Symmetries, optimal system, exact and soliton solutions of (3+1 )-dimensional Gardner-KP equation

doi:10.1016/j.joes.2022.04.035

Abstract

Abstract:

In this research article, the (3+1 )-dimensional nonlinear Gardner-Kadomtsov-Petviashvili (Gardner-KP) equation which depicts the nonlinear modulation of periodic waves, is analyzed through the Lie group-theoretic technique. Considering the Lie invariance condition, we find the symmetry generators. The proposed model yields eight-dimensional Lie algebra. Moreover, an optimal system of sub-algebras is computed, and similarity reductions are made. The considered nonlinear partial differential equation is reduced into ordinary differential equations (ODEs) by utilizing the similarity transformation method (STM), which has the benefit of yielding a large number of accurate traveling wave solutions. These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases, while in other cases, we use the new auxiliary equation method to get its soliton solutions. The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.

Highlights

● Lie symmetry analysis of $(3 + 1)$ -dimensional Gardner-KP equation: weakly nonlinear, dispersive surface waves propagating near-critical depth levels.

● Optimal system of sub-algebras.

● Exact and soliton solutions.

Key words: Gardner-KP equation, Lie symmetry analysis, Optimal system, New auxiliary equation method, Solitary wave solutions

Amjad Hussain, Ashreen Anjum, M. Junaid-U-Rehman, Ilyas Khan, Mariam A. Sameh, Amnah S. Al-Johani. Symmetries, optimal system, exact and soliton solutions of (3+1 )-dimensional Gardner-KP equation[J]. Journal of Ocean Engineering and Science, 2024, 9(2): 178-190.

Amjad Hussain, Ashreen Anjum, M. Junaid-U-Rehman, Ilyas Khan, Mariam A. Sameh, Amnah S. Al-Johani. [J]. Journal of Ocean Engineering and Science, 2024, 9(2): 178-190.

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URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.04.035

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I2/178

Figures/Tables 30

Table 1 Commutator table .

$[Z_{m}, Z_{n}]$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{1}$	0	0	0	0	$3 Z_{1} + 3 Z_{2}$	0	$- 2 Ω Z_{3}$	$- 2 Ω Z_{4}$
$Z_{2}$	0	0	0	0	$Z_{2}$	0	0	0
$Z_{3}$	0	0	0	0	$2 Z_{3}$	$- Z_{4}$	$Z_{2}$	0
$Z_{4}$	0	0	0	0	$2 Z_{4}$	$Z_{3}$	0	$Z_{2}$
$Z_{5}$	$- 3 Z_{1} - 3 Z_{2}$	$- Z_{2}$	$- 2 Z_{3}$	$- 2 Z_{4}$	0	0	$Z_{7}$	$Z_{8}$
$Z_{6}$	0	0	$Z_{4}$	$- Z_{3}$	0	0	$Z_{8}$	$- Z_{7}$
$Z_{7}$	$2 Ω Z_{3}$	0	$- Z_{2}$	0	$- Z_{7}$	$- Z_{8}$	0	0
$Z_{8}$	$2 Ω Z_{4}$	0	0	$- Z_{2}$	$- Z_{8}$	$- Z_{7}$	0	0

Table 1 Commutator table .

$[Z_{m}, Z_{n}]$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{1}$	0	0	0	0	$3 Z_{1} + 3 Z_{2}$	0	$- 2 Ω Z_{3}$	$- 2 Ω Z_{4}$
$Z_{2}$	0	0	0	0	$Z_{2}$	0	0	0
$Z_{3}$	0	0	0	0	$2 Z_{3}$	$- Z_{4}$	$Z_{2}$	0
$Z_{4}$	0	0	0	0	$2 Z_{4}$	$Z_{3}$	0	$Z_{2}$
$Z_{5}$	$- 3 Z_{1} - 3 Z_{2}$	$- Z_{2}$	$- 2 Z_{3}$	$- 2 Z_{4}$	0	0	$Z_{7}$	$Z_{8}$
$Z_{6}$	0	0	$Z_{4}$	$- Z_{3}$	0	0	$Z_{8}$	$- Z_{7}$
$Z_{7}$	$2 Ω Z_{3}$	0	$- Z_{2}$	0	$- Z_{7}$	$- Z_{8}$	0	0
$Z_{8}$	$2 Ω Z_{4}$	0	0	$- Z_{2}$	$- Z_{8}$	$- Z_{7}$	0	0

Table 2 Adjoint representation table.

$A d_{g}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{1}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{2}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{3}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{4}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{5}$	$e^{3}^{ε} Z_{1} + (\frac{- 3}{2} e^{ε} + \frac{3}{2} e^{3}^{ε}) Z_{2}$	$e^{ε} Z_{2}$	$e^{2}^{ε} Z_{3}$	$e^{2}^{ε} Z_{4}$
$Z_{6}$	$Z_{1}$	$Z_{2}$	$\cos (ε) Z_{3} - \sin (ε) Z_{4}$	$\sin (ε) Z_{3} + \cos (ε) Z_{4}$
$Z_{7}$	$- Z_{2} Ω ε^{2} - 2 Z_{3} Ω ε + Z_{1}$	$Z_{2}$	$Z_{2} ε + Z_{3}$	$Z_{4}$
$Z_{8}$	$- Z_{2} Ω ε^{2} - 2 Z_{4} Ω ε + Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{2} ε + Z_{4}$
$A d_{g}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{1}$	$- 3 Z_{1} ε - 3 Z_{2} ε + Z_{5}$	$Z_{6}$	$2 Z_{3} Ω ε + Z_{7}$	$2 Z_{4} Ω ε + Z_{8}$
$Z_{2}$	$- Z_{2} ε + Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{3}$	$- 2 Z_{3} ε + Z_{5}$	$Z_{4} ε + Z_{6}$	$- Z_{2} ε + Z_{7}$	$Z_{8}$
$Z_{4}$	$- 2 Z_{4} ε + Z_{5}$	$Z_{3} ε + Z_{6}$	$Z_{7}$	$- Z_{2} ε + Z_{8}$
$Z_{5}$	$Z_{5}$	$Z_{6}$	$e^{- ε} Z_{6}$	$e^{- ε} Z_{8}$
$Z_{6}$	$Z_{5}$	$Z_{6}$	$\cos (ε) Z_{7} - \sin (ε) Z_{8}$	$\sin (ε) Z_{7} + \cos (ε) Z_{8}$
$Z_{7}$	$Z_{7} ε + Z_{5}$	$Z_{8} ε + Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{8}$	$Z_{8} ε + Z_{5}$	$- Z_{7} ε + Z_{6}$	$Z_{7}$	$Z_{8}$

Table 2 Adjoint representation table.

$A d_{g}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{1}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{2}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{3}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{4}$	$Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{4}$
$Z_{5}$	$e^{3}^{ε} Z_{1} + (\frac{- 3}{2} e^{ε} + \frac{3}{2} e^{3}^{ε}) Z_{2}$	$e^{ε} Z_{2}$	$e^{2}^{ε} Z_{3}$	$e^{2}^{ε} Z_{4}$
$Z_{6}$	$Z_{1}$	$Z_{2}$	$\cos (ε) Z_{3} - \sin (ε) Z_{4}$	$\sin (ε) Z_{3} + \cos (ε) Z_{4}$
$Z_{7}$	$- Z_{2} Ω ε^{2} - 2 Z_{3} Ω ε + Z_{1}$	$Z_{2}$	$Z_{2} ε + Z_{3}$	$Z_{4}$
$Z_{8}$	$- Z_{2} Ω ε^{2} - 2 Z_{4} Ω ε + Z_{1}$	$Z_{2}$	$Z_{3}$	$Z_{2} ε + Z_{4}$
$A d_{g}$	$Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{1}$	$- 3 Z_{1} ε - 3 Z_{2} ε + Z_{5}$	$Z_{6}$	$2 Z_{3} Ω ε + Z_{7}$	$2 Z_{4} Ω ε + Z_{8}$
$Z_{2}$	$- Z_{2} ε + Z_{5}$	$Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{3}$	$- 2 Z_{3} ε + Z_{5}$	$Z_{4} ε + Z_{6}$	$- Z_{2} ε + Z_{7}$	$Z_{8}$
$Z_{4}$	$- 2 Z_{4} ε + Z_{5}$	$Z_{3} ε + Z_{6}$	$Z_{7}$	$- Z_{2} ε + Z_{8}$
$Z_{5}$	$Z_{5}$	$Z_{6}$	$e^{- ε} Z_{6}$	$e^{- ε} Z_{8}$
$Z_{6}$	$Z_{5}$	$Z_{6}$	$\cos (ε) Z_{7} - \sin (ε) Z_{8}$	$\sin (ε) Z_{7} + \cos (ε) Z_{8}$
$Z_{7}$	$Z_{7} ε + Z_{5}$	$Z_{8} ε + Z_{6}$	$Z_{7}$	$Z_{8}$
$Z_{8}$	$Z_{8} ε + Z_{5}$	$- Z_{7} ε + Z_{6}$	$Z_{7}$	$Z_{8}$

Fig.1 2D Graphics.

Fig.2 3D Graphics. Graphical interpretation of

B_{1, 2} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.75

ν = 2.00

γ = 2

μ = 3.00

ω = 4

ν = 2

and

Ω = 1

c = - 90.5

Fig.2 3D Graphics. Graphical interpretation of $B_{1, 2} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.75$ , $ν = 2.00$ , $γ = 2$ , $μ = 3.00$ , $ω = 4$ , $ν = 2$ and $Ω = 1$ , $c = - 90.5$ .

Fig.3 2D Graphics.

Fig.4 3D Graphics. Graphical interpretation of

B_{1, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.75

ν = 2.00

γ = 2

μ = 3.00

ω = 4

ν = 2

Ω = 1

, and

c = - 90.5

Fig.4 3D Graphics. Graphical interpretation of $B_{1, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.75$ , $ν = 2.00$ , $γ = 2$ , $μ = 3.00$ , $ω = 4$ , $ν = 2$ $Ω = 1$ , and $c = - 90.5$ .

Fig.5 2D Graphics.

Fig.6 3D Graphics. Graphical interpretation of

B_{2, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.75

ν = 2.00

γ = 2

μ = 3.00

ω = 4

Ω = 1

, and

c = - 90.5

Fig.6 3D Graphics. Graphical interpretation of $B_{2, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.75$ , $ν = 2.00$ , $γ = 2$ , $μ = 3.00$ , $ω = 4$ , $Ω = 1$ , and $c = - 90.5$ .

Fig.7 2D Graphics.

Fig.8 3D Graphics. Graphical interpretation of

B_{4, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.5

ν = 1

κ_{2} = 1

γ = 1

μ = - 2

ω = 1

Ω = 2

, and

c = 1

Fig.8 3D Graphics. Graphical interpretation of $B_{4, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.5$ , $ν = 1$ , $κ_{2} = 1$ , $γ = 1$ , $μ = - 2$ , $ω = 1$ , $Ω = 2$ , and $c = 1$ .

Fig.9 2D Graphics.

Fig.10 3D Graphics. Graphical interpretation of

B_{4, 2} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.5

ν = 1

κ_{2} = 1

γ = 1

μ = - 2

ω = 1

Ω = 2

, and

c = 1

Fig.10 3D Graphics. Graphical interpretation of $B_{4, 2} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.5$ , $ν = 1$ , $κ_{2} = 1$ , $γ = 1$ , $μ = - 2$ , $ω = 1$ , $Ω = 2$ , and $c = 1$ .

Fig.11 2D Graphics.

Fig.12 3D Graphics. Graphical interpretation of

B_{3, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.05

ν = 3

κ_{2} = 1

γ = 2

μ = 1

ω = 0.75

Ω = 2

, and

c = 2

Fig.12 3D Graphics. Graphical interpretation of $B_{3, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.05$ , $ν = 3$ , $κ_{2} = 1$ , $γ = 2$ , $μ = 1$ , $ω = 0.75$ , $Ω = 2$ , and $c = 2$ .

Fig.13 2D Graphics.

Fig.14 3D Graphics. Graphical interpretation of

B_{5, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{2} = 1

κ_{3} = 2

ν = 3

γ = 4

μ = 5

ω = 1

Ω = 1

, and

c = - 20.5

Fig.14 3D Graphics. Graphical interpretation of $B_{5, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{2} = 1$ , $κ_{3} = 2$ , $ν = 3$ , $γ = 4$ , $μ = 5$ , $ω = 1$ , $Ω = 1$ , and $c = - 20.5$ .

Fig.15 2D Graphics.

Fig.16 3D Graphics. Graphical interpretation of

B_{5, 2} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 2

ν = 3

γ = 4

μ = 5

ω = 1

κ_{2} = 1

Ω = 1

, and

c = - 20.5

Fig.16 3D Graphics. Graphical interpretation of $B_{5, 2} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 2$ , $ν = 3$ , $γ = 4$ , $μ = 5$ , $ω = 1$ , $κ_{2} = 1$ $Ω = 1$ , and $c = - 20.5$ .

Fig.17 2D Graphics.

Fig.18 3D Graphics. Graphical interpretation of

B_{6, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 2.5

ν = 1.00

γ = 2

μ = 1.00

ω = 2

Ω = 1

ν = 1

, and

c = - 4.5

Fig.18 3D Graphics. Graphical interpretation of $B_{6, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 2.5$ , $ν = 1.00$ , $γ = 2$ , $μ = 1.00$ , $ω = 2$ $Ω = 1$ , $ν = 1$ , and $c = - 4.5$ .

Fig.19 2D Graphics.

Fig.20 3D Graphics. Graphical interpretation of

B_{6, 2} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 2.5

ν = 1.00

γ = 2

μ = 1.00

ω = 2

Ω = 1

ν = 1

, and

c = - 4.5

Fig.20 3D Graphics. Graphical interpretation of $B_{6, 2} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 2.5$ , $ν = 1.00$ , $γ = 2$ , $μ = 1.00$ , $ω = 2$ $Ω = 1$ , $ν = 1$ , and $c = - 4.5$ .

Fig.21 2D Graphics.

Fig.22 3D Graphics. Graphical interpretation of

B_{7, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 0.95

ν = 1

κ_{2} = 1

γ = 0

ν = 1

μ = 3

ω = 7

Ω = 2

, and

c = - 44.5

Fig.22 3D Graphics. Graphical interpretation of $B_{7, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 0.95$ , $ν = 1$ , $κ_{2} = 1$ , $γ = 0$ , $ν = 1$ $μ = 3$ , $ω = 7$ , $Ω = 2$ , and $c = - 44.5$ .

Fig.23 2D Graphics.

Fig.24 3D Graphics. Graphical interpretation of

B_{9, 1} (θ, η, ζ, τ)

for

κ_{1} = 1

κ_{3} = 2

ν = 1

κ_{2} = 1

γ = 3

ν = 1

μ = 3

ω = 3

Ω = 1

, and

c = - 18

Fig.24 3D Graphics. Graphical interpretation of $B_{9, 1} (θ, η, ζ, τ)$ for $κ_{1} = 1$ , $κ_{3} = 2$ , $ν = 1$ , $κ_{2} = 1$ , $γ = 3$ , $ν = 1$ , $μ = 3$ , $ω = 3$ , $Ω = 1$ , and $c = - 18$ .

Fig.25 2D Graphics.

Fig.26 3D Graphics. Graphical interpretation of

B_{14, 1} (θ, η, ζ, τ)

for

κ_{1} = 0.75

κ_{3} = 3

ν = 1

κ_{2} = 1

γ = 3

μ = 0

ω = 2

Ω = 1

, and

c = 1.45

Fig.26 3D Graphics. Graphical interpretation of $B_{14, 1} (θ, η, ζ, τ)$ for $κ_{1} = 0.75$ , $κ_{3} = 3$ , $ν = 1$ , $κ_{2} = 1$ , $γ = 3$ , $μ = 0$ , $ω = 2$ , $Ω = 1$ , and $c = 1.45$ .

Fig.27 2D Graphics.

Fig.28 3D Graphics. Graphical interpretation of

B_{15, 1} (θ, η, ζ, τ)

for

κ_{1} = 0.9

κ_{3} = 0.75

ν = 2

κ_{2} = 1

γ = 3

μ = 3

ω = 3

Ω = 2

, and

c = - 57.1

Fig.28 3D Graphics. Graphical interpretation of $B_{15, 1} (θ, η, ζ, τ)$ for $κ_{1} = 0.9$ , $κ_{3} = 0.75$ , $ν = 2$ , $κ_{2} = 1$ , $γ = 3$ , $μ = 3$ , $ω = 3$ , $Ω = 2$ , and $c = - 57.1$ .

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