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(ac) , 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(df),respectively. Figure 2(ac) 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(df) 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.