Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov-Kuznetsov model
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Open Physics 2021; 19: 91–99
Research Article
Khaled El-Rashidy, Aly R. Seadawy*, Saad Althobaiti, and M. M. Makhlouf
Investigation of interactional phenomena and
multi wave solutions of the quantum
hydrodynamic Zakharov–Kuznetsov model
https://doi.org/10.1515/phys-2021-0009 these nonlinear equations is a major area of concern
received September 30, 2020; accepted February 09, 2021 for many mathematicians and physicists because of their
Abstract: The symbolic computation with the ansatz func- important role in understanding the behavior of non-
tion and the logarithmic transformation method are used linear physical phenomena. The Zakharov–Kuznetsov
to obtain a formula for certain exact solutions of the (3 + 1) (Z–K) equation supports stable solitary waves, see ref.
Zakharov–Kuznetsov (Z–K) equation. We use homoclinic [8,9]. This makes the Z–K equation a very attractive
breather, three waves method, and double exponential. model equation for investigating of vortices in geophy-
There is a conflict of results with considerably known sical owe. The Z–K equation determines weak non-linear
results, which indicates the solutions found in this study behavior of ion sound waves that contain electrons with
are new. By selecting appropriate parameter values, 3d hot and cold temperatures in a regular magnetic field,
representations are plotted to establish W-shaped, multi- see ref. [9]. With regard to the exact solutions of the
peak, and kinky breathers solutions. Z–K and (3 + 1)-dimensional Z–K equations, many sol-
ving methods have been used to solve it, see [2,9–14],
Keywords: 02.30Xx soliton solutions, (3 + 1) Zakharov– and have many applications in numerical and analytical
Kuznetsov equation, the logarithmic transformation method approaches in physical sciences [15–21]. An efficient ana-
lytical technique for fractional partial differential equa-
tions occurs in ion-acoustic waves in plasma and warm
plasma [37,38].
1 Introduction Some new methods and important developments in
the search for analysis have been done to investigate
Many physical phenomena originate in various fields of wave solutions for partial non-linear differential equa-
engineering and science, such as fluid dynamics, fiber tions. The results of this manuscript may be completely
optics, quantum mechanics, plasma, and physics. These complementary to in existence manuscripts of literature
physical phenomena were constructed in the form of non- such as: direct algebraic method that is extended and
linear equations [1–7]. To understand the behavior of a modified; techniques’ Seadawy and extended mapping
phenomenon, we need to solve nonlinear equations that method to find solutions for some nonlinear partial dif-
describe the phenomenon, which is often challenging. ferential equations [5]; showing the bi-directional propa-
Many methods have been developed over the years to gation of small-amplitude long-capillary gravity waves
solve such nonlinear equations, many of which are based on the surface of shallow water [22]; bright and dark
on assumptions. Investigating the exact solutions for solitons, solitary wave solutions of higher-order non-
linear differential equations and the elliptic function
[23]; the higher-order non-linear differential equations
with cubic quintic nonlinearity [24]; solitary wave
solutions to the nonlinear-modified KdV dynamical equa-
tion [25]; modified equal–width equations and dispersive
* Corresponding author: Aly R. Seadawy, Mathematics Department, traveling wave solutions of the equal–width [26]; the
Faculty of science, Taibah University, Al-Madinah Al-Munawarah,
exact travelling wave solutions of the SRLW equation
Saudi Arabia, e-mail: aly@ujs.edu.cn
Khaled El-Rashidy, Saad Althobaiti, M. M. Makhlouf: Technology
and modified Liouville equation [27]; fourth-order non-
and Science Department, Ranyah University Collage, Taif University, linear Ablowitz–Kaup–Newell–Segur water wave dyna-
P.O. Box 11099, Taif 21944, Saudi Arabia mical equation [28]; nonlinear wave solutions of the
Open Access. © 2021 Khaled El-Rashidy et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0
International License.
92 Khaled El-Rashidy et al.
Kudryashov–Sinelshchikov dynamical equation [29]; the 2 Interactional phenomena and
third-order NLS equation [30]; approximate solutions of
nonlinear Parabolic equation by using modified varia-
multi-waves solutions
tional iteration algorithm [31]; new solitary wave solu-
Solutions for multiple waves and fractional phenomena
tions of nonlinear Nizhnik–Novikov–Vesselov equation
logarithmic transformation can be shown as
[32]; and Hirota bilinear method was used to study mul-
tiple soliton interactions [33]. ϕ(ξ ) = 2 log( f (ξ ))ξ (4)
The purpose of this work is to find traveling wave
equation (3) has the form:
solutions of (3 + 1) equation-dimensional Z–K using
logarithmic transformation and symbolic computation k ((δ + 1) ν 2 + k 2) f (ξ )2 f (3) (ξ ) + 2k ((δ + 1) ν 2 + k 2
by using the function method, see [10,34–36]: + 2λ2 ) f ′(ξ )3 + 2kμf (ξ ) f ′(ξ )2 + 2ωf (ξ )2 f ′(ξ ) (5)
1 1 − 3k ((δ + 1) ν 2 + k2 + 2λ2 ) f (ξ ) f ′(ξ ) f ″(ξ ) =0
ut + μuux + uxxx + (δ + 1) uxyy + uxzz = 0, (1)
2 2
the function u is a function of the variables x , y , z and
the temporal variable t which is the electrostatic wave 2.1 Type (I): Three waves hypothesis
potential in plasmas, where μ and δ are real constants
coefficient. In this subsection, for the next three waves hypoth-
Consider the solution in its complex form as follows: esis, multiple wave solutions can be expressed for non-
linear equation (5) as follows:
u(x , y , z , t ) = ϕ(ξ ), ξ = kx + νy + λz + ωt , (2)
f (ξ ) = b0 cosh(a1 ξ + a2) + b1 cos(a3 ξ + a4)
where k , ν , λ , and ω are real constants wave numbers and (6)
frequency. + b2 cosh(a5 ξ + a6),
Substituting from equation (2) into equation (1) and where the real constants aj , ( j = 1, … , 6) are determined
separating the imaginary and real parts lead to the ordinary later. Substituting (6) into (5) with the help of collecting
differential equation as: all the parameters for all the exponents of cosh(a1 ξ + a2),
k ((δ + 1) ν 2 + k 2 + 2λ2 ) ϕ″(ξ ) + kμϕ(ξ )2 + 2ωϕ(ξ ) = 0 (3) cos(a3 ξ + a4), cosh(a5 ξ + a6), sinh(a1 ξ + a2), sin(a3 ξ + a4)
and sinh(a5 ξ + a6) functions be zero, which establishes a
For exact solutions to equation (1), we use the loga- set of algebraic equations with respect to above con-
rithmic transformation and function method for equation stants. With help of the Wolfram Mathematica program
(3). Introduction to the set of basic equations is procured software, solving the determination algebraic equations,
in Section 3. A conclusion is carried out in Section 4. we conclude the following cases:
Case 1
ω 4λ2 − k 2
a1 = , a5 = a1 , a3 = 0, b1 = b02 + b22 , ν= , (7)
2 λ 4 k 2 − 6λ2 δ+1
ω 2 ω
substituting from (6), (7), and (4). By taking γ1 = and τ = , we obtain,
2λ 4
k2 − 6λ2 λ k 2 − 6λ2
4
u11(x , y , z , t )
=
τ [b0 sinh (a2 + γ(kx + tω + νy + λz )) + b2 sinh (a6 + γ(kx + tω + νy + λz ))] (8)
b0 cosh (a2 + γ(kx + tω + νy + λz )) + b2 cosh (a6 + γ(kx + tω + νy + λz )) + b02 + b22 cos (a4 )
Figure 1
Case 2
1 i −k 2 + 4λ2 − ν 2
a1 = , a3 = − , a5 = 0, ω = −λ2 k 2 − 6λ2 , δ= , b0 = ib1 , (9)
2 2 ν2
substituting from (6), (9), and (4), we have:
u12 (x , y , z , t ) =
(
2 b1 i sinh a2 + ( kx + tω + νy + λz
2 ) + sinh ( kx + tω + νy + λz
2
+ ia4 )) (10)
(
b2 cosh (a6) + b1 i cosh a2 + ( 2 ) + cosh (
kx + tω + νy + λz kx + tω + νy + λz
2
+ ia4 ))
Investigation of interactional phenomena and multi wave solutions 93
(a) (b)
(c) (d)
(e) (f)
Figure 1: This figure for wave solutions (8) using parameters: k = 3, λ = 1, δ = 5, ω = 4, a2 = 1 , a4 = 30, a6 = 2, b0 = 1 , b2 = 1 , z = 1 , y = 3 .
5
(a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour
solutions.
Figure 2
Case 3
k i k 4k 2
a1 = 0, a3 = , a5 → , b1 = ib2 , ω=− (11)
δλ2 ν 2 + 2λ4 + λ2 ν 2 δλ2 ν 2 + 2λ4 + λ2 ν 2 δν 2 + 2λ2 + ν 2
94 Khaled El-Rashidy et al.
(a) (b)
(c) (d)
(e) (f)
Figure 2: This figure for wave solutions (10) using parameters: a2 = 1 , a4 = 1 , a6 = 2, b2 = 2 2 , b1 = 2 2 , y = 2, k = 4, ν = 2. Wave solutions
appear in (a) and (b) as three dimensions but (c) and (d) represented wave solutions in two dimensions and (e) and (f) a contour solutions.
k
substituting from (6), (11), and (4). By taking γ2 = , we obtain
(δ + 1) λ2 ν 2 + 2λ4
2ib2 γ3 ( sin (a4 + γ2(kx + tω + νy + λz )) − i sin (γ2(kx + tω + νy + λz ) − ia6 ))
u13(x , y , z , t ) (12)
b0 cosh (a2) + b2 (i cos (a4 + γ2(kx + tω + νy + λz )) + cos (γ2(kx + tω + νy + λz ) − ia6 ))
Figure 3
Investigation of interactional phenomena and multi wave solutions 95 (a) (b) (c) (d) (e) (f) Figure 3: This figure for wave solutions (12) using parameters: k = − 4, λ = 2, δ = 1 , a2 = 2, a4 = 1 , a6 = 2, b2 = 2, z = 1 , y = 2, ν = 2, b0 = 1 . (a) and (b) Ploted as wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.
96 Khaled El-Rashidy et al.
(a) (b)
(c) (d)
(e) (f)
2 2
Figure 4: This figure for wave solutions (15) using parameters: a1 = 3 , a3 = 3 . (a) and (b) Represented wave solutions in three dimensions,
(c) and (d) showed wave solutions in two dimensions and (e) and (f) a contour solutions.Investigation of interactional phenomena and multi wave solutions 97
(a) (b)
(c) (d)
(e) (f)
1
Figure 5: This figure for wave solutions (18) using parameters: k = 1 , z = 1 , y = 1 , p1 = 1 , p = 1 , λ = 1 , μ = 3 , ν = 1 , a2 = 1 , a4 = 1 , a6 = − i ,
b0 = 1 , b1 = 1 . (a) and (b) Represented wave solutions in three dimensions, (c) and (d) showed wave solutions in two dimensions and (e) and
(f) a contour solutions.98 Khaled El-Rashidy et al.
2.2 Type (II): Interactional phenomena and
( )
2(kx + tω + νy + λz )
2 2 (e a2 b1 + e a4 b2 ) e 3
exponential form u2 (x , y , z , t ) = (15)
( )
2(kx + tω + νy + λz )
3 (e a2 b1 + e a4 b2 ) e 3
The hypothesis of a double exponential function takes
the form: Figure 4
f (ξ ) = b1 e a1 ξ + a2 + b2 e a3 ξ + a4 , (13)
where the real constants aj ( j = 1, … , 6) are determined
2.3 Type (III): Homoclinic breather approach
later. By substituting (13) into (5) with the help of sym-
bolic accounts having all the coefficients of all powers of
The Homoclinic breather approach function takes the form:
exponential functions zero, we have a system of algebraic
equations. By resolving this system, we get f (ξ ) = e ξ1(−p) + b0 cos (ξ3 p1 ) + b1 e ξ2 p, (16)
a1 =
2
, a3 =
2
, μ = −6 3 ν 2 + 12ν 2 − 11 3 where the real constants aj ( j = 1, … , 6) are determined
3 3 (14) later. Substituting (16) into (4) and by resolving the alge-
+ 22, braic equations, we obtain:
substitution from (13), (14) and (4), we obtain i 6μ 3μ 3μ 9kμ2
a5 → , a1 → , a3 → − , ω → , (17)
λ 2 p1 λ2 p λ2 p λ2
μ
substitution from (16) and (17) using (4). By taking γ3 = λ2
, the wave solution takes the form:
2γ3 ( 6 b0 e a2 p + 3γ3(kx + tω + νy + λz ) sinh ( 6 γ3(kx + tω + νy + λz ) − ia6 p1 ) − 3b1 e(a2 + a4 ) p − 3)
u3(x , y , z , t ) = (18)
b0 e a2 p + 3γ3(kx + tω + νy + λz ) cosh ( 6 γ3(kx + tω + νy + λz ) − ia6 p1 ) + b1 e(a2 + a4 ) p + 1
Figure 5
3 Conclusion [2] Seadawy AR, El-Rashidy K. Water wave solutions of the
coupled system Zakharov–Kuznetsov and generalized coupled
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Acknowledgments: Taif University Researchers Supporting
[7] El-Rashidy K. New traveling wave solutions for the higher
Project number (TURSP-2020/165), Taif University, Taif,
Sharma-Tasso-Olver equation by using extension exponential
Saudi Arabia. rational function method. Results Phys. 2020;20:103066.
[8] Zakharov VE, Kuznetsov EA. On three-dimensional solitons.
Conflict of interest: Authors state no conflict of interest. Soviet Phys. 1974;39:285–8.
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