ISSN 1541-308X, Physics of Wave Phenomena, 2020, Vol. 28, No. 3, pp. 299–304. © Allerton Press, Inc., 2020.
PHYSICS OF 1D AND 2D MEDIA
Optical Solitons with Kudryashov’s Equation
by Lie Symmetry Analysis1
S. Kumara, S. Malika, A. Biswasb, c, d, e, *, Q. Zhouf, L. Morarug, A. K. Alzahranic, and M. R. Belich
aDepartment of Mathematics and Statistics Central University of Punjab, Punjab, Bathinda-151001 India
bDepartment of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL 35762-7500 USA
c
Department of Mathematics, King Abdulaziz University, Jeddah, 21589 Saudi Arabia
dDepartment of Applied Mathematics, National Research Nuclear University, Moscow, 115409 Russia
e
Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, 0008 South Africa
School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan, 430212 People’s Republic of China
g
Faculty of Sciences and Environment, Department of Chemistry, Physics and Environment,
“Dunarea de Jos” University of Galati, Galati, 800008 Romania
hScience Program, Texas A&M University at Qatar, Doha, PO Box 23874 Qatar
*e-mail: biswas.anjan@gmail.com
f
Received December 25, 2019; revised January 21, 2020; accepted January 21, 2020
Abstract—In this work, Kudryashov’s equation is studied with Lie symmetry analysis, which is implemented
to describe the propagation pulses in an optical fiber. The equation is converted into system of ordinary differential equations with similarity transformations. These gave way to bright, dark and singular optical soliton
solutions to the model.
DOI: 10.3103/S1541308X20030127
1. INTRODUCTION
Optical solitons are aggressively pursued with several models that have existed since the past few
decades [1–17]. There are several equations that have
been lately proposed to address the dynamics of soliton propagation through a variety of waveguides such
as optical fibers, crystals, metamaterials, PCF and
magneto-optic waveguides. The most familiar model
is the nonlinear Schrödinger’s equation [2, 7, 11, 15].
A few other models, governing soliton dynamics, are
Biswas–Arshed equation [8, 16], Triki–Biswas equation
[1, 10], and the Radhakrishan–Kundu–Laksmanan
equation [13] that come with different forms of nonlinear
refractive index. Recently, N. Kudryashov proposed a
law of refractive index that led to Kudryashov’s equation
(KE) [9]. KE is used to describe the propagation pulses in
an optical fiber. The solutions to KE have been recovered
with extended trial function [4], F-expansion [5] and
undetermined coefficients [6]. The current paper handles KE by Lie symmetry analysis.
Here, first we generate infinitesimals and Lie symmetries of KE. Then two vector fields are obtained.
With the help of these vector fields, the governing
equation is reduced into system of ordinary differential
equations (ODEs). Then exact optical soliton solu1 The text was submitted by the authors in English.
tions of system of ODEs are recovered. Finally, the
corresponding bright, dark and singular optical soliton
solutions of governing equation are presented.
1.1. Governing Model
The dimensionless form of KE is given by [4–6, 9]
(
iqt + aq xx + b q
2n
+cq +g q
n
−n
+hq
−2 n
) q = 0. (1)
In Eq. (1), the first term indicates the linear temporal evolution and a indicates the coefficients of
chromatic dispersion. The remaining terms are nonlinear and stem from the law of refractive index of an optical
fiber and gives self-phase modulation effect to the model.
At n = 1; by taking g = h = 0 and c = g = h = 0, the model
referred as parabolic law and Kerr law respectively,
which has been extensively studied. Other special cases
are power law and dual-power laws of nonlinearity
that has also been extensively studied. The goal of the
current paper will address Eq. (1), as it stands, extensively and exhaustively by Lie symmetry analysis.
2. LIE SYMMETRY ANALYSIS
In this section, we will employ Lie classical method
[18–20] on Eq. (1) in order to obtain the infinitesimals. To achieve this goal, let us consider
299
300
KUMAR et al.
q( x, t ) = u( x, t ) exp [i v ( x, t )] ,
(2)
where u and v are real-valued functions. Equation (2)
transforms Eq. (1) into imaginary and real portions as
ut + 2aux v x + auv x, x = 0, aux, x − uvt − auv x2
+ (bu2n + cu n + gu −n + hu −2n )u = 0.
(3)
For the system of equations (3), let us consider
one-parameter (ε) transformations as
x* = x + εξ ( x, t, u, v ) + O(ε2 ),
t* = t + ετ ( x, t, u, v ) + O(ε ),
2
(4)
u* = u + εη ( x, t, u, v ) + O(ε2 ),
v * = v + εφ ( x, t, u, v ) + O(ε2 ),
0 = aη − 2auv x φ − uφ −
xx
x
t
(
where ξ, τ, η, and φ are infinitesimals, depending
upon x, t, u, v, have to be determined.
The vector field associated with these transformations is
(5)
V = ξ∂ x + τ∂t + η∂u + φ∂ v .
For system of equations (3), the second prolongations formula [19, 20] are
pr (2)V = V + ηx ∂ + φ x ∂ + ηt ∂ + φ xx ∂ ,
∂u x
∂v x
∂ut
∂v xx
(6)
pr (2)V = V + φ x ∂ + φt ∂ + ηxx ∂ ,
∂v x
∂v t
∂uxx
x
t
x
t
xx
xx
where η , η , φ , φ , φ , and η are extended infinitesimals.
From the invariance conditions pr(2)V(Δ) = 0
whenever Δ = 0 in(3), we have
(
)
0 = aηv xx + auφ xx + 2a ηx v x + φ x ux + ηt ,
av x2 + vt
(7)
) η + (2n + 1) bu + ( n + 1) cu − ( n − 1) gu − (2n − 1) hu  η.
2n
Substituting the values of infinitesimals ηx, ηt, φx,
t, φxx, and ηxx, and by equating the coefficient of var-
−n
n
−2 n
3.1. Case (i): V4
φ
ious derivative terms equal to zero, we get the system
of PDEs. By solving this system, we get
By solving characteristic Eq. (11), we have following similarity variables:
ξ = C2 + tC1, τ = C1, η = 0, φ = C3 + x C4, (8)
2a
where C1, C2, C3, and C4 are arbitrary constants.
Hence Lie algebra of system (3) is spanned by the
infinitesimal generators as
(12)
s = t, u( x, t ) = P (s), v ( x, t ) = x + Q(s),
4at
where P and Q are new dependent variables, which
depends upon s.
Using (12) in (3), we have
V1 = ∂ , V2 = ∂ , V3 = ∂ , V4 = t ∂ + x ∂ . (9)
∂t
∂x
∂v
∂x 2a ∂v
The commutation relations determined of V1, V2,
V3, and V4 are given by
[V1,V2 ] = 0, [V1,V3 ] = 0, [V1,V4 ] = V2,
[V2,V3 ] = 0, [V2,V4 ] = V3 , [V1,V4 ] = 0.
2a
(10)
2
P ' + 1 P = 0,
2s
(
(13)
)
Q ' − bP 2n + cP n + gP −n + hP −2n = 0,
(14)
where prime represents derivative w.r.t. s. From (13),
we have
P ( s ) = k1
(15)
s,
3. SYMMETRY REDUCTION
AND INVARIANT SOLUTIONS
To derive invariant solutions of system (3), we have
to solve the corresponding characteristic equation
given as
where k1 is arbitrary constant.
By substituting (15) into (14) and solving, we have
following cases:
(a) When n = –2
dx = dt = du = dv ,
(11)
ξ
τ
η
φ
where ξ, τ, η, and φ are presented by (8). To solve
characteristic Eq. (11), we will consider two cases of
vector fields: (i) V4 and (ii) V3 + βV2 + μV1, where β and
μ are arbitrary non-zero real numbers.
(16)
4
3
2
hk
Q ( s ) = 1 bs4 + 1 cs2 + gk12 ln s − 1 + k2.
3 k1 2 k1
s
(b) When n = –1
3 2
2
Q ( s ) = 1 bs2 + 2 cs + 2 gk1 s + hk12 ln s + k3. (17)
2 k1 3 k1
PHYSICS OF WAVE PHENOMENA
Vol. 28
No. 3
2020
OPTICAL SOLITONS WITH KUDRYASHOV’S EQUATION
(a) When n = –2, Eq. (26) can be written as
(c) When n = 1
Q (s)
2
= bk1 ln s + 2ck1
3 2
2
4aμ − β2
1 k7 M −3
+
M" = 1
M
4 a2μ4
4 a2β2μ4
− 1 2 bM −3 + cM −1 + gM 3 + hM 5 .
aμ
Integrating (27), we have
2
gs
s+2
+ 1 hs2 + k4. (18)
3 k1
2 k1
(
(d) When n = 2
Q (s) = − −
2
3
bk14
gs
2
+ ck1 ln s + 1 2 + 1 hs4 + k5. (19)
s
2 k1 3 k1
(e) When n ≠ –2, –1, 1, and 2
bk12n 1− n 2ck1n 1− n /2
s −
s
n −1
n−2
(20)
2 gk1−n 1+ n /2 hk1−2n 1+ n
s
s + k6,
+
+
n+2
n +1
where k2, k3, k4, k5, and k6, are arbitrary constants.
Hence, corresponding solution of (1) is given by
Q (s) = −
q ( x, t ) =
2
(
aμ
aμ
2
 k

+ k9 A −  2 72 4 − 4b2  .
 a β μ aμ 
Equation (29) can be written as
4aμ − β
2g
α1 = − 4h 2 , α2 = − 2 , α3 =
,
3aμ
aμ
a2μ4
 k2

α4 = k9, α5 = −  2 72 4 − 4b2  .
 a β μ aμ 
(b) When n = –1, Eq. (26) can be written as
2
Substituting (22) in (3), we have
2
4aμ − β2
1 k7 M −3
M" = 1
M
+
4 a2μ4
4 a2β2μ4
2
3
−1
− 1 2 bM + c + gM + hM ,
aμ
Integrating (32), we have
(2aμ − β + 2aβμ N ') M ' + aβμ MN " = 0, (23)
aβ μ M " + (β − 2aμβ) N ' − a − aβ μ N '  M
(24)
+ β ( bM + cM + gM + hM ) M = 0,
2n
n
2
−n
(
2
−2 n
N =
k7
β − 2aμ
1 dρ + k ,
ρ+
8
2
2
2
2aβμ
2aβμ M

2
4aμ − β2
1 k7 M −3
+
M" = 1
M
4 a2μ4
4 a2β2μ4
1+ 2n
1+ n
1− n
1− 2n
.
− 1 2 bM
+ cM + gM + hM
aμ
)
Vol. 28
)
k72
−2
M
2 4
2 2 4
2
8 aμ
8a β μ
k
g
− 1 2 b ln M + c + M 3 + h M 4 + 10 ,
3
4
2
aμ
(
)
(33)
where k10 is arbitrary constant.
Assuming M(ρ) = A(ρ), k7 = 0 and b = 0; Eq. (33)
transform into Eq. (30), where
Substituting (25) into (24), we have
PHYSICS OF WAVE PHENOMENA
(32)
2
(25)
where k7 and k8 are arbitrary constants.
(
(31)
( M ') = 1 4aμ − β2 M 2 − 1
where prime represents derivative w.r.t. ρ. Double
integral of (23) gives
2
(30)
where
where M and N are new dependent variables, which
depends upon ρ.
2
(29)
( A ')2 = α1 A 4 + α2 A3 + α3 A 2 + α4 A + α5,
ρ = μx − β t, u( x, t ) = M (ρ), v ( x, t ) = x + N (ρ), (22)
β
2 2
2
2
(21)
By solving characteristic Eq. (11), we have following similarity variables:
3
)
( A') = − 4h2 A4 − 2 g2 A3 + 4aμ2 −4β A2
3.2. Case (ii): V3 + βV2 + μV1
2 2
)
3aμ
k1
  2

exp i  x + Q  ,
t
  4at

2
(27)
(M ')
k2
4aμ − β2 2
−2
= 1 2 4 M − 1 2 72 4 M
2
8 aμ
8a β μ
(28)
k9
g 4 h 6
−2
b
1
− 2 − M + c ln M + M + M + ,
2
4
6
8
aμ
where k9 is arbitrary constant.
Assuming M2(ρ) = A(ρ) and c = 0; Eq. (28) transform into
where Q is given by (16)–(20) and s is given by (12).
2
301
4aμ − β
2g
, α3 =
α1 = − h 2 , α2 = −
,
2
2aμ
3aμ
4a2μ4
α4 = − 2c2 , α5 = k10.
aμ
2
(26)
No. 3
2020
(34)
302
KUMAR et al.
Let us assume M(ρ) = A1/n(ρ), we have
(c) When n = 1, Eq. (26) can be written as
2
4aμ − β2
1 k7 M −3
+
M" = 1
M
4 a2μ4
4 a2β2μ4
−1
3
2
− 1 2 bM + cM + g + hM .
aμ
Integrating (35), we have
(
(35)
)
( M ') = 1 4aμ − β2 M 2 − 1
2
2
k7
−2
M
2 4
2 2 4
2
8 aμ
8a β μ
(36)
k11
3
4
b
c
1
− 2 ln M + M + gM + hM + ,
3
2
aμ 4
where k11 is arbitrary constant.
Assuming M(ρ) = A(ρ), k7 = 0 and h = 0; Eq. (36)
transform into Eq. (30),where
(
)
4aμ − β
α1 = − b 2 , α2 = − 2c 2 , α3 =
,
2aμ
3aμ
4a2μ4
2g
α4 = − 2 , α5 = k11.
aμ
(d) When n = 2, Eq. (26) can be written as
2
2
4aμ − β2
1 k7 M −3
+
M" = 1
M
4 a2μ4
4 a2β2μ4
− 1 2 bM 5 + cM 3 + gM −1 + hM −3 .
aμ
Integrating (38), we have
(
(38)
)
( M ') = 1 4aμ − β2 M 2 − 1
2
(37)
2
k7
−2
M
2
8 a2μ4
8 a2β2μ4
(39)
k12
6
4
−2
1
b
c
h
− 2 M + M + g ln M − M
+
,
4
2
8
aμ 6
where k12 is arbitrary constant.
Assuming M2(ρ) = A(ρ) and g = 0; Eq. (39) transform into Eq. (30), where
(
)
4aμ − β
α1 = − 4b 2 , α2 = − 2c2 , α3 =
,
aμ
a2μ4
3aμ
(40)
k72
h
4
α4 = k12, α5 = 2 − 2 2 4 .
aμ a β μ
(e) In the case of n ≠ –2, –1, 1, 2, by integrating
Eq. (26), we have
2
b M
(
aμ 2n + 2
( M ')2 = 1 4aμ − β2 M 2 − 1
2
8 aμ
2 4
2n + 2
2
c M n + 2 + g M 2 − n + h M 2 − 2n
n+2
2−n
2 − 2n
2
k
k
− 1 2 72 4 M −2 + 13 ,
8a β μ
2
where k13 is arbitrary constant.
+
) (41)
(
2
1 A 2/ n A ' 2 = 1 4aμ − β A 2/ n − 1
b A 2/ n+ 2
( )
2
2 4
2
8 aμ
2n
aμ 2n + 2
g
/
2/
2/ n − 2
2
n
n
+
−
1
1
(42)
+ c A
+
A
+ h A
n+2
2−n
2 − 2n
k
k2
− 1 2 72 4 A −2/ n + 13 .
8a β μ
2
)
Multiplying Eq. (42) by n2A2 – 2/n and taking k7 = 0
and k13 = 0, we have
2
bn2
A 4 − 22cn
A3
aμ (n + 1)
aμ (n + 2)
(43)
2 2
2
2
(4aμ − β )n 2
2 gn
hn
.
+
A + 2
A+ 2
4a2μ4
aμ (n − 2)
aμ (n − 1)
Equation (43) is similar to Eq. (30), where
( A')2 = −
2
2
2
bn
, α2 = − 22cn
,
aμ (n + 1)
aμ (n + 2)
2 2
(4aμ − β )n
(44)
,
α3 =
4a2μ4
2
2 gn2
, α5 = 2hn
.
α4 = 2
aμ (n − 2)
aμ (n − 1)
Relation between parameters a, b, c, g, h, and n
of (1) and parameters α1, α2, α3, α4, and α5 of (30) in
order to derive the solutions M(ρ) of (26) is given by
following Table 1.
α1 = −
2
3.3. Soliton Solutions to Equation (30)
To derive the general solution of the equation
(45)
( A′)2 = α1 A 4 + α2 A3 + α3 A 2 + α4 A + α5,
one can use substituting methods [11, 12, 16, 17]. In
[9], the author expressed the general solution of
Eq. (45) in the form of Weierstrass and Jacobi elliptic
function. Here we obtained the bright, dark and singular soliton solutions of Eq. (45).
3.3.1. Bright solitons. Let us assume
α2(4α3α1 − α22 )
,
α12
α2(16α3α1 − 5α22 )
α5 = 1 2
.
256
α13
Then the solution of Eq. (45) is given by
α4 = 1
8
α
A (ρ) = − 1 2 ± 1 6α22 − 16α3α1
4 α1 4α1
 2(8α3α1 − 3α22 )

× sech 1
ρ ± k14  ,
α1
4

PHYSICS OF WAVE PHENOMENA
Vol. 28
No. 3
(46)
2020
OPTICAL SOLITONS WITH KUDRYASHOV’S EQUATION
303
Table 1. Relation between parameters of Eqs. (1) and (30) in order to derive the solutions M(ρ) of (26)
α1
Parameters
α2
α3
α4
α5
M(ρ)
2g
2
aμ
4aμ − β
2 4
aμ
k9
 k2

−  2 72 4 − 4b2 
aμ 
a β μ
A(ρ)
2
n = –2,
c=0
− 4h 2
3aμ
−
n = –1,
k7 = b = 0
− h2
2aμ
−
2g
2
3aμ
4aμ − β
2 4
4a μ
− 2c2
aμ
k10
A(ρ)
n = 1,
k7 = h = 0
− b2
2aμ
− 2c 2
3aμ
4aμ − β
2 4
4a μ
−
2g
2
aμ
k11
A(ρ)
n = 2,
g=0
− 4b 2
3aμ
− 2c2
aμ
4aμ − β
2 4
aμ
k12
4h − k7
2
2 2 4
aμ
aβμ
A(ρ)
2cn2
2
aμ (n + 2)
(4aμ − β )n
2 4
4a μ
hn2
2
aμ (n − 1)
A1/n(ρ)
n ≠ –2, –1, 1, 2,
k7 = k13 = 0
−
bn2
2
aμ (n + 1)
−
2
2
2
2
2
where k14 is arbitrary constant. Hence corresponding
bright soliton solution of Eq. (1) is given by
q ( x, t )
2
 

β − 2aμ 1
= M (ρ) exp i  x +
d ρ + k8  ,
2
2
M
  β 2αβμ

(47)

where ρ = μx – βt and the values of M depending upon
various parameters and the function A as discussed in
Table 1 and A is given by (46).
3.3.2. Dark solitons. Let us assume
2
2
2 gn
2
aμ (n − 2)
Then the solution of Eq. (45) is given by
α
A (ρ) = − 1 2 ± 1 16α3α1 − 6α22
4 α1 4α1
(50)
 2 8α3α1 − 3α22

× csch 1
ρ ± k16  ,
4

α1


where k16 is arbitrary constant. Hence corresponding
bright soliton solution of Eq. (1) is given by
(
)
q ( x, t ) = M (ρ)
 
k7
β − 2aμ
1 d ρ + k  , (51)
× exp i  x +
ρ+
8 
2
2
2αβμ M 2
  β 2αβμ

where ρ = μx – βt and the values of M depending upon
various parameters and the function A as discussed in
Table 1 and A is given by (50).
The constraint conditions or existence criteria for
these solitons are given as
2
α (4α α − α2 )
α (4α α − α2 )
α4 = 1 2 3 21
, α5 = 1 2 3 31
.
8
64
α1
α1
2
2
2
Then the solution of Eq. (45) is given by
α
A (ρ) = − 1 2 ± 1 3α22 − 8α3α1
4 α1 4α1
1 3α22 − 8α3α1

× tanh 
ρ ± k15  ,
α1
4

(48)
(
 
k7
β2 − 2aμ
1 d ρ + k  , (49)
× exp i  x +
ρ+
8 
2
2
2
2αβμ M
  β 2αβμ


where ρ = μx – βt and the values of M depending upon
various parameters and the function A as discussed in
Table 1 and A is given by (48).
3.3.3. Singular solitons. Let us assume
2
2
α2(4α3α1 − α22 )
1 α2(16α3α1 − 5α2 ) .
,
α
=
5
256
α12
α13
Vol. 28
No. 3
(52)
for bright and singular solitons, while for dark solitons
one needs to have
(
)
α1 8α3α1 − 3α22 < 0.
q ( x, t ) = M (ρ)
PHYSICS OF WAVE PHENOMENA
)
α1 8α3α1 − 3α22 > 0,
where k15 is arbitrary constant. Hence corresponding
bright soliton solution of Eq. (1) is given by
α4 = 1
8

(53)
4. CONCLUSIONS
This paper studied KE to secure bright, dark and
singular soliton solutions to the model. The existence
criteria for such solitons are also enumerated. These
constraints guarantee the formation of such solitons.
The results are thus strongly encouraging to pursue
this avenue of research, further along. Later on, the
model will be extended to study soliton dynamics with
KE in birefringent fibers and DWDM topology. Additional optoelectronic devices are also going to be
2020
304
KUMAR et al.
touched base upon. These would be magneto-optic
waveguides, optical metamaterials, optical couplers
and several such. Those results are currently awaited
and they would be reported with time.
FUNDING
Support of CSIR Research Grant 09/1051(0028)/2018EMR-I to one of the authors (S. Malik) for carrying out the
research work is fully acknowledged. The research work of
fourth author (QZ) was supported by the National Natural
Science Foundation of China (Grant nos. 11705130 and
1157149); this author was also sponsored by the Chutian
Scholar Program of Hubei Government in China. The
research work of the seventh author (MRB) was supported
by the grant NPRP 11S-1126-170033 from QNRF and he is
thankful for it.
CONFLICT OF INTEREST
The authors also declare that there is no conflict of
interest.
REFERENCES
1. S. Khan, “Stochastic perturbation of sub-pico second
envelope solitons for Triki–Biswas equation with multiphoton absorption and bandpass filters,” Optik. 183,
174–178 (2019).
https://doi.org/10.1016/j.ijleo.2019.02.065
2. G. E. Astrakharchik and B. A. Malomed, “Dynamics
of one-dimensional quantum droplets,” Phys. Rev. A.
98 (1), 013631 (2018).
https://doi.org/10.1103/PhysRevA.98.013631
3. J. Atai and B. A. Malomed, “Gap solitons in Bragg
gratings with dispersive reflectivity,” Phys. Lett. A. 342
(5-6), 404–412 (2005).
https://doi.org/10.1016/j.physleta.2005.05.081
4. A. Biswas, M. Ekici, A. Sonmezoglu, A. S. Alshomrani,
and M. R. Belic, “Optical solitons with Kudryashov’s
equation by extended trial function,” Optik. 202,
163290 (2020).
https://doi.org/10.1016/j.ijleo.2019.163290
5. A. Biswas, A. Sonmezoglu, M. Ekici, A. S. Alshomrani,
and M. R. Belic, “Optical solitons with Kudryashov’s
equation by F-expansion,” Optik. 199, 163338 (2019).
https://doi.org/10.1016/j.ijleo.2019.163338
6. A. Biswas, J. Vega-Guzmán, M. Ekici, Q. Zhou,
H. Triki, A. S. Alshomrani, and M. R. Belic, “Optical
solitons and conservation laws of Kudryashov’s equation using undetermined coefficients,” Optik. 202,
163417 (2020).
https://doi.org/10.1016/j.ijleo.2019.163417
7. K. W. Chow, B. A. Malomed, and K. Nakkeeran, “Exact solitary- and periodic-wave modes in coupled equa-
tions with saturable nonlinearity,” Phys. Lett. A.
359 (1), 37–41 (2006).
https://doi.org/10.1016/j.physleta.2006.05.082
8. M. Ekici and A. Sonmezoglu, “Optical solitons with
Biswas–Arshed equation by extended trial function
method,” Optik. 177, 13–20 (2019).
https://doi.org/10.1016/j.ijleo.2018.09.134
9. N. A. Kudryashov, “A generalized model for description of propagation pulses in optical fiber,” Optik. 189,
42–52 (2019).
https://doi.org/10.1016/j.ijleo.2019.05.069
10. N. A. Kudryashov, “First integral and general solution
of traveling wave reduction for the Triki–Biswas equation,” Optik. 181, 338–342 (2019).
https://doi.org/10.1016/j.ijleo.2019.03.087
11. N. A. Kudryashov, “Construction of nonlinear differential equations for description of propagation pulses in
optical fiber,” Optik. 192, 162964 (2019).
https://doi.org/10.1016/j.ijleo.2019.162964
12. N. A. Kudryashov, “On types of nonlinear nonintegrable equations with exact solutions,” Phys. Lett. A.
155 (4-5), 269–275 (1991).
https://doi.org/10.1016/0375-9601(91)90481-M
13. N. A. Kudryashov, D. V. Safonova, and A. Biswas,
“Painlevé analysis and a solution to the traveling wave
reduction of the Radhakrishnan–Kundu–Lakshmanan equation,” Regul. Chaot. Dyn. 24 (6), 607–614
(2019).
https://doi.org/10.1134/S1560354719060029
14. N. K. Vitanov, “Application of simplest equations of
Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial
nonlinearity,” Commun. Nonlin. Sci. Numer. Simul.
15 (8), 2050–2060 (2010).
https://doi.org/10.1016/j.cnsns.2009.08.011
15. Y. Qiu, B. A. Malomed, D. Mihalache, X. Zhu, J. Peng,
and Y. He, “Generation of stable multi-vortex clusters
in a dissipative medium with anti-cubic nonlinearity,”
Phys. Lett. A. 383 (22), 2579–2583 (2019).
https://doi.org/10.1016/j.physleta.2019.05.022
16. Y. Yildirim, “Optical solitons to Biswas–Arshed model
in birefringent fibers using modified simple equation
architecture,” Optik. 182, 1149–1162 (2019).
https://doi.org/10.1016/j.ijleo.2019.02.013
17. J. Zhang, X. Wei, and Y. Lu, “A generalized (G'/G)-expansion method and its applications,” Phys. Lett. A. 372
(20), 3653–3658 (2008).
https://doi.org/10.1016/j.physleta.2008.02.027
18. L. V. Ovsiannikov, Group Analysis of Differential Equations (Academic, New York, 1982).
19. G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations (Springer Science & Business Media, 2008).
20. P. J. Olver, Applications of Lie Groups to Differential
Equations (Springer Science & Business Media, 2000).
PHYSICS OF WAVE PHENOMENA
Vol. 28
No. 3
2020