Table 1
Particular cases of equation (1) found in ocean dynamics and related fields.
| Variation of κi | PDEs | Special form |
|---|---|---|
| κ1 = κ2 = 1, κ3 = κ4 = κ5 = κ6 = 0, z = x | ut + uxxx + 6uux = 0 | KdV equation [26] |
| κ1 = 1, κ2 = a1, κ3 = a5, κ5 = 0, κ4 + κ6 = 0, z = x | ut + a1 uxxx + + a3uxxxxx + a4vy + a5uxxy + a6(uxv + uvx) + a7(uuxx)x + , 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 = x | ut + + + b3ux + b4uy + b5Vyy = 0, Vx = u | (2+1)-D gBK equation [28] |
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.
Table 2
Visual analysis of the KDKK and BK equations for the free parameters.
| Fig. No. | Free parameters with intervals |
|---|---|
| Fig. 1 | κ1 = 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. 2 | κ1 = 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. 3 | κ1 = 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. 4 | κ1 = 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. 5 | κ1 = 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. 6 | κ1 = 1, κ2 = 1, κ3 = 1, κ4 = –1, κ5 = 1, κ6 = 1, m1 = i, m2 = m3 = 1, n1 = n3 = 1, w2 = 1, δ1 = δ2 = 1, , z = 3 for –30 ≤ x ≤ 30, –30 ≤ y ≤ 30 |
| Fig. 7 | κ1 = 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. 8 | κ1 = 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. 9 | κ1 = 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. 10 | κ1 = 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 |
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.