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Generalized KdV Equation: Novel Nature Oceanic, M-lump and Physical Collision Waves

Hajar Farhan ISMAEL

Hajar Farhan ISMAEL

  • Department of Mathematics, College of Science, University of ZakhoZakho, Iraq
  • Department of Computer Science, College of Science, Knowledge UniversityErbil, Iraq
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ISMAEL, Hajar Farhan
  
Jun 06, 2025
Generalized KdV Equation: Novel Nature Oceanic, M-lump and Physical Collision Waves's Cover Image
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Published Online: Jun 06, 2025
Page range: 217 - 223
Received: Nov 30, 2024
Accepted: Feb 10, 2025
DOI: https://doi.org/10.2478/ama-2025-0026
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© 2025 Hajar Farhan ISMAEL, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Fig. 1.

Graphs of one-M-lump wave when z = 2, t = 2, a1=15 , b1 = 0.5, c1 = 0.5, d1 = 0.5, β = 2, γ = –1.
Graphs of one-M-lump wave when z = 2, t = 2, a1=15 , b1 = 0.5, c1 = 0.5, d1 = 0.5, β = 2, γ = –1.

Fig. 2.

Plot of Eq. (12) for z = 2, a1=15 , b1 = 0.5, c1 = 0.5, d1 = 0.5, β = 2, γ = –1
Plot of Eq. (12) for z = 2, a1=15 , b1 = 0.5, c1 = 0.5, d1 = 0.5, β = 2, γ = –1

Fig. 3.

Plots of 2-M-lump wave when z = 2, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13  , c1=13 , d1=13 , c2=14 , d2=14 , β = 2, γ = –1.
Plots of 2-M-lump wave when z = 2, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13  , c1=13 , d1=13 , c2=14 , d2=14 , β = 2, γ = –1.

Fig. 4.

Plots of 3-M-lump wave when z = 2, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , a3=14 , b3=14 , c1 = 0.5, d1 = 0.5, c2=15 , d2=15 , c3=16 , d3=16 , β = 2, γ = –1
Plots of 3-M-lump wave when z = 2, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , a3=14 , b3=14 , c1 = 0.5, d1 = 0.5, c2=15 , d2=15 , c3=16 , d3=16 , β = 2, γ = –1

Fig. 5.

Plots of M-lump with soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , k3 = 1, l3 = 1, j3 = 2, λ3 = 20, β = 2, γ = –1
Plots of M-lump with soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , k3 = 1, l3 = 1, j3 = 2, λ3 = 20, β = 2, γ = –1

Fig. 6.

Graphs of M-lump with a 2-soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, c1=13 , d1=13 , k3 = 1, l3 = 1, j3 = 2, λ3 = 10, k4 = 1, l4 = 2, j4 = 3, λ4 = 20, β = 1, γ = –5
Graphs of M-lump with a 2-soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, c1=13 , d1=13 , k3 = 1, l3 = 1, j3 = 2, λ3 = 10, k4 = 1, l4 = 2, j4 = 3, λ4 = 20, β = 1, γ = –5

Fig. 7.

Plots of 2-M-lump with soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , c1=14 , d1=14 , c2=15 , d2=15 , k5 = 1, l5 = 1, j5 = 2, λ5 = 20, β = 1, γ = –5
Plots of 2-M-lump with soliton solution when z = 1, t = 2, a1 = 0.5, b1 = 0.5, a2=13 , b2=13 , c1=14 , d1=14 , c2=15 , d2=15 , k5 = 1, l5 = 1, j5 = 2, λ5 = 20, β = 1, γ = –5

Fig. 8.

Graphs of complex one-soliton wave are plotted for z = 1, t = 2, k1 = 1, l1 = 1, j1 = 2, α1 = 1, β = 1, γ = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part
Graphs of complex one-soliton wave are plotted for z = 1, t = 2, k1 = 1, l1 = 1, j1 = 2, α1 = 1, β = 1, γ = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part

Fig. 9.

Graphs of complex two-soliton wave are plotted for z = 1,t = 2, k1 = 1, k2 = 1, l1 = 1, l2 = 0.5, j1 = 0.5, j2 = 0.5, α1 = 3, β2 = 1, γ1 = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part
Graphs of complex two-soliton wave are plotted for z = 1,t = 2, k1 = 1, k2 = 1, l1 = 1, l2 = 0.5, j1 = 0.5, j2 = 0.5, α1 = 3, β2 = 1, γ1 = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part

Fig. 10.

Graphs of complex four-soliton wave are plotted for z = 1, t = 2, k1 = 1, k2 = 1, k3 = 2, l1 = 1, l2 = 0.5, l3=13 , j1 = 0.5, j2 = 0.5, j3=13 , α1 = 3, α2 = 6, α3 = 9, β = 1, γ = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part
Graphs of complex four-soliton wave are plotted for z = 1, t = 2, k1 = 1, k2 = 1, k3 = 2, l1 = 1, l2 = 0.5, l3=13 , j1 = 0.5, j2 = 0.5, j3=13 , α1 = 3, α2 = 6, α3 = 9, β = 1, γ = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part

Fig. 11.

Graphs of complex four-soliton wave are plotted for z = 1, t = 2, k1 = 1, k2 = 1, k3 = 2, k4 = 2, l1 = 1, l2 = 0.5, l3=13 , l4=14 , j1 = 0.5, j2 = 0.5, j3=14 , j4=14 , α1 = 3, β2 = 6, α3 = 9, α4 = 12, β = 1, β = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part
Graphs of complex four-soliton wave are plotted for z = 1, t = 2, k1 = 1, k2 = 1, k3 = 2, k4 = 2, l1 = 1, l2 = 0.5, l3=13 , l4=14 , j1 = 0.5, j2 = 0.5, j3=14 , j4=14 , α1 = 3, β2 = 6, α3 = 9, α4 = 12, β = 1, β = 2: a) Real part, b) Imaginary part, c) Contour plot of real part, d) Contour plot of imaginary part
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