Solitary waves are referred to as localized gravity waves whose coherence are maintained and thus ensure their visibility through properties of nonlinear hydrodynamics. A solitary wave usually possess a finite amplitude and also propagates with a constant speed as well as constant shape. Moreover, solitons are a special class of solitary waves possessing an elastic scattering property: they are found to constantly retain their shapes along with speed even after experiencing collision with one another. The results secured for (3 + 1)D-gZKe (3) in this paper are solitonic as can be seen in their various graphical representations by considering some particular cases of n. Therefore, we depicted the dynamics of bright soliton solution (36) with
Fig. 1 in the structure of 3D plot, density plot and 2D plots with dissimilar values of the employed constant parameters as ϵ=1, κ=1, γ=10, α=1, a=105, b=0.5, c=20, σ=1 for n=1 with −5≤y, z≤5 where t=0 and x=0.5. Besides,
Fig. 2 further gives the streaming pattern of (36) in 3D, density alongside 2D plots for n=2 with unalike values of the included parameters as ϵ=0, κ=1, γ=10, α=1, a=105 b=1.5, c=20, σ=1 where −5≤y, z≤5 as well as t=1 and x=2. Furthermore, bright soliton solution (37) is represented with 3D, density and 2D plots form in
Fig. 3 with diverse values of the included parameters as ϵ=1, κ=1, γ=10, α=1, a=105 b=0.4, c=20, σ=1 for n=1 with −5≤y, z≤5 where t=1 and x=1. Solitary wave solution (37) is also exhibited graphically in
Fig. 4 in 3D plot, density plot alongside 2D plot with varying values of the parameters as ϵ=0, κ=1, γ=10, α=1, a=105 b=1.8, c=20, σ=1 for n=2 where −5≤y, z≤5 with t=−1 and x=2. We further depicted soliton solution (48) via 3D plot, density plot together with 2D plot in
Fig. 5 with diverse parameter values as ϵ=0, β=0, κ=4, γ=10, α=8, a=−15, b=5, c=300, σ=1 for n=1 where −5≤y, z≤5 with t=5 and x=1.
Fig. 6 shows the streaming behaviour of (48) with 3D plot, density plot as well as 2D plot format with varying values of the involved parameters as ϵ=0, κ=−40, β=0, γ=100, α=80, a=10 b=5, c=300, σ=10 for n=2 where −5≤y, z≤5 with t=10 and x=−10. Singular soliton solution (49) is depicted with 3D plot, density plot and 2D plot in
Fig. 7 with unalike values of parameters as ϵ=1, κ=1, β=0.6, γ=10, α=1, a=105 b=20, c=20, σ=10 for n=1 where −5≤y, z≤5 with t=1 and x=2. Moreover, we displayed the dynamics of the singular solution in
Fig. 8 in 3D, density and 2D plots for n=2 with varying values of parameters as ϵ=0, κ=1, β=0.9, γ=10, α=1, a=105 b=20, c=20, σ=1 where −5≤y, z≤5 with t=3 and x=−4. Cnoidal wave solution (59) of (3) which is a periodic soliton solution is represented with 3D, density together with 2D plots in
Fig. 9 with dissimilar values of parameters as ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 with −10≤y, z≤10 whereas t=1 and x=0. Besides,
Fig. 10 depicted (59) further with 3D plot, density plot and 2D plot with diverse values of parameters as ϵ=1, κ=1, β=1.01, γ=0, α=1, p=−1 q=1, r=1, σ=1 with −15≤y, z≤15 where t=1 and x=1. Snoidal wave solution (62) (periodic soliton) is considered and depicted in
Fig. 11 in 3D, density and 2D format with varying values of parameters as ϵ=1, κ=1, β=1, γ=0, α=0, p=−1 q=1, r=1, σ=1 where −6≤y, z≤6 with t=1 and x=2. In addition, periodic soliton solution (62) is further represented in
Fig. 12 via 3D plot, density plot as well as 2D plot with dissimilar values of the parameters as ϵ=1, κ=1, β=1.01, γ=0, α=1, p=−1 q=1, r=1, σ=1 where −10≤y, z≤10 alongside t=−1 and x=6. The depiction of dnoidal wave solution (65) is given via
Fig. 13 in 3D, density as well as 2D plots with unalike parametric values ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 whereas −5≤y, z≤5 together with variables t=1.3 and x=0. Finally, we exhibited the wave influence of solitary wave solution (65) in
Fig. 14 via 3D plot, density along with 2D plots using diverse involved parameters with assigned values ϵ=1, κ=1, β=1, γ=0, α=1, p=−1 q=1, r=1, σ=1 whereas −3≤y, z≤3 where we have variables t=−0.4 with x=1.