 1 − m2 , m → 1.
F (ϑ)
= ds (ϑ) = csch (ϑ) , P = 1, Q = 2m2 − 1, R = −m2 1 − m2 , m → 1
F (ϑ)
= nc (ϑ) = sec (ϑ) , P = 1 − m2 , Q = 2m2 − 1, R = −m2 , m → 0
F (ϑ)
= ns (ϑ) = csc (ϑ) , P = 1, Q = −(1 + m2 ), R = m2 , m → 0
(6)
Step-3: The strategic equations are easily obtained as long as the solution structure (4) along with Eq.(5) is put in Eq.(2). Thus, the important results µi emerged from these strategic equations are obtained. If we use the results µi in Eq.(4), solitons in an optical fiber communication system are recovered by Eq.(6). 2
2.1
KERR LAW
The CQ-NLSE equation in polarization-preserving fibers with the kerr law is: iqt + iaqxxx + bqxxxx + c |q|2 q = 0.
(7)
In the model (7), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficient c provides the Kerr media effect which is known as the basic case of fiber nonlinearity. Most optical fibers which has been quite popular recently comply with this law nonlinearity. The CQ-NLSE equation in coupled vector form without FWM is: iut + ia1 uxxx + b1 uxxxx + c1 |u|2 + d1 |v|2 u = 0, ivt + ia2 vxxx + b2 vxxxx + c2 |v|2 + d2 |u|2 v = 0
(8)
of
with the constants cj , dj which assure the existence of the self–phase modulation (SPM) and cross-phase modulation (XPM) sequentially.
u(x, t) = U1 (ϑ)eiϕ1 (x,t) v(x, t) = U2 (ϑ)eiϕ2 (x,t) .
ro
Solitons with the CQ-NLSE in coupled vector form without FWM along with the F -expansion principle are recovered in this section. Firstly, the strategic equations are easily employed as
(9)
ϑ = x − pt
re
while the amplitude component is as
-p
In Eq.(9), the phase component is shown as Uj (ϑ) and the velocity is given by
ϕj (x, t) = −kj x + wj t + ζj
(10) (11)
The real and imaginary equations are (4)
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + cj Uj3 + dj Uj U˜j2 = 0,
ur na
bj Uj
lP
where the parameters kj , ζj , wj sequentially correspond to the frequency, phase and wave number.
(aj − 4 bj kj ) Uj′′ − p + 3 aj kj 2 − 4 bj kj 3 Uj = 0
(12)
(13)
respectively as long as Eq.(9) is put in Eqs.(8)-(13). In order to use the balancing rule, the important results emerged from Eq.(12) by using of U˜j = Uj are as (4)
bj Uj
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + (cj + dj ) Uj3 = 0,
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(14)
aj = 4 bj kj ,
p = −3 aj kj 2 + 4 bj kj 3 .
(15)
Uj = µ0 + µ1 Fj + µ2 Fj2
(16)
Eq.(4) can be given as which comes from Eq.(14) in consequence of using balancing rule.
3
The strategic equations are easily obtained as long as Eq.(16) along with Eq.(5) is put in Eq.(14). Thus, the important results are as (27 cj 3 kj 6 + 81 cj 2 dj kj 6 + 81 cj dj 2 kj 6 + 27 dj 3 kj 6 + 540 P Rcj 3 kj 2 +1620 P Rcj 2 dj kj 2 + 1620 P Rcj dj 2 kj 2 + 540 P Rdj 3 kj 2 − 180 Q2 cj 3 kj 2 −540 Q2cj 2 dj kj 2 − 540 Q2cj dj 2 kj 2 − 180 Q2dj 3 kj 2 + 1800 P QRcj 3 +5400 P QRcj 2 dj + 5400 P QRcj dj 2 + 1800 P QRdj 3 − 400 Q3cj 3 −1200 Q3cj 2 dj − 1200 Q3cj dj 2 − 400 Q3dj 3 )f 6 + (4860 aj cj 2 kj 5
+9720 aj cj dj kj 5 + 4860 aj dj 2 kj 5 + 32400 P Raj cj 2 kj + 64800 P Raj cj dj kj +32400 P Raj dj 2 kj − 10800 Q2aj cj 2 kj − 21600 Q2aj cj dj kj − 10800 Q2aj dj 2 kj )f 4 + 291600 aj 2 cj kj 4 + 291600 aj 2 dj kj 4 f 2 + 5832000 aj 3 kj 3 = 0,
wj = −
−3 f 2cj kj 2 − 3 f 2 dj kj 2 + 10 f 2Qcj + 10 f 2 Qdj − 180 aj kj , 30f (cj + dj )
ro
µ0 =
f 2 (cj + dj ) , µ1 = 0, µ2 = f P, 120
of
bj = −
1 (−13 f 4 cj 2 kj 4 − 26 f 4 cj dj kj 4 − 13 f 4dj 2 kj 4 (600 cj + 600 dj ) f 2
-p
+360 f 4P Rcj 2 + 720 f 4 P Rcj dj + 360 f 4P Rdj 2 − 120 f 4Q2 cj 2 − 240 f 4Q2 cj dj −120 f 4Q2 dj 2 − 1560 f 2aj cj kj 3 − 1560 f 2aj dj kj 3 − 64800 aj 2 kj 2 ).
lP
re
If we use the results (17) along with Eq.(6) in Eq.(16), dark solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 − 20 f 2c1 − 20 f 2 d1 − 180 a1k1 i(−k x+w t+ζ ) 30f (c1 + d1 ) 1 1 1 e u(x, t) = +f tanh2 x + 3 a1 k1 2 − 4 b1 k1 3 t −3 f 2c2 k2 2 − 3 f 2 d2 k2 2 − 20 f 2c2 − 20 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f tanh2 x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
Similarly, singular solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 − 20 f 2c1 − 20 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = +f coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t
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i(−k x+w t+ζ ) 1 1 1 e
−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 − 20 f 2c2 − 20 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f coth2 x + 3 a2 k2 2 − 4 b2 k2 3 t
Similarly, bright solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 + 10 f 2c1 + 10 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = −f sech2 x + 3 a1 k1 2 − 4 b1 k1 3 t 4
i(−k x+w t+ζ ) 1 1 1 e
(17)
(18)
(19)
(20)
(21)
(22)
Similarly, singular solitons are obtained as −3 f 2 c1 k1 2 − 3 f 2 d1 k1 2 + 10 f 2c1 + 10 f 2 d1 − 180 a1k1 30f (c1 + d1 ) u(x, t) = +f csch2 x + 3 a1 k1 2 − 4 b1 k1 3 t
i(−k x+w t+ζ ) 1 1 1 e
−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 + 10 f 2c2 + 10 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = +f csch2 x + 3 a2 k2 2 − 4 b2 k2 3 t
2.2
(23)
(24)
(25)
of
PARABOLIC LAW
ro
The CQ-NLSE equation in polarization-preserving fibers with the parabolic law is given by 2 4 iqt + iaqxxx + bqxxxx + c |q| + d |q| q = 0.
(26)
-p
In the model (26), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficients c, d provide the parabolic nonlinearity effect which has been quite popular recently and is known as a generalized state of kerr fiber nonlinearity.
re
The CQ-NLSE equation in coupled vector form without FWM is: iut + ia1 uxxx + b1 uxxxx + c1 |u|2 + d1 |v|2 u + e1 |u|4 + g1 |u|2 |v|2 + h1 |v|4 u = 0,
lP
2 2 4 2 2 4 ivt + ia2 vxxx + b2 vxxxx + c2 |v| + d2 |u| v + e2 |v| + g2 |v| |u| + h2 |u| v = 0
(27)
with the constants cj ,ej and dj , gj , hj which assure SPM and XPM sequentially.
ur na
The important result is as
(4)
bj Uj
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj + (cj + dj ) Uj3 + (ej + gj + hj ) Uj5 = 0
(28)
as long as Eq.(9) is put in Eq.(27) along with U˜j = Uj .
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−3 f 2c2 k2 2 − 3 f 2 d2 k2 2 + 10 f 2c2 + 10 f 2 d2 − 180 a2k2 i(−k x+w t+ζ ) 30f (c2 + d2 ) 2 2 2 e v(x, t) = −f sech2 x + 3 a2 k2 2 − 4 b2 k2 3 t
Eq.(4) can be given as
Uj = µ0 + µ1 Fj
(29)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(29) along with Eq.(5) is put in Eq.(28). Thus, the important results are as bj = −
6 P aj kj + cj µ21 + dj µ1 2 , µ0 = 0, 4P −3 kj 2 + 5 Q
−3 ej kj 2 − 3 gj kj 2 − 3 hj kj 2 + 5 Qej + 5 Qgj + 5 Qhj µ41 5
+ (−6 P cj − 6 P dj ) µ21 − 36 P 2 aj kj = 0,
wj = −
1 (−6 P aj kj 5 + µ1 2 cj kj 4 + µ1 2 dj kj 4 2 4P −3 kj + 5 Q
+20 P Qaj kj 3 − 6µ1 2 Qcj kj 2 − 6 µ1 2 Qdj kj 2 + 72 P 2 Raj kj
−54 P Q2aj kj + 12 µ1 2 P Rcj + 12 µ1 2 P Rdj + µ1 2 Q2 cj + µ1 2 Q2 dj ).
(30)
If we use the results (30) along with Eq.(6) in Eq.(29), dark solitons are obtained as µ1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t ×e
=
f 2 (c1 + d1 )(k14 + 12k12 + 16) − 2a1 k1 (3k14 + 20k12 + 72) t+ζ1 4 (3k12 + 10)
µ1 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
i−k2 x+
f 2 (c2 + d2 )(k24 + 12k22 + 16) − 2a2 k2 (3k24 + 20k22 + 72) t+ζ2 4 (3k22 + 10)
Similarly, singular solitons are obtained as
×e
=
f 2 (c1 + d1 )(k14 + 12k12 + 16) − 2a1 k1 (3k14 + 20k12 + 72) t+ζ1 4 (3k12 + 10)
µ1 coth x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
v(x, t)
i−k1 x+
re
µ1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
i−k2 x+
(31)
(32)
lP
u(x, t) =
ro
v(x, t)
i−k1 x+
of
-p
u(x, t) =
f 2 (c2 + d2 )(k24 + 12k22 + 16) − 2a2 k2 (3k24 + 20k22 + 72) t+ζ2 4 (3k22 + 10)
(33)
(34)
Similarly, bright solitons are obtained as
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u(x, t) =
v(x, t)
=
µ1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
i−k1 x−
f 2 (c1 + d1 )(k14 − 6k12 + 1) + 2a1 k1 (3k14 − 10k12 + 27) t+ζ1 4 (3k12 − 5)
µ1 sech x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
i−k2 x−
f 2 (c2 + d2 )(k24 − 6k22 + 1) + 2a2 k2 (3k24 − 10k22 + 27 t+ζ2 4 (3k22 − 5) 6
(35)
(36)
Similarly, singular solitons are obtained as = µ1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t
u(x, t)
×e
i−k1 x+
×e
i−k2 x+
f 2 (c2 + d2 )(k24 − 6k22 + 1) − 2a2 k2 (3k24 − 10k22 + 27 t+ζ2 4 (3k22 − 5)
(37)
(38)
QUADRATIC-CUBIC LAW
of
2.3
f 2 (c1 + d1 )(k14 − 6k12 + 1) − 2a1 k1 (3k14 − 10k12 + 27) t+ζ1 4 (3k12 − 5)
= µ1 csch x + 3 a2 k2 2 − 4 b2 k2 3 t
v(x, t)
(39)
ro
The CQ-NLSE equation in polarization-preserving fibers with the quadratic-cubic law is given by 2 iqt + iaqxxx + bqxxxx + c |q| + d |q| q = 0.
-p
In the model (39), the constant a is coefficient of the 3OD while the constant b assures the existence of the 4OD. Also, the coefficients c, d provide the quadratic-cubic law effect.
q 2 2 2 2 |v| + |u| + vu∗ + v ∗ u + d2 |v| + e2 |u| v = 0
(40)
lP
ivt + ia2 vxxx + b2 vxxxx + c2 v
re
The CQ-NLSE in coupled vector form without FWM is: q iut + ia1 uxxx + b1 uxxxx + c1 u |u|2 + |v|2 + uv ∗ + u∗ v + d1 |u|2 + e1 |v|2 u = 0,
with the constants dj and ej which assure SPM and XPM sequentially. Also, the constant cj assures the existence of the quadratic media in birefringent fibers where the first two terms correspond to SPM and XPM sequentially while the last two terms refer to 4WM.
ur na
The important result is as (4)
bj Uj
+ 3aj k − 6bj k 2 Uj′′ − w + aj k 3 − bj k 4 Uj + 2cj Uj2 + (dj + ej ) Uj3 = 0
(41)
as long as Eq.(9) is put in Eq.(41) along with
U˜j = Uj ,
ϕ(x, t) = −kx + wt + ζ.
(42)
Uj = µ0 + µ1 Fj + µ2 Fj2
(43)
Eq.(4) can be given as
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which comes from Eq.(41) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(43) along with Eq.(5) is put in Eq.(41). Thus, the important results are as (27 k 6 dj 3 + 81 k 6dj 2 ej + 81 k 6 dj ej 2 + 27 k 6ej 3 + 540 P Rk 2dj 3 +1620 P Rk 2dj 2 ej + 1620 P Rk 2dj ej 2 + 540 P Rk 2ej 3 − 180 Q2k 2 dj 3 −540 Q2k 2 dj 2 ej − 540 Q2k 2 dj ej 2 − 180 Q2 k 2 ej 3 + 1800 P QRdj 3
+5400 P QRdj 2 ej + 5400 P QRdj ej 2 + 1800 P QRej 3 − 400 Q3dj 3 −1200 Q3dj 2 ej − 1200 Q3dj ej 2 − 400 Q3ej 3 )f 6 + (270 k 4 cj dj 2 7
+540 k 4cj dj ej + 270 k 4cj ej 2 − 5400 P Rcj dj 2 − 10800 P Rcj dj ej
+(4860 k 5aj dj 2 + 9720 k 5aj dj ej + 4860 k 5aj ej 2 + 32400 P Rkaj dj 2
+64800 P Rkaj dj ej + 32400 P Rkaj ej 2 − 10800 Q2kaj dj 2 − 21600 Q2kaj dj ej −10800 Q2kaj ej 2 )f 4 + 291600 k 4aj 2 dj + 291600 k 4aj 2 ej f 2 + (32400 k 3aj cj dj +32400 k 3aj cj ej − 4000 cj 3 )f 3 + 972000 k 2 aj 2 cj f + 5832000 k 3aj 3 = 0
w=−
1 (−13 f 4 k 4 dj 2 − 26 f 4k 4 dj ej − 13 f 4 k 4 ej 2 (600 dj + 600 ej ) f 2
+360 f 4P Rdj 2 + 720 f 4P Rdj ej + 360 f 4P Rej 2 − 120 f 4Q2 dj 2 − 240 f 4Q2 dj ej −120 f 4Q2 ej 2 − 1560 f 2k 3 aj dj − 1560 f 2k 3 aj ej + 800 f 2 cj 2 − 64800 k 2aj 2 ),
bj = −
f 2 (dj + ej ) , µ1 = 0, µ2 = f P. 120
of
−3 f 2 k 2 dj − 3 f 2 k 2 ej + 10 f 2 Qdj + 10 f 2 Qej − 20 f cj − 180 aj k , 30f (dj + ej )
ro
µ0 =
re
-p
If we use the results (44) along with Eq.(6) in Eq.(43), dark solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 − 20 f 2 d1 − 20 f 2e1 − 20f c1 − 180 a1k i(−kx+wt+ζ) 30f (d1 + e1 ) e u(x, t) = +f tanh2 x + 3 a1 k 2 − 4 b1 k 3 t
lP
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 − 20 f 2 d2 − 20 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = +f tanh2 x + 3 a2 k 2 − 4 b2 k 3 t
ur na
Similarly, singular solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 − 20 f 2 d1 − 20 f 2e1 − 20f c1 − 180 a1k 30f (d1 + e1 ) u(x, t) = +f coth2 x + 3 a1 k 2 − 4 b1 k 3 t
i(−kx+wt+ζ) e
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 − 20 f 2 d2 − 20 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = 2 2 3 +f coth x + 3 a2 k − 4 b2 k t
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
−5400 P Rcj ej 2 + 1800 Q2cj dj 2 + 3600 Q2cj dj ej + 1800 Q2cj ej 2 )f 5
Similarly, bright solitons are obtained as −3 f 2d1 k 2 − 3 f 2 e1 k 2 + 10 f 2 d1 + 10 f 2e1 − 20f c1 − 180 a1k 30f (d1 + e1 ) u(x, t) = −f sech2 x + 3 a1 k 2 − 4 b1 k 3 t
i(−kx+wt+ζ) e
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 + 10 f 2 d2 + 10 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = 2 2 3 −f sech x + 3 a2 k − 4 b2 k t
8
(44)
(45)
(46)
(47)
(48)
(49)
(50)
× ei(−kx+wt+ζ)
−3 f 2 d2 k 2 − 3 f 2 e2 k 2 + 10 f 2 d2 + 10 f 2 e2 − 20f c2 − 180 a2k i(−kx+wt+ζ) 30f (d2 + e2 ) e v(x, t) = +f csch2 x + 3 a2 k 2 − 4 b2 k 3 t
2.4
(51)
(52)
NON-LOCAL LAW
The CQ-NLSE equation in polarization-preserving fibers with the non-local law is given by 2 iqt + iaqxxx + bqxxxx + c |q| q = 0.
(53)
of
xx
In the model (53), the constant a is coefficient of the 3OD while the constants b assure the existence of the 4OD. Also, the coefficient c provides the non-local law effect.
ro
The CQ-NLSE equation in coupled vector form without FWM is: 2 2 u = 0, + d1 |v| iut + ia1 uxxx + b1 uxxxx + c1 |u| xx
2 ivt + ia2 vxxx + b2 vxxxx + c2 |v|
-p
xx
xx
2 v=0 + d2 |u| xx
(54)
re
with the constants cj and dj which assure SPM and XPM sequentially. The important result is as
+ 3aj kj − 6bj kj 2 Uj′′ − wj + aj kj 3 − bj kj 4 Uj
lP
(4)
bj Uj
ur na
+2 (cj + dj ) Uj Uj′
2
+ 2 (cj + dj ) Uj2 Uj′′ = 0
(55)
as long as Eq.(9) is put in Eq.(55) along with U˜j = Uj . Eq.(4) can be given as
Uj = µ0 + µ1 Fj
(56)
which comes from Eq.(55) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(56) along with Eq.(5) is put in Eq.(55). Thus, the important results are as 3aj kj , µ0 = 0, bj = − 2 −3 kj 2 + Q
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Similarly, singular solitons are obtained as −3 f 2 d1 k 2 − 3 f 2 e1 k 2 + 10 f 2 d1 + 10 f 2 e1 − 20f c1 − 180 a1 k 30f (d1 + e1 ) u(x, t) = +f csch2 x + 3 a1 k 2 − 4 b1 k 3 t
µ1 = ±
wj = −
s
6P aj kj , −3 cj kj 2 − 3 dj kj 2 + Qcj + Qdj
−3 kj 4 + 2 Qkj 2 + 12 RP − 3 Q2 aj kj 2 −3 kj 2 + Q
9
(57)
If we use the results (57) along with Eq.(6) in Eq.(56), dark solitons are obtained as
± −
2
3 c1 k1
3 k1 4 + 4 k1 2 a1 k1 t+ζ1 2 3 k1 2 + 2
×e
v(x, t)
= ±
i−k1 x−
s
−
3 c2 k2
2
i−k2 x−
6a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 + 2 c2 + 2 d2
Similarly, singular solitons are obtained as
= ±
−
3 c1 k1
±
−
re
3 c2 k2
×e
(60)
lP
=
2
6a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 + 2 c2 + 2 d2
ur na
v(x, t)
i−k1 x−
s
6a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 + 2 c1 + 2 d1
3 k1 4 + 4 k1 2 a1 k1 t+ζ1 2 3 k1 2 + 2
×e
2
(59)
-p
u(x, t)
s
(58)
3 k2 4 + 4 k2 2 a2 k2 t+ζ2 2 3 k2 2 + 2
×e
6a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 + 2 c1 + 2 d1
of
s
ro
u(x, t) =
i−k2 x−
3 k2 4 + 4 k2 2 a2 k2 t+ζ2 2 3 k2 2 + 2
(61)
Similarly, bright solitons are obtained as
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
u(x, t) =
±
s
3 c1 k1
×e
2
i−k1 x−
6a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 − c1 − d1 3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
10
(62)
v(x, t)
=
±
s
3 c2 k2
2
6a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 − c2 − d2 3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
×e
i−k2 x−
(63)
Similarly, singular solitons are obtained as
3 c1 k1
= ±
−
×e
i−k2 x−
(64)
6a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t 2 + 3 d2 k2 − c2 − d2
3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
(65)
RICCATI EQUATION TECHNIQUE: A QUICK OVERVIEW
lP
3
3 c2 k2
2
re
v(x, t)
i−k1 x−
s
6a1 k1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t 2 + 3 d1 k1 − c1 − d1
3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
×e
2
of
± −
ro
s
-p
u(x, t) =
The steps of applying the method to any governing model are as follows : Step-1: A model can be taken as
ur na
S (u, ut , ux , utt , uxt , uxx , ...) = 0.
(66)
The important equation emerged from Eq.(66) is as
by using of
G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
(67)
u(x, t) = U (ϑ), ϑ = x − pt.
(68)
Step-2: The solution structure of Eq.(67) is considered as
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
U (ϑ) =
N X
Ai V i (ϑ)
(69)
i=0
and also the number N comes from Eq.(67) in consequence of using balancing condition. The function V (ϑ) holds the Riccati equation as V ′ (ϑ) = S2 V 2 (ϑ) + S1 V (ϑ) + S0 , S2 6= 0 (70)
11
V (ϑ) =
−
S1 1 − , µ=0 2S2 S2 ϑ + ϑ0
(71)
where µ = S12 − 4S0 S2 and ϑ0 is an arbitrary real constant.
3.1
ro
of
Step-3: The strategic equations are easily obtained as long as the solution structure (69) along with Eq.(70) is put in Eq.(67). Thus, the important results Ai ,S2 , S1 , S0 emerged from these strategic equations are obtained. If we use the results Ai ,S2 , S1 , S0 in Eq.(69), solitons in an optical fiber communication system are recovered by Eq.(71).
KERR LAW
-p
Solitons with the CQ-NLSE in coupled vector form without FWM along with the riccati function principle are recovered in this section. Eq.(69) can be given as
Uj = A0 + A1 Vj + A2 Vj2
(72)
re
which comes from Eq.(14) in consequence of using balancing rule.
Result-1:
A2 5 bj S1 2 − 6 bj kj 2 + 3 aj kj 2
20S2 bj
ur na
A0 =
lP
The strategic equations are easily obtained as long as Eq.(72) along with Eq.(70) is put in Eq.(14). Thus, the important results are as
S0 =
, A1 =
A2 S1 , S2
120 S24 bj + A2 2 dj 5 bj S1 2 − 6 bj kj 2 + 3 aj kj , cj = − , 20S2 bj A2 2 kj 2 119 bj 2 kj 2 − 119 aj bj kj + 36 aj 2 wj = − 25bj
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
where S2 , S1 and S0 are constants. The strategic solutions of Eq.(70) are as √ √ µ µ S1 − tanh ϑ + ϑ0 , µ > 0 V (ϑ) = − 2S2 2S2 2 √ √ µ µ S1 V (ϑ) = − − coth ϑ + ϑ0 , µ > 0 2S2 2S2 2 √ √ S1 −µ −µ + tan ϑ + ϑ0 , µ < 0 V (ϑ) = − 2S2 2S2 2 √ √ S1 −µ −µ − cot ϑ + ϑ0 , µ < 0 V (ϑ) = − 2S2 2S2 2
and
µ = S12 − 4S0 S2 = −
3kj (−2 kj bj + aj ) . 5bj
(73)
If we use the results (73) along with Eq.(71) in Eq.(72), dark solitons are obtained as
u(x, t) =
3 A2 k1 (2 b1 k1 − a1 + (a1 − 2b1 k1 )) − 20 S2 2 b1 r 15k1 (−2 b1 k1 + a1 ) 2 − x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh 100b1 ×ei(−k1 x+w1 t+ζ1 )
12
(74)
v(x, t)
=
3 A2 k2 (2 b2 k2 − a2 + (a2 − 2b2 k2 )) − 20 S2 2 b2 r 15k2 (−2 b2 k2 + a2 ) 2 3 2 x + 3 a2 k2 − 4 b2 k2 t + ϑ0 − ×tanh 100b2
×ei(−k2 x+w2 t+ζ2 )
(75)
with bj kj (−2 bj kj + aj ) < 0. Similarly, singular solitons are obtained as
-p
×ei(−k1 x+w1 t+ζ1 )
3 A2 k2 (2 b2 k2 − a2 + (a2 − 2b2 k2 )) − 20 S2 2 b2 = r 15k2 (−2 b2 k2 + a2 ) 2 3 2 x + 3 a2 k2 − 4 b2 k2 t + ϑ0 − ×coth 100b2
re
v(x, t)
of
3 A2 k1 (2 b1 k1 − a1 + (a1 − 2b1 k1 )) − 20 S2 2 b1 r 15k1 (−2 b1k1 + a1 ) 2 ×coth − x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 100b1
ro
u(x, t) =
lP
×ei(−k2 x+w2 t+ζ2 )
with
(76)
(77)
Result-2:
ur na
bj kj (−2 bj kj + aj ) < 0.
A0 =
A1 =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
A2 5 bj S1 2 − 6 bj kj 2 + 3 aj kj + 40 f 60 S2 2 bj
,
f 120 S24 bj + A2 2 dj A2 S1 , S0 = , cj = − , S2 b j S2 A2 2
160 f 2 + −80 bj S1 2 − 120 bj kj 2 + 60 aj kj f + 10 S1 4 bj 2 + 30 S1 2 bj 2 kj 2 +36 bj 2 kj 4 − 15 S1 2 aj kj bj − 36 aj bj kj 3 + 9 aj 2 kj 2 = 0,
wj =
kj (90 S1 2 bj 2 kj + 56 bj 2 kj 3 − 45 S1 2 aj bj 20bj
−56 aj bj kj 2 − 360 f bj kj + 9 aj 2 kj + 180 f aj ) and µ = S12 − 4S0 S2 = − 13
−bj S1 2 + 4 f . bj
(78)
If we use the results (78) along with Eq.(71) in Eq.(72), dark solitons are obtained as
u(x, t) =
A2 10 S1 2 b1 + 6 b1 k1 2 − 3 a1 k1 − 40 f + 60f − 15S1 2 b1 − 60 S2 2 b1 s 2 b + 4 f −S 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh2 − 4b1
A2 10 S1 2 b2 + 6 b2 k2 2 − 3 a2 k2 − 40 f + 60f − 15S1 2 b2 − 60 S2 2 b2 s 2 b + 4 f −S 2 1 2 3 2 ×tanh − x + 3 a2 k2 − 4 b2 k2 t + ϑ0 4b2
×ei(−k1 x+w1 t+ζ1 )
×ei(−k2 x+w2 t+ζ2 )
with
re
Similarly, singular solitons are obtained as
lP
A2 10 S1 2 b1 + 6 b1 k1 2 − 3 a1 k1 − 40 f + 60f − 15S1 2 b1 − 60 S2 2 b1 s = 2 b + 4 f −S 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×coth2 − 4b1
ur na
ei(−k1 x+w1 t+ζ1 )
v(x, t)
=
A2 10 S1 2 b2 + 6 b2 k2 2 − 3 a2 k2 − 40 f + 60f − 15S1 2 b2 − 60 S2 2 b2 q 2 ×coth2 − −S1 4bb22+4 f x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 ×ei(−k2 x+w2 t+ζ2 )
with
3.2
PARABOLIC LAW
(80)
-p
bj kj −S1 2 bj + 4 f < 0.
u(x, t)
of
=
(79)
ro
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(81)
(82)
bj kj −S1 2 bj + 4 f < 0.
Eq.(69) can be given as Uj = A0 + A1 Vj
(83)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(83) along with Eq.(70) is put in Eq.(28). Thus, 14
the important results are as S1 = 2
A1 4 gj + A1 4 hj + 24 S2 4 bj A0 S2 , , ej = − A1 A1 4
cj =
1 (40 A0 2 S2 4 bj − 40 A1 2 S0 S2 3 bj A1 4
+12 A1 2 S2 2 bj kj 2 − 6 A1 2 S2 2 aj kj − A1 4 dj ),
wj =
1 (16 A0 4 S2 4 bj − 32 A0 2 A1 2 S0 S2 3 bj A1 4
+12 A0 2 A1 2 S2 2 bj kj 2 + 16 A1 4 S0 2 S2 2 bj − 12 A1 4 S0 S2 bj kj 2
and µ=
S12
− 4S0 S2 =
4S2 A0 2 S2 − A1 2 S0 A1 2
.
of
+A1 4 bj kj 4 − 6 A0 2 A1 2 S2 2 aj kj + 6 A1 4 S0 S2 aj kj − A1 4 aj kj 3 )
(84)
A0 2 S2 − A1 2 S0 S2 s
×tanh
S2 A0 2 S2 − A1 2 S0 A1 2
=
−
s
×tanh
S2 A0 2 S2 − A1 2 S0 A1 2
×ei(−k2 x+w2 t+ζ2 )
with
Similarly, singular solitons are obtained as
u(x, t)
= −
s
x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0
(85) (86) (87)
A0 2 S2 − A1 2 S0 S2
ur na
v(x, t)
lP
×ei(−k1 x+w1 t+ζ1 )
s
-p
−
re
u(x, t) =
s
ro
If we use the results (84) along with Eq.(71) in Eq.(83), dark solitons are obtained as
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(88)
x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0
(89) (90)
S2 A0 2 S2 − A1 2 S0 > 0.
A0 2 S2 − A1 2 S0 S2 (91) s
×coth
S2 A0 2 S2 − A1 2 S0 A1
2
×ei(−k1 x+w1 t+ζ1 ) 15
x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0
(92) (93)
= −
v(x, t)
s
A0 2 S2 − A1 2 S0 S2 (94) s
×coth
S2 A0 2 S2 − A1 2 S0 A1
2
×ei(−k2 x+w2 t+ζ2 ) with
3.3
x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0
(95) (96)
S2 A0 2 S2 − A1 2 S0 > 0.
QUADRATIC-CUBIC LAW
Eq.(69) can be given as
of
Uj = A0 + A1 Vj + A2 Vj2 which comes from Eq.(41) in consequence of using balancing rule.
(97)
ro
The strategic equations are easily obtained as long as Eq.(97) along with Eq.(70) is put in Eq.(41). Thus, the important results are as
w=
-p
Result-1:
1 (k 4 A2 4 bj − 6 k 2 A1 2 A2 2 S2 2 bj + 24 k 2 A2 4 S0 S2 bj A2 4
re
+A1 4 S2 4 bj − 8 A1 2 A2 2 S0 S2 3 bj + 16 A2 4 S0 2 S2 2 bj
3S2 2 6 k 2 A2 2 bj − 5 A1 2 S2 2 bj + 20 A2 2 S0 S2 bj − 3 kA2 2 aj A2 3
A0 =
and
,
A2 S0 A1 S2 120 S24 bj + A2 2 ej , , S1 = , dj = − S2 A2 A2 2
ur na
cj =
lP
−k 3 A2 4 aj + 3 kA1 2 A2 2 S2 2 aj − 12 kA2 4 S0 S2 aj )
µ=
S12
− 4S0 S2 =
S2 A1 2 S2 − 4 A2 2 S0 A2 2
.
(98)
If we use the results (98) along with Eq.(71) in Eq.(97), dark solitons are obtained as
Jo
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u(x, t) =
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S S − 4 A A S 0 2 2 1 2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 ×tanh2 4A2 2 ×ei(−kx+wt+ζ)
16
(99)
v(x, t)
=
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S S − 4 A S A 0 2 2 2 1 ×tanh2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 4A2 2
×ei(−kx+wt+ζ) with
(100)
S2 A1 2 S2 − 4 A2 2 S0 > 0.
Similarly, singular solitons are obtained as
of
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 4A2 S2 s 2 2 S2 A1 S2 − 4 A2 S0 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 ×coth2 4A2 2
ro
u(x, t) =
re
4 A2 2 S0 − A1 2 S2 + A1 2 S2 − 4A2 2 S0 = 4A2 S2 s 2 2 S S − 4 A A S 0 2 2 1 2 ×coth2 x + 3 a1 k 2 − 4 b1 k 3 t + ϑ0 4A2 2
lP
v(x, t)
3.4
ur na
×ei(−kx+wt+ζ)
with
(101)
-p
×ei(−kx+wt+ζ)
(102)
S2 A1 2 S2 − 4 A2 2 S0 > 0.
NON-LOCAL LAW
Eq.(69) can be given as
Uj = A0 + A1 Vj
(103)
which comes from Eq.(55) in consequence of using balancing rule.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The strategic equations are easily obtained as long as Eq.(103) along with Eq.(70) is put in Eq.(55). Thus, the important results are as A0 =
cj = −
S1 2 bj + 6 bj kj 2 − 3 aj kj A1 S1 , S0 = , 2S2 4S2 bj
A1 2 dj + 4 S2 2 bj , wj = −kj 3 (−bj kj + aj ) A1 2
and µ = S12 − 4S0 S2 =
3kj (−2 bj kj + aj ) . bj
17
(104)
If we use the results (104) along with Eq.(71) in Eq.(103), dark solitons are obtained as
u(x, t)
s
3A21 k1 (−2 b1 k1 + a1 ) 4S22 b1 s 3k (−2 b k + a ) 1 1 1 1 x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 ×tanh 4b1
= −
×ei(−k1 x+w1 t+ζ1 )
of
3A21 k2 (−2 b2 k2 + a2 ) 4S22 b2 s 3k2 (−2 b2 k2 + a2 ) ×tanh x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 4b2
= −
×ei(−k2 x+w2 t+ζ2 ) with
re
Similarly, singular solitons are obtained as s
3A21 k1 (−2 b1 k1 + a1 ) 4S22 b1 s 3k (−2 b k + a ) 1 1 1 1 ×coth x + 3 a1 k1 2 − 4 b1 k1 3 t + ϑ0 4b1
lP
−
(106)
-p
bj kj (−2 bj kj + aj ) > 0.
u(x, t) =
(105)
s
ro
v(x, t)
(107)
ur na
×ei(−k1 x+w1 t+ζ1 )
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
=
s
3A21 k2 (−2 b2 k2 + a2 ) 4S22 b2 s 3k2 (−2 b2k2 + a2 ) ×coth x + 3 a2 k2 2 − 4 b2 k2 3 t + ϑ0 4b2 −
×ei(−k2 x+w2 t+ζ2 )
with bj kj (−2 bj kj + aj ) > 0.
4
SINE-GORDON EQUATION: A QUICK BRUSH-UP
The steps of applying the method to any governing model are as follows:
18
(108)
Step-1: A model can be taken as S (u, ut , ux , utt , uxt , uxx , ...) = 0.
(109)
The important equation emerged from Eq.(109) is as G (U, U ′ , U ′′ , U ′′′ , · · · ) = 0
(110)
u(x, t) = U (ϑ), ϑ = x − pt.
(111)
by using of
Step-2: The solution structure of Eq.(110) is considered as U (ϑ) =
N X
cosi−1 (V (ϑ))[Bi sin(V (ϑ)) + Ai cos(V (ϑ))] + A0
(112)
i=1
of
and also the number N comes from Eq.(110) in consequence of using balancing condition. The function V (ϑ) holds V ′ (ϑ) = sin (V (ϑ)) .
ro
(113)
The elementary solutions of Eq.(113) are given as
-p
sin (V (ϑ)) = sech (ϑ) or sin (V (ϑ)) = icsh (ϑ) ,
re
cos (V (ϑ)) = tanh (ϑ) or cos (V (ϑ)) = coth (ϑ)
(114)
4.1
lP
Step-3: The strategic equations are easily obtained as long as the solution structure (112) along with Eq.(113) is put in Eq.(110). Thus, the important results Ai ,Bi , A0 emerged from these strategic equations are obtained. If we use the results Ai ,Bi , A0 in Eq.(112), solitons in an optical fiber communication system are recovered by Eq.(114).
KERR LAW
ur na
Solitons with the CQ-NLSE in coupled vector form without FWM along with sine-Gordon function principle are recovered in this section. Eq.(112) can be given as
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + cos(Vj (ϑ)) [B2 sin(Vj (ϑ)) + A2 cos(Vj (ϑ))] + A0
(115)
which comes from Eq.(14) in consequence of using balancing rule.
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
The strategic equations are easily obtained as long as Eq.(115) along with Eq.(113) is put in Eq.(14). Thus, the important results are as Result-1:
A0 = 0, A1 = 0, A2 = 0, B1 = 0,
B2 = ±
s
180aj kj 3aj kj , , bj = 2 3 kj 2 + 5 3 kj + 5 (cj + dj ) 2
19
(116)
If we put Eq.(116) along with Eq.(114) in Eq.(115), one can get combo dark-bright solitons
u(x, t)
= ±
s
180a1k1 3 k1 + 5 (c1 + d1 ) 2
× tanh x + 3 a1 k1 2 − 4 b1 k1 3 t sech x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t)
=
±
a1 k1 i−k1 x−
s
3 k1 4 + 10 k1 2 − 33 2 3 k1 2 + 5
t+ζ1
(117)
180a2 k2j 3 k2 + 5 (c2 + d2 ) 2
of
×e
×e
a2 k2 i−k2 x−
3 k2 4 + 10 k2 2 − 33 2 3 k2 2 + 5
u(x, t) =
± −
t+ζ2
(118)
re
Similarly, singular solitons are obtained as s
-p
ro
× tanh x + 3 a2 k2 2 − 4 b2 k2 3 t sech x + 3 a2 k2 2 − 4 b2 k2 3 t
180a1k1 3 k1 + 5 (c1 + d1 ) 2
a1 k1 i−k1 x−
3 k1 4 + 10 k1 2 − 33 2 3 k1 2 + 5
t+ζ1
(119)
ur na
×e
lP
× coth x + 3 a1 k1 2 − 4 b1 k1 3 t csch x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
aj kj 3 kj 4 + 10 kj 2 − 33 wj = − . 2 3 kj 2 + 5
=
s
± −
180a2 k2j 3 k2 + 5 (c2 + d2 ) 2
× coth x + 3 a2 k2 2 − 4 b2 k2 3 t csch x + 3 a2 k2 2 − 4 b2 k2 3 t a2 k2 3 k2 4 + 10 k2 2 − 33 i−k2 x− 2 3 k2 2 + 5
×e
t+ζ2
(120)
Result-2: A1 = 0, B1 = 0, B2 = 0,
aj kj 3 kj 4 − 20 kj 2 + 192 3aj kj bj = , w = − j 2 3 kj 2 − 10 2 3 kj 2 − 10 20
A0 = ±
180aj kj − , A2 = ± 3 kj 2 − 10 (cj + dj )
s
−
180aj kj . 3 kj 2 − 10 (cj + dj )
(121)
If we put Eq.(121) along with Eq.(114) in Eq.(115), one can get dark solitons
u(x, t) =
s s 180a1 k1 180a1 k1 ± − ± − 3 k1 2 − 10 (c1 + d1 ) 3 k1 2 − 10 (c1 + d1 )
× tanh2
x + 3 a1 k1 2 − 4 b1 k1 3 t
a1 k1 3 k1 4 − 20 k1 2 + 192 i−k1 x− 2 3 k1 2 − 10
t+ζ1
(122)
of
×e
× tanh2
x + 3 a2 k2 2 − 4 b2 k2 3 t
×e
a2 k2 i−k2 x−
3 k2 4 − 20 k2 2 + 192 2 3 k2 2 − 10
× coth2
x + 3 a1 k1 2 − 4 b1 k1 3 t
×e
v(x, t)
=
t+ζ2
s s 180a1 k1 180a1 k1 ± − ± − 2 2 3 k1 − 10 (c1 + d1 ) 3 k1 − 10 (c1 + d1 )
ur na
u(x, t) =
a1 k1 i−k1 x−
3 k1 4 − 20 k1 2 + 192 2 3 k1 2 − 10
× coth2
×e
3 k2 4 − 20 k2 2 + 192 2 3 k2 2 − 10 21
t+ζ1
x + 3 a2 k2 2 − 4 b2 k2 3 t
a2 k2 i−k2 x−
(123)
s s 180a2 k2j 180a2 k2j ± − ± − 2 2 3 k2 − 10 (c2 + d2 ) 3 k2 − 10 (c2 + d2 )
lP
Similarly, combo singular solitons are obtained as
ro
-p
=
s s 180a2 k2j 180a2 k2j ± − ± − 3 k2 2 − 10 (c2 + d2 ) 3 k2 2 − 10 (c2 + d2 )
re
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
(124)
t+ζ2
(125)
Result-3: A1 = 0, B1 = 0, bj =
wj = −
3 kj 4 − 5 kj 2 + 12 aj kj 6 kj 2 − 5
3aj kj , 6 kj 2 − 5 s
, A0 = ± −
s
90aj kj A2 = ± − , B2 = ± 2 6 kj − 5 (cj + dj )
s
90aj kj , 6 kj 2 − 5 (cj + dj ) 6 kj
2
90aj kj , − 5 (cj + dj )
(126)
If we put Eq.(126) along with Eq.(114) in Eq.(115), one can get combo dark-bright solitons
6 k1 2 − 5
-p
3 k1 4 − 5 k1 2 + 12
t+ζ1
lP
×e
a1 k1 i−k1 x−
s s 90a2 k2j 90a2 k2j ± − ± 2 2 6 k2 − 5 (c2 + d2 ) 6 k2 − 5 (c2 + d2 ) ×sech x + 3 a2 k2 2 − 4 b2 k2 3 t tanh x + 3 a2 k2 2 − 4 b2 k2 3 t s 90a2 k2j 3 2 2 ± − 6 k 2 − 5 (c + d ) tanh x + 3 a2 k2 − 4 b2 k2 t 2 2 2
ur na =
×e
a2 k2 i−k2 x−
re
v(x, t)
of
s s 90a1 k1 90a1 k1 ± − ± 2 2 6 k1 − 5 (c1 + d1 ) 6 k1 − 5 (c1 + d1 ) ×sech x + 3 a1 k1 2 − 4 b1 k1 3 t tanh x + 3 a1 k1 2 − 4 b1 k1 3 t s 90a1 k1 3 2 2 ± − 6 k 2 − 5 (c + d ) tanh x + 3 a1 k1 − 4 b1 k1 t 1 1 1
ro
u(x, t) =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
3 k2 4 − 5 k2 2 + 12 6 k2 2 − 5
Similarly, combo singular solitons are obtained as
22
(127)
t+ζ2
(128)
u(x, t)
i−k1 x−
a1 k1 3 k1 4 − 5 k1 2 + 12 6 k1 2 − 5
t+ζ1
(129)
of
×e
ro
4.2
PARABOLIC LAW
6 k2 2 − 5
t+ζ2
ur na
Eq.(112) can be given as
3 k2 4 − 5 k2 2 + 12
lP
×e
a2 k2 i−k2 x−
re
-p
v(x, t)
s s 90a2 k2j 90a2 k2j ± − ± − 2 2 − 5 (c + d ) 6 k 6 k 2 2 2 2 − 5 (c2 + d2 ) = ×csch x + 3 a2 k2 2 − 4 b2 k2 3 t coth x + 3 a2 k2 2 − 4 b2 k2 3 t s 90a2 k2j 3 2 2 ± − 6 k 2 − 5 (c + d ) coth x + 3 a2 k2 − 4 b2 k2 t 2 2 2
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + A0
(130)
(131)
which comes from Eq.(28) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(131) along with Eq.(113) is put in Eq.(28). Thus, the important results are as Result-1:
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s s 90a1 k1 90a1 k1 ± − ± − 2 2 6 k1 − 5 (c1 + d1 ) 6 k1 − 5 (c1 + d1 ) = ×csch x + 3 a1 k1 2 − 4 b1 k1 3 t coth x + 3 a1 k1 2 − 4 b1 k1 3 t s 90a1 k1 3 2 2 ± − 6 k 2 − 5 (c + d ) coth x + 3 a1 k1 − 4 b1 k1 t 1 1 1
wj = bj kj 4 − aj kj 3 + 12 bj kj 2 − 6 aj kj + 16 bj
A0 = 0, B1 = 0, A1 = ±
ej = −
s
12 bj kj 2 − 6 aj kj + 40 bj , cj + dj
1 (36 bj 2 gj kj 4 + 36 bj 2 hj kj 4 (−6bj kj2 + 3aj kj − 20bj )2
−36 aj bj gj kj 3 − 36 aj bj hj kj 3 + 9 aj 2 gj kj 2 + 9 aj 2 hj kj 2 23
+240 bj 2 gj kj 2 + 240 bj 2 hj kj 2 − 120 aj bj gj kj − 120 aj bj hj kj +400 bj 2 gj + 400 bj 2 hj + 6 bj cj 2 + 12 bj cj dj + 6 bj dj 2 )
(132)
If we put Eq.(132) along with Eq.(114) in Eq.(131), one can get dark solitons
= ±
s
12 b1 k1 2 − 6 a1 k1 + 40 b1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t c1 + d1
×ei(−k1 x+(b1 k1
v(x, t)
= ±
s
−a1 k1 3 +12 b1 k1 2 −6 a1 k1 +16 b1 )t+ζ1 )
4
12 b2 k2 2 − 6 a2 k2 + 40 b2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t c2 + d2
×ei(−k2 x+(b2 k2
4
−a2 k2 3 +12 b2 k2 2 −6 a2 k2 +16 b2 )t+ζ2 )
= ±
12 b1k1 2 − 6 a1 k1 + 40 b1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t c1 + d1
=
±
−a1 k1 3 +12 b1 k1 2 −6 a1 k1 +16 b1 )t+ζ1 )
×ei(−k2 x+(b2 k2
4
−a2 k2 3 +12 b2 k2 2 −6 a2 k2 +16 b2 )t+ζ2 )
ur na
Result-2: A1 = ±
(135)
12 b2k2 2 − 6 a2 k2 + 40 b2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t c2 + d2
lP
v(x, t)
s
4
re
×ei(−k1 x+(b1 k1
-p
u(x, t)
(134)
ro
Similarly, singular solitons are obtained as s
(133)
of
u(x, t)
s
45aj kj , B1 = ± − 2 6 kj + 5 (cj + dj )
A0 = 0, wj = −
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ej = −
s
−
aj kj 3 kj 4 + 5 kj 2 + 27 2
6 kj + 5
6 kj
2
, bj =
(136)
45aj kj , + 5 (cj + dj ) 3aj kj 6 kj 2 + 5
1 (−12 cj 2 kj 2 − 24 cj dj kj 2 − 12 dj 2 kj 2 225 aj kj
+225 aj gj kj + 225 aj hj kj − 10 cj 2 − 20 cj dj − 10 dj 2 ).
If we put Eq.(137) along with Eq.(114) in Eq.(131), one can get combo dark-bright solitons
24
(137)
u(x, t) =
s 45a1 k1 3 2 ± − 6 k 2 + 5 (c + d ) sech x + 3 a1 k1 − 4 b1 k1 t 1 1 1
×e
=
a1 k1 i−k1 x−
3 k1 4 + 5 k1 2 + 27 6 k1 2 + 5
t+ζ1
(138)
s 45a2 k2 ± − tanh x + 3 a2 k2 2 − 4 b2 k2 3 t 2 6 k2 + 5 (c2 + d2 )
6 k2 2 + 5
t+ζ2
lP
s 45a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t ± − 2 6 k1 + 5 (c1 + d1 )
ur na
s 45a1 k1 2 3 ± csch x + 3 a k − 4 b k t 1 1 1 1 2 6 k1 + 5 (c1 + d1 )
×e
=
a1 k1 i−k1 x−
3 k1 4 + 5 k1 2 + 27 6 k1 2 + 5
×e
3 k2 4 + 5 k2 2 + 27 6 k2 2 + 5
25
t+ζ1
s 45a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t 2 ± 6 k2 + 5 (c2 + d2 ) a2 k2 i−k2 x−
(139)
s 45a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t ± − 2 6 k2 + 5 (c2 + d2 )
ro
3 k2 4 + 5 k2 2 + 27
Similarly, combo singular solitons are obtained as
v(x, t)
-p
×e
a2 k2 i−k2 x−
re
=
s 45a2 k2 3 2 ± − 6 k 2 + 5 (c + d ) sech x + 3 a2 k2 − 4 b2 k2 t 2 2 2
u(x, t)
of
v(x, t)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s 45a1 k1 ± − tanh x + 3 a1 k1 2 − 4 b1 k1 3 t 2 6 k1 + 5 (c1 + d1 )
(140)
t+ζ2
(141)
4.3
QUADRATIC-CUBIC LAW
Eq.(112) can be given as Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + cos(Vj (ϑ)) [B2 sin(Vj (ϑ)) + A2 cos(Vj (ϑ))] + A0
(142)
which comes from Eq.(41) in consequence of using balancing rule. The strategic equations are easily obtained as long as Eq.(142) along with Eq.(113) is put in Eq.(41). Thus, the important results are as Result-1: k 3 aj − 12 kaj + w , A2 = ± bj = k 4 − 24 k 2 + 16
−
120 (k 3 aj − 12 kaj + w) (dj + ej ) (k 4 − 24 k 2 + 16)
s
3 (3 k 5 aj − 20 k 3 aj + 6 k 2 w + 192 kaj − 20 w) (dj + ej ) 40 (k 4 − 24 k 2 + 16) ( k 3 aj − 12 kaj + w)
2
re
-p
If we put Eq.(143) along with Eq.(114) in Eq.(142), one can get dark solitons s s 3 120 k a − 12 ka + w 120 k 3 a1 − 12 ka1 + w 1 1 ± − ± − (d1 + e1 ) (k 4 − 24 k 2 + 16) (d1 + e1 ) (k 4 − 24 k 2 + 16) u(x, t) = × tanh2 x + 3 a1 k1 2 − 4 b1 k1 3 t s s 3 120 k a − 12 ka + w 120 k 3 a2 − 12 ka2 + w 2 2 ± − ± − (d2 + e2 ) (k 4 − 24 k 2 + 16) (d2 + e2 ) (k 4 − 24 k 2 + 16) × tanh2 x + 3 a2 k2 2 − 4 b2 k2 3 t
ur na
lP
v(x, t) =
Similarly, combo singular solitons are obtained as s s 120 k 3 a1 − 12 ka1 + w 120 k 3 a1 − 12 ka1 + w ± − ± − (d1 + e1 ) (k 4 − 24 k 2 + 16) (d1 + e1 ) (k 4 − 24 k 2 + 16) u(x, t) = × coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t
v(x, t) =
Result-2:
s s 3 120 k a − 12 ka + w 120 k 3 a2 − 12 ka2 + w 2 2 ± − ± − (d2 + e2 ) (k 4 − 24 k 2 + 16) (d2 + e2 ) (k 4 − 24 k 2 + 16)
(143)
ro
−
of
120 ( k 3 aj − 12 kaj + w) , A1 = 0, B1 = 0, B2 = 0, (dj + ej ) (k 4 − 24 k 2 + 16)
A0 = ± −
cj = ±
s
s
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
× coth2 x + 3 a2 k2 2 − 4 b2 k2 3 t s
ei(−kx+wt+ζ)
(144)
ei(−kx+wt+ζ)
(145)
ei(−kx+wt+ζ)
(146)
ei(−kx+wt+ζ)
(147)
s 30 k 3 aj − 90 kaj + 30 w 30 k 3 aj − 90 kaj + 30 w A0 = ± − , A2 = ± − 4 2 (dj + ej ) (k − 6 k + 1) (dj + ej ) (k 4 − 6 k 2 + 1)
26
A1 = 0, B1 = 0, B2 = ±
cj = ±
s
−
k 3 aj − 3 kaj + w 30 (k 3 aj − 3 kaj + w) , b = j (dj + ej ) (k 4 − 6 k 2 + 1) k4 − 6 k2 + 1 2
3 (dj + ej ) (3 k 5 aj − 5 k 3 aj + 6 k 2 w + 12 kaj − 5 w) . 40 (k 4 − 6 k 2 + 1) ( k 3 aj − 3 kaj + w)
(148)
If we put Eq.(148) along with Eq.(114) in Eq.(142), one can get combo dark-bright solitons
ro
of
u(x, t) =
s s 30 k 3 a1 − 3ka1 + w 30 k 3 a1 − 3ka1 + w ± ± − (d1 + e1 ) (k 4 − 6 k 2 + 1) (d1 + e1 ) (k 4 − 6 k 2 + 1) ×sech x + 3 a1 k1 2 − 4 b1 k1 3 t tanh x + 3 a1 k1 2 − 4 b1 k1 3 t s 3 ± − 30 k a1 − 3ka1 + w tanh2 x + 3 a k 2 − 4 b k 3 t 1 1 1 1 (d1 + e1 ) (k 4 − 6 k 2 + 1)
-p
×ei(−kx+wt+ζ)
re
=
ur na
lP
v(x, t)
s s 3 30 k 3 a2 − 3ka2 + w 30 k a − 3ka + w 2 2 ± − ± (d2 + e2 ) (k 4 − 6 k 2 + 1) (d2 + e2 ) (k 4 − 6 k 2 + 1) ×sech x + 3 a2 k2 2 − 4 b2 k2 3 t tanh x + 3 a2 k2 2 − 4 b2 k2 3 t s 3 ± − 30 k a2 − 3ka2 + w tanh2 x + 3 a k 2 − 4 b k 3 t 2 2 2 2 (d2 + e2 ) (k 4 − 6 k 2 + 1)
(149)
×ei(−kx+wt+ζ)
(150)
Similarly, combo singular solitons are obtained as s s 3 30 k a − 3ka + w 30 k 3 a1 − 3ka1 + w 1 1 ± − ± − (d1 + e1 ) (k 4 − 6 k 2 + 1) (d1 + e1 ) (k 4 − 6 k 2 + 1) = ×csch x + 3 a1 k1 2 − 4 b1 k1 3 t coth x + 3 a1 k1 2 − 4 b1 k1 3 t s 30 k 3 a1 − 3ka1 + w coth2 x + 3 a1 k1 2 − 4 b1 k1 3 t ± − (d1 + e1 ) (k 4 − 6 k 2 + 1)
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
u(x, t)
×ei(−kx+wt+ζ)
(151)
27
v(x, t)
s s 30 k 3 a2 − 3ka2 + w 30 k 3 a2 − 3ka2 + w ± − ± − (d2 + e2 ) (k 4 − 6 k 2 + 1) (d2 + e2 ) (k 4 − 6 k 2 + 1) = ×csch x + 3 a2 k2 2 − 4 b2 k2 3 t coth x + 3 a2 k2 2 − 4 b2 k2 3 t s 3 ± − 30 k a2 − 3ka2 + w coth2 x + 3 a k 2 − 4 b k 3 t 2 2 2 2 (d2 + e2 ) (k 4 − 6 k 2 + 1) ×ei(−kx+wt+ζ)
(152)
NON-LOCAL LAW
Eq.(112) can be given as
which comes from Eq.(55) in consequence of using balancing rule.
(153)
ro
Uj (ϑ) = B1 sin(Vj (ϑ)) + A1 cos(Vj (ϑ)) + A0
of
4.4
-p
The strategic equations are easily obtained as long as Eq.(153) along with Eq.(113) is put in Eq.(55). Thus, the important results are as Result-1: s
re
3 kj 2 + 4 aj kj 3 6aj kj , wj = − − 3 kj 2 + 2 (cj + dj ) 2 3 kj 2 + 2
lP
A1 = ±
A0 = 0, B1 = 0, bj =
3aj kj . 2 3 kj 2 + 2
(154)
ur na
If we put Eq.(154) along with Eq.(114) in Eq.(153), one can get dark solitons
u(x, t) =
±
s
−
3 k1
2
×e
v(x, t)
=
i−k1 x−
s
± −
3 k2
×e
i−k2 x−
6a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t + 2 (c1 + d1 )
3 k1 2 + 4 a1 k1 3 t+ζ1 2 3 k1 2 + 2
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
2
(155)
6a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t + 2 (c2 + d2 ) 3 k2 2 + 4 a2 k2 3 t+ζ2 2 3 k2 2 + 2
Similarly, singular solitons are obtained as
28
(156)
=
± −
6a1 k1 coth x + 3 a1 k1 2 − 4 b1 k1 3 t + 2 (c1 + d1 )
2
3 k1
3 k1 2 + 4 a1 k1 3 t+ζ1 2 3 k1 2 + 2
×e
v(x, t)
= ±
i−k1 x−
s
−
3 k2
6a2 k2 coth x + 3 a2 k2 2 − 4 b2 k2 3 t + 2 (c2 + d2 )
2
3 k2 2 + 4 a2 k2 3 t+ζ2 2 3 k2 2 + 2
×e
(157)
i−k2 x−
(158)
of
u(x, t)
Result-2:
3 kj 4 − 2 kj 2 + 3 aj kj 6aj kj , wj = − 3 kj 2 − 1 (cj + dj ) 2 3 kj 2 − 1
ro
s
A0 = 0, A1 = 0,
bj =
3aj kj . 2 3 kj 2 − 1
-p
B1 = ±
(159)
±
6a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t − 1 (c1 + d1 )
i−k1 x−
3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
(160)
ur na
×e
3 k1
2
lP
u(x, t) =
s
re
If we put Eq.(159) along with Eq.(114) in Eq.(153), one can get bright solitons
u(x, t) =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
s
±
s
3 k2
2
6a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t − 1 (c2 + d2 )
×e
i−k2 x−
3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
(161)
Similarly, singular solitons are obtained as
u(x, t) =
s
± −
×e
3 k1
i−k1 x−
2
6a1 k1 csch x + 3 a1 k1 2 − 4 b1 k1 3 t − 1 (c1 + d1 ) 3 k1 4 − 2 k1 2 + 3 a1 k1 t+ζ1 2 3 k1 2 − 1
29
(162)
s
u(x, t) =
± −
3 k2
×e
i−k2 x−
Result-3:
2
6a2 k2 csch x + 3 a2 k2 2 − 4 b2 k2 3 t − 1 (c2 + d2 ) 3 k2 4 − 2 k2 2 + 3 a2 k2 t+ζ2 2 3 k2 2 − 1
s
3aj kj A1 = ± − , B1 = ± 6 kj 2 + 1 (cj + dj )
s
(163)
3aj kj , 6 kj 2 + 1 (cj + dj )
3 kj 2 + 1 aj kj 3 3aj kj A0 = 0, bj = , wj = − . 6 kj 2 + 1 6 kj 2 + 1
of
(164)
If we put Eq.(164) along with Eq.(114) in Eq.(153), one can get combo dark-bright solitons
3 k1 2 + 1 a1 k1 3 6 k1 2 + 1
t+ζ1
s 3a2 k2 tanh x + 3 a2 k2 2 − 4 b2 k2 3 t ± − 2 6 k2 + 1 (c2 + d2 )
ur na =
s 3a2 k2 sech x + 3 a2 k2 2 − 4 b2 k2 3 t 2 ± 6 k2 + 1 (c2 + d2 )
×e
i−k2 x−
3 k2 2 + 1 a2 k2 3 6 k2 2 + 1
Similarly, combo singular solitons are obtained as
30
lP
×e
i−k1 x−
re
s 3a1 k1 sech x + 3 a1 k1 2 − 4 b1 k1 3 t 2 ± 6 k1 + 1 (c1 + d1 )
v(x, t)
ro
s 3a1 k1 tanh x + 3 a1 k1 2 − 4 b1 k1 3 t ± − 2 6 k1 + 1 (c1 + d1 )
-p
u(x, t) =
Jo
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(165)
t+ζ2
(166)
u(x, t)
=
s 3a1 k1 3 2 ± − 6 k 2 + 1 (c + d ) csch x + 3 a1 k1 − 4 b1 k1 t 1 1 1
×e
=
3 k1 2 + 1 a1 k1 3 6 k1 2 + 1
t+ζ1
(167)
s 3a2 k2 ± − coth x + 3 a2 k2 2 − 4 b2 k2 3 t 2 6 k2 + 1 (c2 + d2 )
×e
i−k2 x−
3 k2 2 + 1 a2 k2 3 6 k2 2 + 1
t+ζ2
(168)
re
CONCLUSIONS
-p
ro
s 3a2 k2 3 2 ± − 6 k 2 + 1 (c + d ) csch x + 3 a2 k2 − 4 b2 k2 t 2 2 2
5
of
v(x, t)
i−k1 x−
ur na
lP
This paper, for the first time, introduced CQ-NLSEs in birefringent fibers with Kerr, parabolic, quadratic-cubic and non-local nonlinear laws. The three integration schemes implemented in this paper together revealed bright, dark and singular optical solitons to the models. The existence criteria for these solitons are also enumerated. Thus, the results laid the foundation for further future activities with the models. Later, these models will be extended to study soliton dynamics for DWDM networks as the fundamental governing models. Furthermore, studies with fractional temporal evolution in birefringent fibers for the models will also be considered. Later, Langevin equation will be derived for stochastic perturbation of optical solitons in birefringent fibers. These studies and many more advanced research activities will be conducted in future and the results will be gradually disseminated. Those knowledge–hungry readers must stay tuned... ACKNOWLEDGEMENTS
The research work of the fifth author (MRB) was supported by the grant NPRP 8-028-1-001 from QNRF and he is thankful for it. The authors also declare that there is no conflict of interest.
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s 3a1 k1 ± − coth x + 3 a1 k1 2 − 4 b1 k1 3 t 2 6 k1 + 1 (c1 + d1 )
31
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