4.1. Graphical explanation of the gKM obtained solutions
The 3D and 2D graphical illustrations of the attained solutions to the
KMN equation considering particular values of the free parameters are
presented to show the solution’s behavior. Moreover, the 3D surface
graphics are visualized to show the spatiotemporal variation of the
obtained optical wave solutions. Surface plots for the real and
imaginary parts and modulus of the optical solution\(Q_{1}\left(x,y=1,t\right)\) are shown in Fig.
1(a –c) , respectively. Each of the real and imaginary parts of\(Q_{1}\left(x,y=1,t\right)\) specifies a periodic soliton. On the
other hand, the modulus of the mentioned solution indicates a dark
soliton (see Fig. 1(a) ). Such types of behaviors mentioned
above can be confirmed from their 2D cross sectional line plots at\(t=0\), which are depicted in Fig. 1(d –f),respectively. Figure 2(a –c) displays also the 3D
graphical representation of the real and imaginary parts, and modulus of
the optical soliton \(Q_{3}\left(x,y=1,t\right)\), whereasFig. 2(d –f) displays, respectively, 2D line plots of
the real and imaginary parts, and modulus of the mentioned solution at\(t=0\). The real and imaginary parts of\(Q_{3}\left(x,y=1,t\right)\) demonstrate singular solitons, which
are shown in Fig. 2(a, b), respectively, whereas\({|Q}_{3}\left(x,y=1,t\right)|\) also designates a singular soliton
(see Fig. 2(c) ). Such types of singular soliton solutions are
confirmed by their 2D plots at \(t=0\) shown in Fig. 2(d–f),respectively. The remaining of the gKM extracted optical solutions
represent the identical physical characteristics that we have mentioned
above.