Fig 1.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (20), when α = 1, β = −1, l = 1, γ = −1, ν = 1, μ = −1, A = 0, B = 1.
Fig 2.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (21), when α = 1, β = 10, l = 1, γ = 1, ν = 4, μ = 5, A = 0 and B = 1.
Fig 3.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (23), when α = 1, β = −0.5, l = 1, γ = −1, ν = 1, μ = −1, A = 1, B = 0.
Fig 4.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (25), when α = 1, β = −0.5, l = 1, γ = −1, ν = 1, μ = −1, A = 1, B = 0.
Fig 5.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (28), when α = 1, β = −0.5, l = 1, γ = −1, ν = 1, μ = −1, A = 1, B = 0.
Fig 6.
(a) 3D surface plot, (b) density plot and (c) 2D combined plots of (29), when α = −0.01, β = −0.01, γ = 6, ν = 2, μ = 2, A = 1, B = 0.
Table 1.
Comparison between Aslan [32] solutions and our obtained solutions for the mBBM equation.
Table 2.
Comparison between Naeem et al. [43] solutions and our results for the mBBM equation.
Fig 7.
Anti-kink wave profile of (20): Effects of the parameter β on the wave profile for the values of α = −0.5, l = 1, ω = 5, γ = 1, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 8.
Kink wave profile of (20): Effects of the parameter β on the wave profile (20) for the values of α = −0.5, l = 1, ω = 5, γ = −1, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 9.
Anti-Kink wave profile of (20): Effects of the parameter α on the wave profile (20) for the values of β = 1, l = 1, ω = 5, γ = −1, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 10.
Kink wave profile of (20): Effects of the parameter α on the wave profile (20) for the values of β = −1, l = 1, ω = 5, γ = 1, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 11.
Kink wave profile of (20): Effects of the parameter α on the wave profile (20) for the values of β = −1, l = 1, ω = 5, γ = −1, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 12.
Kink wave profile of (20): Effects of the parameter γ on the wave profile (20) for the values of α = 1, β = −1, l = 0.2, ω = 5, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 13.
Anti-Kink wave profile of (20): Effects of the parameter γ on the wave profile (20) for the values of α = 1, β = 1, l = 0.2, ω = 5, μ = 2, ν = 3, A = 0, B = 1 at time t = 1.
Fig 14.
(a) Phase portrait with equilibria at of the system (34), and (b) 3D plot of the Hamiltonian (35), for α = 1, β = −1, l = 1, ω = 1, γ = 1.
Fig 15.
(a) Phase portrait with equilibria at of the system (34), and (b) 3D plot of the Hamiltonian (35), for α = −1, β = −1, l = 1, ω = 2, γ = −1.
Fig 16.
(a) Phase portrait with equilibrium at (0, 0) of the system (34), and (b) 3D plot of the Hamiltonian (35), for α = −1, β = −1, l = 1, ω = 1, γ = 1.
Fig 17.
(a) Phase portrait with equilibrium at (0, 0) of the system (34), and (b) 3D plot of the Hamiltonian (35), for α = −1, β = −1, l = 1, ω = 1 γ = −1.