Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

doi:10.1016/j.joes.2022.04.012

Abstract

Abstract:

The nonlinear evolution equations have a wide range of applications, more precisely in physics, biology, chemistry and engineering fields. This domain serves as a point of interest to a large extent in the world's mathematical community. Thus, this paper purveys an analytical study of a generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model. We further reduce the equation using the subalgebras obtained. Besides, more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique. Consequently, we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright, dark, periodic (cnoidal and snoidal), compact-type as well as singular solitons. The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time, are outlined with suitable diagrams. This can be of interest to oceanographers and ocean engineers for future analysis. Furthermore, to visualize the dynamics of the results found, we present the graphic display of each of the solutions. Conclusively, we construct conservation laws of the understudy equation via the application of Noether's theorem.

Highlights

● We study cnoidal and snoidal wave solutions via optimal system of one-dimensional sub-algebras for a generalized extended (2+1)-D quantum ZK equation with power-law non-linearity in oceanography and ocean engineering.

● We obtain non-topological soliton, cnoidal and snoidal wave and hyperbolic solutions.

● We derive conservation laws using Noether's theorem.

● We highlight the applications of our results (cnoidal and snoidal waves) in oceanography and ocean engineering.

Key words: A generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation, Lie point symmetries, Optimal system of subalgebras, Cnoidal and snoidal waves, Extended Jacobi function expansion technique, Conservation laws

Oke Davies Adeyemo. Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering[J]. Journal of Ocean Engineering and Science, 2024, 9(2): 126-153.

Oke Davies Adeyemo. [J]. Journal of Ocean Engineering and Science, 2024, 9(2): 126-153.

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URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.04.012

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I2/126

Figures/Tables 51

Table 1 Commutator table of the Lie algebra of 3D-gextQZKe (1.4).

[ $Υ_{μ}, Υ_{υ}$ ]	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$	Empty Cell
$Υ_{1}$	0	0	0	$n Υ_{1}$
$Υ_{2}$	0	0	0	$n Υ_{2}$
$Υ_{3}$	0	0	0	0
$Υ_{4}$	$- n Υ_{1}$	$- n Υ_{2}$	0	0

Table 1 Commutator table of the Lie algebra of 3D-gextQZKe (1.4).

[ $Υ_{μ}, Υ_{υ}$ ]	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$	Empty Cell
$Υ_{1}$	0	0	0	$n Υ_{1}$
$Υ_{2}$	0	0	0	$n Υ_{2}$
$Υ_{3}$	0	0	0	0
$Υ_{4}$	$- n Υ_{1}$	$- n Υ_{2}$	0	0

Table 2 Adjoint representation table of the Lie algebra of 3D-gextQZKe (1.4).

Ad	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$	Empty Cell
$Υ_{1}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$- n ε Υ_{1} + Υ_{4}$
$Υ_{2}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$- n ε Υ_{1} + Υ_{4}$
$Υ_{3}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$
$Υ_{4}$	$e^{n ε} Υ_{1}$	$e^{n ε} Υ_{2}$	$Υ_{3}$	$Υ_{4}$

Table 2 Adjoint representation table of the Lie algebra of 3D-gextQZKe (1.4).

Ad	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$	Empty Cell
$Υ_{1}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$- n ε Υ_{1} + Υ_{4}$
$Υ_{2}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$- n ε Υ_{1} + Υ_{4}$
$Υ_{3}$	$Υ_{1}$	$Υ_{2}$	$Υ_{3}$	$Υ_{4}$
$Υ_{4}$	$e^{n ε} Υ_{1}$	$e^{n ε} Υ_{2}$	$Υ_{3}$	$Υ_{4}$

Table 3 Summary of optimal system of (1+2)-D gKPle (1.4).

Cases	Element selections	Optimal System	Empty Cell
1.	$a_{1} = 0$ , $a_{2} = 0$ , $a_{3} = 0$ , $a_{4} \neq 0$	$Υ_{4}$
2.	$a_{1} = 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} + β Υ_{2}$
3.	$a_{1} \neq 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} - Υ_{1} + β Υ_{2}$
4.	$a_{1} \neq 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} + Υ_{1} + β Υ_{2}$

Table 3 Summary of optimal system of (1+2)-D gKPle (1.4).

Cases	Element selections	Optimal System	Empty Cell
1.	$a_{1} = 0$ , $a_{2} = 0$ , $a_{3} = 0$ , $a_{4} \neq 0$	$Υ_{4}$
2.	$a_{1} = 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} + β Υ_{2}$
3.	$a_{1} \neq 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} - Υ_{1} + β Υ_{2}$
4.	$a_{1} \neq 0$ , $a_{2} \neq 0$ , $a_{3} \neq 0$ , $a_{4} = 0$	$Υ_{3} + Υ_{1} + β Υ_{2}$

Table 4 Reduction summary for 3D-gextQZKe (1.4) through the dual and triple Lie vectors.

Vectors	Variables transform	Symmetries	Reduced equation
$Υ_{3} + β Υ_{2}$	$f = x$ , $g = y - β t$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$α G_{g} + c G_{g g g} + a G^{n} G_{f} + d G_{f g g}$
			$+ e G_{f f g} + b G_{f f f} = 0$
$Υ_{3} - Υ_{1} + β Υ_{2}$	$f = t + x$ , $g = y + β t$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$(c + β (d + β (b β + e))) G_{g g g} + d G_{f g g}$
			$+ a β G^{n} G_{g} + a G^{n} G_{f} + 2 e β G_{f g g} + G_{f}$
			$+ 3 b β^{2} G_{f g g} + e G_{f f g} + 3 b β G_{f f g} + b G_{f f f} = 0$
$Υ_{3} + Υ_{1} + β Υ_{2}$	$f = y - β t$ , $g = y - β x$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$- β G_{f} - a β G^{n} G_{g} + (c - d β) G_{g g g}$
			$+ e β^{2} G_{g g g} - b β^{3} G_{g g g} + 3 c G_{f g g} - 2 d β G_{f g g}$
			$+ e β^{2} G_{f g g} + 3 c G_{f g g} - d β G_{f f g} + c G_{f f f} = 0$

Table 4 Reduction summary for 3D-gextQZKe (1.4) through the dual and triple Lie vectors.

Vectors	Variables transform	Symmetries	Reduced equation
$Υ_{3} + β Υ_{2}$	$f = x$ , $g = y - β t$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$α G_{g} + c G_{g g g} + a G^{n} G_{f} + d G_{f g g}$
			$+ e G_{f f g} + b G_{f f f} = 0$
$Υ_{3} - Υ_{1} + β Υ_{2}$	$f = t + x$ , $g = y + β t$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$(c + β (d + β (b β + e))) G_{g g g} + d G_{f g g}$
			$+ a β G^{n} G_{g} + a G^{n} G_{f} + 2 e β G_{f g g} + G_{f}$
			$+ 3 b β^{2} G_{f g g} + e G_{f f g} + 3 b β G_{f f g} + b G_{f f f} = 0$
$Υ_{3} + Υ_{1} + β Υ_{2}$	$f = y - β t$ , $g = y - β x$	$Γ_{1} = \frac{\partial}{\partial f}$ , $Γ_{2} = \frac{\partial}{\partial g}$	$- β G_{f} - a β G^{n} G_{g} + (c - d β) G_{g g g}$
			$+ e β^{2} G_{g g g} - b β^{3} G_{g g g} + 3 c G_{f g g} - 2 d β G_{f g g}$
			$+ e β^{2} G_{f g g} + 3 c G_{f g g} - d β G_{f f g} + c G_{f f f} = 0$

Fig.1 Bright soliton wave profile of (3.21),

- 10 \leq x, y \leq 10

t = 0

for

n = 1

Fig.1 Bright soliton wave profile of (3.21), $- 10 \leq x, y \leq 10$ at $t = 0$ for $n = 1$ .

Fig.2 Bright soliton wave profile of (3.21),

- 10 \leq x, y \leq 10

t = 0

for

n = 2

Fig.2 Bright soliton wave profile of (3.21), $- 10 \leq x, y \leq 10$ at $t = 0$ for $n = 2$ .

Fig.3 Dark soliton wave profile of (3.29),

- 10 \leq x, y \leq 10

t = 1

for

n = 1

Fig.3 Dark soliton wave profile of (3.29), $- 10 \leq x, y \leq 10$ at $t = 1$ for $n = 1$ .

Fig.4 Bright soliton wave profile of (3.29),

- 7 \leq x, y \leq 7

t = 1

for

n = 2

Fig.4 Bright soliton wave profile of (3.29), $- 7 \leq x, y \leq 7$ at $t = 1$ for $n = 2$ .

Fig.5 Multi-soliton wave structure of (3.30),

- 7 \leq x, y \leq 7

t = - 2

for

n = 1

Fig.5 Multi-soliton wave structure of (3.30), $- 7 \leq x, y \leq 7$ at $t = - 2$ for $n = 1$ .

Fig.6 Multi-soliton wave structure of (3.30),

- 7 \leq x, y \leq 7

t = - 2

for

n = 2

Fig.6 Multi-soliton wave structure of (3.30), $- 7 \leq x, y \leq 7$ at $t = - 2$ for $n = 2$ .

Fig.7 Bright soliton wave profile of (3.33),

- 10 \leq x, y \leq 10

t = 0.3

for

n = 1

Fig.7 Bright soliton wave profile of (3.33), $- 10 \leq x, y \leq 10$ at $t = 0.3$ for $n = 1$ .

Fig.8 Bright soliton wave profile of (3.33),

- 10 \leq x, y \leq 10

t = 0.3

for

n = 2

Fig.8 Bright soliton wave profile of (3.33), $- 10 \leq x, y \leq 10$ at $t = 0.3$ for $n = 2$ .

Fig.9 Multi-soliton wave profile of (3.34),

- 7 \leq x, y \leq 7

t = - 0.8

for

n = 1

Fig.9 Multi-soliton wave profile of (3.34), $- 7 \leq x, y \leq 7$ at $t = - 0.8$ for $n = 1$ .

Fig.10 Multi-soliton wave profile of (3.34),

- 7 \leq x, y \leq 7

t = - 0.8

for

n = 2

Fig.10 Multi-soliton wave profile of (3.34), $- 7 \leq x, y \leq 7$ at $t = - 0.8$ for $n = 2$ .

Fig.11 Smooth periodic wave profile of (4.43) with

- 10 \leq x, y \leq 10

t = - 1

Fig.11 Smooth periodic wave profile of (4.43) with $- 10 \leq x, y \leq 10$ at $t = - 1$ .

Fig.12 Smooth periodic wave profile of (4.44) with

- 10 \leq x, y \leq 10

t = - 3

Fig.12 Smooth periodic wave profile of (4.44) with $- 10 \leq x, y \leq 10$ at $t = - 3$ .

Fig.13 Smooth periodic wave profile of solution (4.48),

- 13 \leq x, y \leq 13

t = 2

Fig.13 Smooth periodic wave profile of solution (4.48), $- 13 \leq x, y \leq 13$ at $t = 2$ .

Fig.14 Smooth periodic wave profile of solution (4.49),

- 10 \leq x, y \leq 10

t = 0

Fig.14 Smooth periodic wave profile of solution (4.49), $- 10 \leq x, y \leq 10$ at $t = 0$ .

Fig.15 Smooth periodic wave profile of (4.53),

- 12 \leq x, y \leq 12

t = - 5

Fig.15 Smooth periodic wave profile of (4.53), $- 12 \leq x, y \leq 12$ at $t = - 5$ .

Fig.16 Singular periodic wave profile of (4.54),

- 15 \leq x, y \leq 15

t = 10

Fig.16 Singular periodic wave profile of (4.54), $- 15 \leq x, y \leq 15$ at $t = 10$ .

Fig.17 Smooth periodic wave profile of (4.58),

- 12 \leq x, y \leq 12

t = 4

Fig.17 Smooth periodic wave profile of (4.58), $- 12 \leq x, y \leq 12$ at $t = 4$ .

Fig.18 Singular periodic wave profile of (4.59),

- 15 \leq x, y \leq 15

t = 2.3

Fig.18 Singular periodic wave profile of (4.59), $- 15 \leq x, y \leq 15$ at $t = 2.3$ .

Fig.19 Smooth periodic wave profile of (4.64) with

- 10 \leq x, y \leq 10

t = - 4.2

Fig.19 Smooth periodic wave profile of (4.64) with $- 10 \leq x, y \leq 10$ at $t = - 4.2$ .

Fig.20 Smooth periodic wave profile of (4.65) with

- 13 \leq x, y \leq 13

t = - 3.7

Fig.20 Smooth periodic wave profile of (4.65) with $- 13 \leq x, y \leq 13$ at $t = - 3.7$ .

Fig.21 Bell shape wave profile of (4.66) with

- 10 \leq x, y \leq 10

t = 6.2

Fig.21 Bell shape wave profile of (4.66) with $- 10 \leq x, y \leq 10$ at $t = 6.2$ .

Fig.22 Periodic wave profile of (4.70) with

- 10 \leq x, y \leq 10

t = 20

Fig.22 Periodic wave profile of (4.70) with $- 10 \leq x, y \leq 10$ at $t = 20$ .

Fig.23 Bell shape wave profile of (4.71) with

- 10 \leq x, y \leq 10

t = - 20

Fig.23 Bell shape wave profile of (4.71) with $- 10 \leq x, y \leq 10$ at $t = - 20$ .

Fig.24 Smooth periodic wave profile of (4.76) with

- 12 \leq x, y \leq 12

t = 8.3

Fig.24 Smooth periodic wave profile of (4.76) with $- 12 \leq x, y \leq 12$ at $t = 8.3$ .

Fig.25 Singular periodic wave profile of (4.77) with

- 15 \leq x, y \leq 15

t = - 0.6

Fig.25 Singular periodic wave profile of (4.77) with $- 15 \leq x, y \leq 15$ at $t = - 0.6$ .

Fig.26 Singular periodic wave profile of (4.78) with

- 15 \leq x, y \leq 15

t = - 0.6

Fig.26 Singular periodic wave profile of (4.78) with $- 15 \leq x, y \leq 15$ at $t = - 0.6$ .

Fig.27 Smooth periodic wave profile of (4.83) with

- 12 \leq x, y \leq 12

t = 0.2

Fig.27 Smooth periodic wave profile of (4.83) with $- 12 \leq x, y \leq 12$ at $t = 0.2$ .

Fig.28 Singular periodic wave profile of (4.84) with

- 15 \leq x, y \leq 15

t = 1.8

Fig.28 Singular periodic wave profile of (4.84) with $- 15 \leq x, y \leq 15$ at $t = 1.8$ .

Fig.29 Singular periodic wave profile of (4.85) with

- 10 \leq x, y \leq 10

t = - 0.5

Fig.29 Singular periodic wave profile of (4.85) with $- 10 \leq x, y \leq 10$ at $t = - 0.5$ .

Fig.30 Periodic wave profile of (4.91) with

- 10 \leq x, y \leq 10

t = - 10

Fig.30 Periodic wave profile of (4.91) with $- 10 \leq x, y \leq 10$ at $t = - 10$ .

Fig.31 Periodic wave profile of (4.92) with

- 13 \leq x, y \leq 13

t = 9

Fig.31 Periodic wave profile of (4.92) with $- 13 \leq x, y \leq 13$ at $t = 9$ .

Fig.32 Periodic wave profile of (4.93) with

- 10 \leq x, y \leq 10

t = - 6

Fig.32 Periodic wave profile of (4.93) with $- 10 \leq x, y \leq 10$ at $t = - 6$ .

Fig.33 Periodic wave profile of (4.94) with

- 10 \leq x, y \leq 10

t = 0.7

Fig.33 Periodic wave profile of (4.94) with $- 10 \leq x, y \leq 10$ at $t = 0.7$ .

Fig.34 Periodic wave profile of (4.99) with

- 10 \leq x, y \leq 10

t = 1

Fig.34 Periodic wave profile of (4.99) with $- 10 \leq x, y \leq 10$ at $t = 1$ .

Fig.35 Periodic wave profile of (4.100) with

- 8 \leq x, y \leq 8

t = - 2

Fig.35 Periodic wave profile of (4.100) with $- 8 \leq x, y \leq 8$ at $t = - 2$ .

Fig.36 Singular periodic wave profile of (4.101) with

- 10 \leq x, y \leq 10

t = 12

Fig.36 Singular periodic wave profile of (4.101) with $- 10 \leq x, y \leq 10$ at $t = 12$ .

Fig.37 Smooth periodic wave profile of (4.107) with

- 10 \leq x, y \leq 10

t = 0.3

Fig.37 Smooth periodic wave profile of (4.107) with $- 10 \leq x, y \leq 10$ at $t = 0.3$ .

Fig.38 Smooth periodic wave profile of (4.108) with

- 10 \leq x, y \leq 10

t = 0.22

Fig.38 Smooth periodic wave profile of (4.108) with $- 10 \leq x, y \leq 10$ at $t = 0.22$ .

Fig.39 Singular periodic wave profile of (4.109) with

- 15 \leq x, y \leq 15

t = 5.6

Fig.39 Singular periodic wave profile of (4.109) with $- 15 \leq x, y \leq 15$ at $t = 5.6$ .

Fig.40 Singular periodic wave profile of (4.110) with

- 15 \leq x, y \leq 15

t = - 2.4

Fig.40 Singular periodic wave profile of (4.110) with $- 15 \leq x, y \leq 15$ at $t = - 2.4$ .

Fig.41 Smooth periodic wave profile of (4.115) with

- 12 \leq x, y \leq 12

t = 20

Fig.41 Smooth periodic wave profile of (4.115) with $- 12 \leq x, y \leq 12$ at $t = 20$ .

Fig.42 Smooth periodic wave profile of (4.116) with

- 15 \leq x, y \leq 15

t = 30

Fig.42 Smooth periodic wave profile of (4.116) with $- 15 \leq x, y \leq 15$ at $t = 30$ .

Fig.43 Singular periodic wave profile of (4.117) with

- 1 \leq x, y \leq 1

t = 0

Fig.43 Singular periodic wave profile of (4.117) with $- 1 \leq x, y \leq 1$ at $t = 0$ .

Fig.44 Ocean's surface solitary wave triggered by atmosphere lower boundary's decreasing pressures [69].

Fig.45 Schematic diagram of the life cycle of nonlinear internal waves in the region southeast of Hainan Island [70].

Fig.46 Russell's solitary-wave solitons occur in shallow water channels, (a) as shown in a laboratory reconstruction of Russell's original observation. The inset is a typical localized solution of the Korteweg-de Vries equation for extremely shallow water. (b) In deep water, waves propagate (from left to right), whereas water particles orbit in circles around their average position. The diameter of the circular orbits shrinks gradually with increasing distance from the surface down to half the wavelength

L

. In shallow waters with depth

h < L / 2

, the orbits become elliptical and essentially flat at the bottom of the tank or seabed [72].

Fig.47 A simplified concept of the main wave transformation and attenuation processes which must be considered by coastal engineers in designing coastal defence schemes [78].

References 80

[47]	D.M. Mothibi, C.M. Khalique. Symmetry (Basel), 7 (2015), pp. 949-961 DOI: 10.3390/sym7020949
[48]	G.W. Wang, T.Z. Xu, S. Johnson. Astrophys. Space Sci., 349 (2014), pp. 317-327 DOI: 10.1007/s10509-013-1659-z
[49]	A. Wazwaz. Phys. Scr., 85 (2012), p. 025006 DOI: 10.1088/0031-8949/85/02/025006
[50]	Y.L. Jiang, Y. Lu, C. Chen. J. Nonlinear Math. Phys., 23 (2016), pp. 157-166
[51]	G.W. Bluman, S. Kumei. Symmetries and Differential Equations, Applied Mathematical Science. Springer, Berlin (1989)
[52]	J.G. O'Hara, C. Sophocleous, P.G.L. Leach. J. Eng. Math., 82 (2013), pp. 67-75 DOI: 10.1007/s10665-012-9595-4
[53]	T. Chaolu, G. Bluman. J. Math. Anal. Appl., 411 (2014), pp. 281-296
[54]	R.K. Gupta, K. Singh. Commun. Nonlinear. Sci. Numer. Simul., 16 (2011), pp. 4189-4196
[55]	R. Jiwari, V. Kumar, K. Ram, S.A. Ali. Int. J. Numer. Method. H., 27 (2017), pp. 1332-1350 DOI: 10.1108/hff-04-2016-0145
[56]	R.K. Gupta, V. Kumar, R. Jiwari. Nonlinear Dyn., 79 (2014), pp. 455-464 DOI: 10.1007/978-3-662-45937-9_45
[57]	L. Kaur, R.K. Gupta. Math. Method. Appl. Sci., 36 (2013), pp. 584-600 DOI: 10.1002/mma.2617
[58]	C.M. Khalique, O.D. Adeyemo. Mathematics, 8 (2020), p. 1692
[59]	A. SjÃ¶berg. Appl. Math. Comput., 184 (2007), pp. 608-616
[60]	E. Godlewski, P.A. Raviart. Numer. Math., 97 (2004), pp. 81-130
[61]	K.S. Al-Ghafri. Symmetry (Basel), 12 (2020), p. 219 DOI: 10.3390/sym12020219
[62]	Y. Zhou, M. Wang, Z. Li. Phys. Lett. A, 216 (1996), pp. 67-75
[63]	I.S. Gradshteyn, I.M. Ryzhik. Table of Integrals, Series and Products, 7th ed. Academic Press, New York (2007)
[64]	M. Abramowitz, I.A. Stegun. National Bureau of Standards Applied Mathematics Series Vol. 55, U.S.A Government Printing Office, Washington, D.C. (1964)
[65]	A.A. Gaber. J. Ocean Eng. Sci., 6 (2021), pp. 292-298, DOI: https://www.sciencedirect.com/science/article/pii/S2468013321000140/pdfft?md5=575daca414f0ce49f4400358ba1cd655&pid=1-s2.0-S2468013321000140-main.pdf
[66]	M.R.A. Fahim, P.R. Kundu, M.E. Islam, M.A. Akbar. J. Ocean Eng. Sci.(2021) Doi:10.1016/j.joes.2021.08.009
[1]	O.D. Adeyemo, C.M. Khalique. Symmetry (Basel), 14 (2022), p. 83 DOI: 10.3390/sym14010083
[2]	X.Y. Gao. Appl. Math. Lett., 91 (2019), pp. 165-172
[3]	O.D. Adeyemo, T. Motsepa, C.M. Khalique. Alex. Eng. J., 61 (2022), pp. 185-194
[4]	C.M. Khalique, O.D. Adeyemo. Results Phys., 18 (2020), p. 103197
[5]	X.X. Du, B. Tian, Q.X. Qu, Y.Q. Yuan, X.H. Zhao. solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fract., 134 (2020), p. 109709
[6]	C.R. Zhang, B. Tian, Q.X. Qu, L. Liu, H.Y. Tian. Z. Angew. Math. Phys., 71 (2020), pp. 1-19
[7]	C.M. Khalique, O.D. Adeyemo. J. Ocean Eng. Sci.(2021), p. 106393.
[8]	A.H. Salas, C.A. Gomez. Math. Probl. Eng. (2010), p. 2010
[9]	M.S. Osman, A.M. Wazwaz. Math. Method. Appl. Sci., 42 (2019), pp. 6277-6283 DOI: 10.1002/mma.5721
[10]	M.S. Osman. Nonlinear Dynam., 96 (2019), pp. 1491-1496 DOI: 10.1007/s11071-019-04866-1
[11]	M.G. Hafez, M. Kauser, M.T. Akter. J. Math. Comput. Sci., 4 (2014), pp. 2582-2593
[12]	M.A. Akbar, N.H.M. Ali. Springerplus, 3 (2014), p. 344
[13]	N.A. Kudryashov. Optik (Stuttg), 183 (2019), pp. 642-649
[14]	C.L. Zheng, J.P. Fang. Chaos Soliton Fract., 27 (2006), pp. 1321-1327
[15]	X.Y. Wen. Appl. Math. Comput., 217 (2010), pp. 1367-1375
[16]	A.M. Wazwaz. Partial Differential Equations. CRC Press, Boca Raton, Florida, USA (2002)
[17]	C. Chun, R. Sakthivel. Comput. Phys. Commun., 181 (2010), pp. 1021-1024
[18]	C.H. Gu. Soliton Theory and Its Application. Zhejiang Science and Technology Press, Zhejiang (1990)
[19]	X. Zeng, D.S. Wang. Appl. Math. Comput., 212 (2009), pp. 296-304
[67]	A. Ali, A.R. Seadawy. J. Ocean Eng. Sci., 6 (2021), pp. 85-98, DOI: https://www.sciencedirect.com/science/article/pii/S246801332030053X/pdfft?md5=5ac1b5598d04897d3fad771408eecb58&pid=1-s2.0-S246801332030053X-main.pdf
[68]	R. Grimshaw, D. Takagi, Y. Ma, A. Stewart, Lecture 16: Solitary waves-geophysical fluid dynamics, [Online]. Available: https://gfd.whoi.edu/wp-content/uploads/sites/18/2018/03/lecture16-roger136604.pdf.
[69]	D. Augustin, M.B. CÃ©sar. Open J. Mar. Sci., 5 (2014), p. 45, DOI: 10.4236/ojms.2015.51005.
[70]	J. Liang, X.M. Li, J. Sha, T. Jia. J. Phys. Oceanogr., 49 (2019), pp. 2133-2145 DOI: 10.1175/jpo-d-18-0231.1
[71]	J.R. Apel. An atlas of oceanic internal solitary waves, 322 (2002), pp. 1-40
[72]	A.N. Bogdanov, C. Panagopoulos, The emergence of magnetic skyrmions, 2020, ArXiv preprint arXiv:2003.09836.
[73]	H. Fuchs, W.H. Hager. J. Waterw. Port Coast. Ocean Eng., 141 (2015), p. 04015004 DOI: 10.1061/(ASCE)WW.1943-5460.0000294
[74]	H.V. Hafen, J. Stolle, N. Goseberg, I. Nistor. Coast. Eng. Proc., 36 (2020), p. 29, DOI: 10.9753/icce.v36v.waves.29.
[75]	T.F. Duda, J.C. Preisig. IEEE J. Ocean.Eng, 24 (1999), pp. 16-32
[76]	J. Zhou, X. Zhang, P.H. Rogers. J. Acoust. Soc., 90 (1991), pp. 2042-2054 DOI: 10.1121/1.401632
[77]	A.C. Warn-Varnas, S.A. Chin-Bing, D.B. King. Surv.Geophys, 24 (2003), pp. 39-79
[78]	Shallow-water wave theory, Available: http://www.coastalwiki.org/wiki/shallow-waterwavetheory. accessed: 15 01, 2022,
[79]	E. Noether. Nachr. v. d. Ges. d. Wiss. zu GÃ¶ttingen, 2 (1918), pp. 235-257
[80]	N.H. Ibragimov, A.H. Kara, F.M. Mahomed. Nonlinear Dynam., 15 (1998), pp. 115-136
[20]	A.J.M. Jawad, M. Mirzazadeh, A. Biswas. Pramana, 83 (2014), pp. 457-471
[21]	N.A. Kudryashov, N.B. Loguinova. Appl. Math. Comput., 205 (2008), pp. 396-402
[22]	R. Hirota. The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
[23]	L.V. Ovsiannikov. Group Analysis of Differential Equations. Academic Press, New York (1982)
[24]	G.W. Bluman, S. Kumei. Symmetries and Differential Equations. Springer-Verlag, New York (1989)
[25]	P.J. Olver. Applications of Lie Groups to Differential Equations, second ed. Springer-Verlag, Berlin (1993)
[26]	N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Vols 1-3, CRC Press, Boca Raton, Florida, 1994-1996.
[27]	N.H. Ibragimov. Elementary Lie Group Analysis and Ordinary Differential Equations. John Wiley & Sons, Chichester, NY (1999)
[28]	L. Zhang, C.M. Khalique. Discrete and Continuous dynamical systems Series S, 11 (4) (2018), pp. 777-790
[29]	M. Wang, X. Li, J. Zhang. Phys. Lett. A, 24 (2005), pp. 1257-1268
[30]	V.B. Matveev, M.A. Salle. Darboux Transformations and Solitons. Springer, New York (1991)
[31]	Y. Chen, Z. Yan. Chaos Solitons Fract., 26 (2005), pp. 399-406
[32]	N.A. Kudryashov. Chaos Solitons Fract., 24 (2005), pp. 1217-1231
[33]	J.H. He, X.H. Wu. Chaos Solitons Fract., 30 (2006), pp. 700-708
[34]	R. Sakthivel, C. Chun, J. Lee. Z. Naturforsch. A, 65 (2010), pp. 633-640 DOI: 10.1515/zna-2010-8-903
[35]	A.M. Wazwaz. Appl. Math. Comput., 169 (2005), pp. 321-338
[36]	J. Hu. Phys. Lett. A, 287 (2001), pp. 81-89
[37]	A. Biswas, A.J.M. Jawad, W.N. Manrakhan. Opt. Laser Technol., 44 (2012), pp. 2265-2269
[38]	A.M. Wazwaz. J. Nat. Sci. Math., 1 (2007), pp. 1-13
[39]	V.E. Zakharov, E.A. Kuznetsov. Zhurnal Eksp. Teoret.Fiz, 66 (1974), pp. 594-597
[40]	S.A. Khan, W. Masood. Phys.Plasmas, 15 (2008), p. 062301
[41]	B.S. Ahmed, E. Zerrad, A Math. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 14 (2013), pp. 281-286
[42]	E.V. Krishnan, A. Biswas. Phys. wave phenom., 18 (2010), pp. 256-261
[43]	M.A. Abdou. Nonlinear Sci. Lett. B, 1 (2011), pp. 99-110
[44]	A.M. Wazwaz. Commun. Nonlinear Sci. Numer. Simul., 10 (2005), pp. 597-606
[45]	A.M. Wazwaz. the modified ZK equation, and its generalized forms, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), pp. 1039-1047
[46]	A. Biswas. Phys. Lett. A, 373 (2009), pp. 2931-2934

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