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15Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 231-241. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220206
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34K99, 35C08
A Study of Different Wave Structures of the (2 + 1)-dimensional Chiral Schrödinger Equation
K. Hosseini, M. Mirzazadeh, K. Dehingia, A. Das, S. Salahshour
In the present paper, the authors are interested in studying a famous nonlinear PDE referred to as the (2 + 1)-dimensional chiral Schrödinger (2D-CS) equation with applications in mathematical physics. In this respect, the real and imaginary portions of the 2D-CS equation are firstly derived through a traveling wave transformation. Different wave structures of the 2D-CS equation, classified as bright and dark solitons, are then retrieved using the modified Kudryashov (MK) method and the symbolic computation package. In the end, the dynamics of soliton solutions is investigated formally by representing a series of 3D-plots.
Keywords: (2 + 1)-dimensional chiral Schrödinger equation, traveling wave transformation, modified Kudryashov method, different wave structures
Received September 17, 2021 Accepted April 16, 2022
Kamyar Hosseini [email protected]
Department of Mathematics, Rasht Branch, Islamic Azad University P.O. Box 41335-3516 Rasht, Iran
Department of Mathematics, Near East University TRNC Mersin 10, Nicosia 99138, Turkey
Mohammad Mirzazadeh [email protected]
Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan
P.C. 44891-63157 Rudsar-Vajargah, Iran
Kaushik Dehingia kaushikdehingia17@gmail. com
Department of Mathematics, Sonari College Sonari 785690, Assam, India
Anusmita Das [email protected]
Department of Mathematics, Gauhati University Guwahati 781014, Assam, India
Soheil Salahshour
Faculty of Engineering and Natural Sciences, Bahcesehir University Istanbul 34349, Turkey
1. Introduction
The search for different wave structures of the (2 + 1)-dimensional chiral Schrodinger equation [1-7]
iut + c1 (uxx + uyy) + i (c2 (uuX — u*ux) + C3 (uu* — u*uy)) u = 0 (1-1)
has achieved much attention in recent years and has been the topic of many studies. The origin of the 2D-CS equation goes back to its 1-dimensional version that was proposed by Nishino et al. [8] as a reduction of a model describing the edge states of the fractional quantum Hall effect [9]. Many scholars have spent their efforts to study the 2D-CS equation. Hosseini and Mirzazadeh [5] applied the Jacobi method to derive solitons and other solutions of the 2D-CS equation. Osman et al. [6] found a group of exact solutions of the 2D-CS equation using the Fan sub-equation method. Rezazadeh et al. [7] employed the extended rational sine-cosine/sinh-cosh methods to obtain traveling wave solutions of the 2D-CS equation. Very recently, Sulaiman and his colleagues [10] considered the 2D-CS equation with variable coefficients and obtained its complex wave solutions through a series of test functions. It is worth mentioning that ut is the evolution term, c1 is the coefficient of dispersion terms, and c2 and c3 are the coefficients of nonlinear terms. Additionally, Eq. (1-1) cannot possess the Painleve test [1] and such a property indicates the significance of investigating its exact solutions.
The present article investigates different wave structures of the 2D-CS equation using the modified Kudryashov method [11-14]. This method utilizes a special finite series solution in the form [15]
Ni K(e) \i-W K (e) ,1 — K2(e)\ , ,
= "o + L (.TTi^)) r + 'TTT^))' or bN *
which is different from that considered in its conventional version [16-19], namely,
N
U (e) = a0 + ^ aiKi(e), aN = 0.
i=1
Such a selection provides other exact solutions of nonlinear PDEs which cannot be derived by the Kudryashov method. To address recent applications of the modified Kudryashov method in finding exact solutions of nonlinear PDEs, Hosseini et al. [11] adopted this method to look for exact solutions of a nonlinear high-order Schrodinger equation. Hosseini et al. [12] also applied this method to seek exact solutions of a 2D nonlinear Schrodinger system. More articles can be found in [20-29].
This paper is organized as follows: Section 2 presents an outline of the modified Kudryashov method along with several useful remarks. In Section 3, the real and imaginary portions of the 2D-CS equation are derived and then different wave structures of the 2D-CS equation are retrieved using the modified Kudryashov method. Besides, the dynamics of soliton solutions is investigated by representing a series of 3D-plots. Finally, Section 4 gives a review of the results.
2. Modified Kudryashov method
This section aims to present an outline of the modified Kudryashov method to retrieve different wave structures of nonlinear ODEs. To this end, the following nonlinear ODE is considered:
O(U, U', U", ...) = 0, U = U(e), (2.1)
where ' =
The KM method supposes that the solution of Eq. (2.1) can be represented as K(e) \i-W K(e) ,1 - K2(e)\ 1 /
^)=«o + E(vTT^yJ ^TrMo+6<TT^yJ' -NortN^0, (2.2)
where a0, ai, i = 1, 2, ..., N, and bi, i = 1, 2, ..., N are unknowns, N is established by the balance approach, and K(e) is a function of the form
A (e) = {A - B)sinh(e) + {A + B)cosh(e)' (2'3)
which solves the following Jacobi equation:
(K'(e))2 = K2(e) (1 - 4ABK2(e)).
By plugging Eqs. (2.2) and (2.3) into Eq. (2.1) and reorganizing the terms, we will reach a nonlinear algebraic set whose solution leads to different wave structures of Eq. (2.1).
Remark 1. When the balance number is N =1 and b1 = a2, Eq. (2.2) can be written as
TT. . K(e) 1 - K2(e)
U(e) = a0 + a, l + R2{e) + «2 l + K2{ey °i or °2 ^ 0,
and so
(7(e) =-+ 1-, or o2 ^ 0. (2.4)
Remark 2. By considering Eq. (2.3), Eq. (2.4) can be written as
U(e) = ((2A2a0 + 2A2a2 - 2B2a0 - 2B2a2) sinh(e) cosh(e) + (Aa1 - Ba1) sinh(e) +
+ (2A2a0 + 2A2a2 + 2B2a0 + 2B2a2) (cosh(e))2 + (Aa1 + Ba1) cosh(e) - A2a0 - A2a2 + +2ABa0 + 2ABa2 - B2a0 - B2a2 + a0 - a2) / ((2A2 - 2B2) cosh(e) sinh(e) +
+ (2A2 +2B2) (cosh(e))2 - A2 + 2AB - B2 + 1), a1 or a2 = 0. (2.5)
Remark 3. By considering
ao = T b, «1=0, a2 = ±b,
from Eq. (2.5), one has
8bB2
Ult2(t) = ±-
' 1'2Vw (8B4 - 2) sinh(e) cosh(e) + (-8B4 - 2) (cosh(e))2 + 4B4 + 1' Remark 4. By considering
B = 0, a0 = 0, a1 = ±b, a2 = 0,
from Eq. (2.5), one has
bA(sinh(e) + cosh(e))
U-Ui) = ±7
~'~2J42 sinh(e) cosh(e) + 2A2(cosh(e))2 — A2 + 1' RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(2), 231 241
Remark 5. By considering
B = 0, a0 = 0, a! = 0, a2 = ±b,
from Eq. (2.5), one has
r b(A4+ 4A2 sinh(e) cosh(e) - 1)
= ± A4 + 4A2(cosh(e))2 — 2A2 + 1 '
Remark 6. By considering
from Eq. (2.5), one has
B = 0, a0 =0, a! = ±b, 0,2 = c,
(A2c + c) sinh(e) + (A2c - c) cosh(e) ± bA Ur>8^' = (A2 - 1) sinh(e) + (A2 + l)cosh(e) '
3. 2D-CS equation and its different wave structures
This section aims to derive different wave structures of the 2D-CS equation through the modified Kudryashov method. For this purpose, the following traveling wave transformation is used:
u(x,y,t)= U (e)ei(K2 x+^2y+^2t), e = x + X1y — /11, (3.1)
where k2 and X2 are frequencies in the x- and y-directions, /2 is the wave frequency, /1 is the wave velocity, and the other parameters are real constants. The traveling wave transformation (3.1) reduces the 2D-CS equation to
(¡¡2 U (e)
ci («1 + A?) ~ + ciAi + ^2) U(e) + 2 (c2k2 + c3A2) Us(e) = 0, (3.2)
where the wave velocity is /1 = 2c1 (k1 k2 + A1A2).
Owing to the terms d ¿¿f* and U3(e), we find the balance number as N = 1. Accordingly, the nontrivial solution of Eq. (3.2) can be expressed as
K (e) 1 — K 2(e) , . .
U(e) = a,0 + ai1 + K2(e) + a'2i + A-2(e)' ai or a2 + °> (3-3)
where a0, a1, and a2 are unknowns. By plugging Eq. (3.3) into Eq. (3.2) and reorganizing the terms, we will reach a nonlinear algebraic set in the form
—2a3 C2K2 — 2a0 C3X2 — 6a0 a2C2 K2 — 6a2 a2C3 X2 — daoa^^ — 6aoa2C3X2 — 2a2 C2K2 —
o OOOO
— 2a2C3X2 + aoCK + aoC1 X2 + a2C1 K2 + a2^X2 + oO/2 + a2^2 = 0, —6a2 a1 c2k2 — 6a2 a1 c3X2 — 12ao a1 a2c2 k2 — 12aoa1a2c3X2 — 6a1 a2c2 k2 —
2 2 2 2 »2
—6a1a2c3X2 — a1c1K1 + a1 c1k.2 — a1c1 X1 + a1c1X2 + a1/2 = 0,
3 3 2 2 2 2 2
—6ao c2K2 — 6ao c3 X2 — 6aoa2c2K2 — 6a0a2c3X2 — 6aoalC2K2 — 6aoa1C3X2 + 6aoa2c2K2+
+6aoa2c3X2 — 6af a2c2 k2 — 6af a2c3 X2 + 6a2c2K2 + 6a2c3 X2 + 3ao c1K2 + 3aoc1 X2+ +8a2 c1kI + a2c1 k2 + 8a2 c1X^ + a2 c1X^ + 3ao/2 + a2 /12 = 0,
8ABa1 c1k21 + 8ABa1 c1X21 - 12a0 12a0 aiC3A2 - 2afC2K2 - 2a1 C3A2 + 12aia2C2K2+
O OOOO
+12a1 a2 c3A2 + 6a1c1 k^ + 2a1 c1k2 + 6a1 c1A1 + 2a1 c^ + 2a1^2 = 0, -48ABa2c1 k2 - 48ABa2c1 A1 - 6aj]c2k2 - 6a0c3A2 + 6a0a2c2k2 + 6a0a2c3A2 - 6a0a2c2K2--6a0a2c3A2 + 6a0a2c2k2 + 6a0a2c3A2 + 6a2a2c2K2 + 6a2a2c3A2 - 6a3c2k2 - 6a:]c3A2+ +3a0c1 k2 + 3a0c1A2 - 8a2c1 k2 - a2c1K2 - 8a2c1A2 - a2c1A2 + 3a0¡2 - a2^2 = 0, -24ABa1 c1k2 - 24ABa1
C1 A2 - 6a0a1C2K2 - 6a1alCзA2 + 12a0a1a2C2K2 + 12a0a1 a2C3A2--6a1 a2c2k2 - 6a1 a2c3A2 - a1c1 k1 + a1c1K2 - a1 c1A2 + a1 c1A2 + a1 ¡2 = 0, 16ABa2c1k1 + 16ABa2c1A1 - 2a3c2k2 - 2a0c3A2 + 6a0a2c2K2 + 6a0a2c3A2 - 6a0a2c2k2-
Q Q Q OOOO
-6a0a2C3A2 + 2a^C2K2 + 2a^A2 + a0C1 K2 + a0C1A2 - a2C1K2 - a2C1 A2 + a0^2 - a2¡2 = 0.
Through employing the Maple software, from the above system, the following cases are generated: Case 1.
1
— Ci KKi — c
-ClA 2
A = ——, fln = =Fi/--—1--
2B' 0 V c2k2 + c3A2
, =0, a2 = ±\ -
c^K2 Ci A2 C2K2 + C3 A2
l2 = 4c1Ki - c2K2 + 4c2Ai - c2A2-Thus, the following soliton solutions to the 2D-CS equation are derived:
U12(x, y, t) =
= ±8< —
ci ci A2
c2K2 + c3 A2
B2 / ((8B4 — 2) sinh(K2x + Aiy — fi2t) cosh(/i1 x + Aiy — iit)+
+ (-8B4 - 2) (cosh(K2x + Aly - |t))2 + 4B4 + 1) t),
where
. \\\ 2 2 j \ 2 \2 ci Ki ci Ai h1 = 2c1{k1k2 +x^), ¿t2 = - c^v, + - c^, -———
c2 K2 + c3 A2
Case 2.
< 0.
5 = 0, ao = 0, a1 = ±2,/-ClK? ClA" ~ .--"-2 « ..2 1 „ \2
c2K2 + c3 A2
-, a2 = 0, n2 = c-iKi - + qAi - CiA2.
Therefore, the following soliton solutions to the 2D-CS equation are obtained:
Usa{x, y, t) = ±2W —
cik2 ci A2
~ci Ki c
c2K2 + c3 A2
A(sinh(^ix + Aiy - iit) +
+ cosh(Kix + A2y - iit))I (2A2 sinh(Kix + Aiy - ¡iit) cosh(/iix + Aiy - iit) +
+2A2(cosh(Kix + A2y - |it))2 - A2 + 1) ^2*+^+^t),
where
. \\\ 2 2 \ 2 \ 2 Ci K-t C1X1
h1=2c1{k1k2 + x^), 1^2 = 0^-0^2 + 0^-^x2, ——-—h-< 0.
C2K2 + C3 X2
Case 3.
C1K1 + C1X2 ,, _ „ (n,„2 \ ,„2 , o \2 1 \2>
5 = 0, a,0 = 0, a, = 0, a2 = ± \ ——-n2 = ~ci
V C2K2 + C3X2
Consequently, the following soliton solutions to the 2D-CS equation are obtained:
\/-c¿SaI + 4A2 sinh(Klx + XlV - ^t) cœh(Klx + XlV - ^t) - 1)
/ \ V 2 2 3 2
u5,6(x, y, t) = ± A4 + 4A2(cosh(Kia. + Xly - ^t))2 - 2A2 + 1 X
x ei{K2x+X2V+^2l) ,
where
c k2 +1- c X2
^ = 2cl{klk2 + X,X2), (jl2 = -Cl (2k? + + 2A? + Ai), 1 1 11 < 0.
C2K2 + C3 X2
Case 4.
5 = 0, a,0 = 0, a1 = ± W---a2
—c1k\ — c1Xf / c1 k2 + c1 X\
c2k2 + c3A2 ' V 4c2 k2 + 4c3A2'
^-v-' v-v-'
b c
»2 = ~\ («1 + 2«a + Xl + 2A!) Cl" Accordingly, the following exact solutions to the 2D-CS equation are achieved:
(.A2c + c) sinh(K1 x + A1y — ¡i1t) + (A2 c — c) cosh(Kix + Aiy — lit) i bA
u7,8V, t) =
where
(A2 — 1) sinh(K1 x + X1v — i1t) + (A2 + 1) cosh(K1x + X1y — i1t)
x ei(K2x+X2V+^2l)
b= 1-0,4-0, Xf
C2K2 + C3X2 C1K^ + c1Xf
4c2k2 + 4c3 X2 ' I1 = 2c1 (K1K2 + X1X2),
»2 = -\ («1 + 2«2 + Xl + 2Ai) Cl-
Remark. It should be noted that, by applying the Kudryashov method, one arrives at a nonlinear algebraic set in the form
2a0c2K2 + 2a0c3A2 — a0c1K2 — a0c1A2 — aoM2 = 0
6a0aiC2K2 + 6a0aiCoA2 + a^i«! — a^iK + a^Ai — aiCiA2 — a^2 = 0, 6aoa2C2«2 + 6aoaic3A2 = 0, —8ABaiciK2i — 8ABaiciA2 + 2afc2K2 + 2a0c3A2 = 0,
where its solution gives
/ -4ABc1k1 - 4ABc-, A? 2 2 ~ ~
on = 0, a1 = ±t------, a«2 =ci«i -c^+c^l -c^o.
y c2K2 + c3 A2
Subsequently, the following soliton solutions to the 2D-CS equation are obtained:
U 0(x, y, t) = ±_-_c2k2 + c3A2_ei(K2X: + X2y+H2t) ^
1,2 ' ' (A — B) sinh(K1x + Xxy — fj,^) + (A + B) cos1i(k1x + Xxy — fj,^) '
where
^ = 2c1(K1K? + AiA2),
2222 M2 = c1K1 - c1K2 + c1A1 - c1 A2 >
< 0.
-4ABc1k21 - 4ABc1A1
c2K2 + c3 A2
-4ABc1Kf-4ABc1Xl
Now, the authors are interested in analyzing the dynamics of soliton solutions derived above by representing a series of 3D-plots. To this end, the first soliton derived using the MK method has been plotted in Figure 1 for B = 1, c1 = 1, c2 = 1, c3 = 1, k1 = 0.5, k2 = —0.5, A1 = 1, A2 = 1, (a) t = 0 and (b) t = 1. Figure 2 represents the fifth soliton obtained through the MK method for A = 1, c1 = 1, c2 = 1, c3 = 1, k1 = —0.25, k2 = —0.25, A1 = 1, A2 = 1, (a) t = 0 and (b) t = 1. The first soliton derived using the Kudryashov method has been portrayed in Figure 3 for A = 2, B = 1, c1 = 1, c2 = 1, c3 = 1, k1 = 0.5, k2 = —0.5, A1 = 1, A2 = 1, (a) t = 0 and (b) t = 1. It should be stated that the first, second, and third figures signify the bright, dark, and bright solitons, respectively. Furthermore, the MK method is capable of retrieving both bright and dark solitons.
As a specific feature, the bright and dark solitons derived using the MK method move in opposite directions. To show such a feature, the following figures are considered.
4. Conclusion
In the present paper, the (2 + 1)-dimensional chiral Schrodinger equation with applications in mathematical physics was considered and explored successfully. First, a traveling wave transformation was adopted to derive the real and imaginary portions of the 2D-CS equation, then the second-order nonlinear ODE in the real domain was solved through the modified Kudryashov method and the symbolic computation package. As an achievement, several bright and dark solitons to the 2D-CS equation were formally extracted. In the end, the dynamics of soliton solutions was examined by establishing a series of 3D-plots.
Fig. 1. The first soliton derived using the MK method for B = 1, ci = 1, c2 = 1, c3 = 1, ki = 0.5, k2 = —0.5, Ai = 1, A2 = 1, (a) t = 0 and (b) t =1
Fig. 2. The fifth soliton obtained through the MK method for A =1, ci = 1, c2 = 1, c3 = 1, ki = —0.25, k2 = —0.25, Ai = 1, A2 = 1, (a) t = 0 and (b) t =1
Fig. 3. The first soliton derived using the Kudryashov method for A = 2, B =1, ci = 1, c2 = 1, c3 = 1, ki = 0.5, k2 = —0.5, Ai = 1, A2 = 1, (a) t = 0 and (b) t =1
X X
(a) (b)
Fig. 4. (a) The first soliton derived using the MK method for B = 1, cx = 1, c2 = 1, c3 = 1, k1 = 0.5, k2 = —0.5, = 1, X2 = 1, and y = 0; (b) The fifth soliton obtained through the MK method for A = 1,
c1 = 1, c2 = 1, c3 = 1, k1 = —0.25, k2 = —0.25, At = 1, A2 = 1, and y = 0
Data availability
The authors declare that this research is purely theoretical and is not associated with any data.
Conflict of interest
The authors declare no conflict of interest.
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