Skip to main content
Have a personal or library account? Click to login
Dynamics of solitons, multi-lumps, and their interactions of the Konopelchenko-Dubrovsky-Kaup-Kupershmidt and Bogoyavlensky-Konopelchenko model in (3+1)-dimensions using the modern advanced approach Cover

Dynamics of solitons, multi-lumps, and their interactions of the Konopelchenko-Dubrovsky-Kaup-Kupershmidt and Bogoyavlensky-Konopelchenko model in (3+1)-dimensions using the modern advanced approach

International Journal of Mathematics and Computer in Engineering
AHEAD OF PRINT
By:  and    
Open Access
|Jun 2026
Figures & tables

Figures & Tables

Fig. 1

Wave profiles of the lump wave solution for equation (15): (a)-(c) are the 3D profiles with density plots; (d)-(f) are the contour plots; and (g)-(i) are the wave propagation along the x-axis.

Fig. 2

Wave profiles of the lump wave solution for equation (18) with different values of z: (a)-(c) are the 3D profiles with density plots; (d)-(f) are the contour plots; and (g)-(i) are the wave propagation along the x-axis.

Fig. 3

The 3D profiles with density plots show the collisions of the single-lump and solitary-wave solutions to equation (22) using different parameters.

Fig. 4

Collisions of single-lump and solitary-wave solutions of equation (25): (a)-(c) are the 3D profiles with density plots; and (d)-(f) are the corresponding 2D wave propagation along the x-axis.

Fig. 5

Collision of periodic and lump wave solutions of equation (29): (a)-(c) display 3D profiles with density plots; (d)-(f) display contour plots; and (g)-(i) display 2D wave profiles.

Fig. 6

Collision of periodic and lump wave solutions of equation (32): (a)-(c) are the 3D profiles with density plots; (d)-(f) are the contour plots; and (g)-(i) are the corresponding 2D wave propagation, respectively.

Fig. 7

Collision of periodic and lump wave solutions of equation (35): 3D profiles with density plots are shown in (a)-(c); contour plots are shown in (d)-(f); and 2D wave propagation along the x-axis is shown in (g)-(i).

Fig. 8

Collision of solitary, periodic, and lump wave solutions of equation (39): (a)-(c) display 3D profiles with density plots; (d)-(f) display contour plots; and (g)-(i) display 2D wave propagation along the x-axis.

Fig. 9

Collision of solitary, periodic, and lump wave solutions of equation (42): (a)-(c) display 3D profiles with density plots; (d)-(f) display contour plots; and (g)-(i) display 2D wave propagation along the x-axis.

Fig. 10

Collision between lump and breather wave solutions of equation (46): (a)-(c) are the 3D profiles with density plots; (d)-(f) are the contour plots; and (g)-(i) are the corresponding 2D wave propagation, respectively.

Fig. 11

Phase portrait graph of the dynamic system (50) using different parameter values.

Fig. 12

Hamiltonian function graphics for the dynamical system (51) for κ1 = 0.3, κ2 = 0.5, κ3 = 0.8, κ4 = 0.8, κ5 = 0.5, κ6 = 0.3, σ = 1.2, τ = 1.1, μ = –0.8, and ρ = –0.5.

Fig. 13

Hamiltonian function graphics for the dynamical system (51) for κ1 = 0.3, κ2 = 0.5, κ3 = 0.3, κ4 = 0.8, κ5 = 0.5, κ6 = 0.3, σ = 1.2, τ = 1.1, μ = 0.8, and ρ = 0.5.

Fig. 14

Hamiltonian function graphics for the dynamical system (51) for κ1 = 0.3, κ2 = –0.5, κ3 = 0.8, κ4 = 0.8, κ5 = –0.5, κ6 = 0.3, σ = 0.9, τ = –0.5, μ = –0.4, and ρ = –0.5.

Fig. 15

Hamiltonian function graphics for the dynamical system (51) for κ1 = 0.3, κ2 = 0.5, κ3 = 0.8, κ4 = 0.8, κ5 = 0.5, κ6 = 0.3, σ = –0.9, τ = 0.5, μ = –0.4, and ρ = –0.5.

Fig. 16

Sensitivity analysis of the dynamical system (50) with different initial conditions.

Particular cases of equation (1) found in ocean dynamics and related fields_

Variation of κiPDEsSpecial form
κ1 = κ2 = 1, κ3 = κ4 = κ5 = κ6 = 0, z = xut + uxxx + 6uux = 0KdV equation [26]
κ1 = 1, κ2 = a1, κ3 = a5, κ5 = 0, κ4 + κ6 = 0, z = xut + a1 uxxx + 12a2(u2)x{1 \over 2}{a_2}{\left( {{u^2}} \right)_x} + a3uxxxxx + a4vy + a5uxxy + a6(uxv + uvx) + a7(uuxx)x + 13a8(u3)x=0{1 \over 3}{a_8}{\left( {{u^3}} \right)_x} = 0, vx = uy, for a2 = 6a1, a6 = 3a5, a3 = a4 = a7 = a8 = 0(2+1)-D gKDKK equation [27]
κ1 = 1, κ2 = b1, κ3 = b2, κ4 = b3, κ5 = b4, κ6 = 0, b5 = 0, z = xut + 6b1(uux+16uxxx)6{b_1}\left( {u{u_x} + {1 \over 6}{u_{xxx}}} \right) + 3b2(13uxxy+uuy+uxvy)3{b_2}\left( {{1 \over 3}{u_{xxy}} + u{u_y} + {u_x}{v_y}} \right) + b3ux + b4uy + b5Vyy = 0, Vx = u(2+1)-D gBK equation [28]

Visual analysis of the KDKK and BK equations for the free parameters_

Fig. No.Free parameters with intervals
Fig. 1κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –2, κ5 = –1, κ6 = –2, l1 = 1, m2 = 1, n2 = 1, w1 = 1, δ1 = δ2 = 1, η = 1, z = 1 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 2κ1 = 2, κ2 = 2, κ3 = 1, κ4 = –2, κ5 = –1, κ6 = 1, l1 = l2 = 1, n1 = 1, w1 = w2 = 1, δ1 = –1, δ2 = 1, η = 1, t = 0 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 3κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –2, κ5 = κ6 = 1, l3 = 1, n1 = n2 = n3 = 1, w2 = 1, δ1 = 1, δ2 = 1, η = 1, z = 1, t = 0 for – 15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 4κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –1, κ5 = –1, κ6 = 1, l1 = l3 = 1, l2 = –1, n1 = –3, n2 = n3 = 1, δ1 = δ2 = 1, η = 1, z = 5 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 5κ1 = 1, κ2 = –1, κ3 = 1, κ4 = –3, κ5 = κ6 = 1, m2 = m3 = 1, n3 = 1, w2 = 1, δ1 = δ2 = 1, η = 1 for –15 ≤ x ≤ 15, –15 ≤ y ≤ 15
Fig. 6κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –1, κ5 = 1, κ6 = 1, m1 = i, m2 = m3 = 1, n1 = n3 = 1, w2 = 1, δ1 = δ2 = 1, η=−12\eta = - {1 \over 2}, z = 3 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 7κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –2, κ5 = –1, κ6 = 1, m2 = m3 = 1, n1 = i, n3 = –1, w2 = 1, δ1 = δ2 = 1, η = 1, z = 1 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 8κ1 = 1, κ2 = 0.5, κ3 = 1, κ4 = –4.5, κ5 = –1, κ6 = 1, m3 = –0.2, m4 = 0.4, n1 = 0.3, n2 = –i, w1 = 0.1, w2 = 0.1, w3 = 0.2, w4 = 0.5, δ1 = δ2 = 1.5, η = –0.6, z = 0 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
Fig. 9κ1 = 2, κ2 = 0.2, κ3 = 1, κ4 = –2, κ5 = –1, κ6 = 1, m3 = –1, m4 = 0.5, w1 = 0.8, w2 = 0.1, w3 = 0.6, w4 = 0.3, δ1 = δ2 = 1.5, η = –0.3, z = 0 for –40 ≤ x ≤ 20, –40 ≤ y ≤ 20
Fig. 10κ1 = 0.2, κ2 = 0.4, κ3 = 0.5, κ4 = –3, κ5 = –1, κ6 = 1, m3 = –0.2, m4 = 0.4, n1 = 0.3, n2 = –i, w1 = 0.1, w2 = 0.2, w3 = 0.3, w4 = 0.4, δ1 = δ2 = δ3 = δ4 = 1.5, η = –0.6, z = 0 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30
DOI: https://doi.org/10.2478/ijmce-2026-0017 | Journal eISSN: 2956-7068
Journal RSS Feed
Language: English
Submitted on: Oct 2, 2025
Accepted on: Feb 27, 2026
Published on: Jun 2, 2026
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Analytical solutions,
KDKK and BK model,
Painlevé analysis,
Hirota bilinear method,
solitons,
symbolic computation work
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2026 Sachin Kumar, Jaionto Karmokar, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT