4.3. Graphical explanation of the time fractional oblique wave solutions
The time fractional oblique wave solutions to the KMN equation given by Eq. (1.1) are reported by executing the gKM and NAEM. Several useful other forms of oblique optical solutions to the KMN equation are obtained via the gKM and NAEM. To illustrate the effectiveness of the gKM and NAEM generated solutions with fractionality and obliqueness, some of the optical solutions attained in this article are displayed graphically along with their physical explanations. The physical explanation of the attained optical solutions, obtained by the gKM specified by (3.11) is described first.
The effects of fractional parameter on the obtained analytic solution\(Q_{1}(x,y=1,t;\tau)\) are provided in Fig. 6 with the particular choice of the free parameters \(p=1\),\(q=1\),\(\ \theta=45\), \(k=1\), \(d=1\),\(a=3.5\),\(\ b_{1}=1\) and \(\theta_{0}=0\). Figure 6(ad) demonstrates the surface profiles of\(\left|Q_{1}(x,y=1,t;\tau,\theta=45)\ \right|\) for \(\tau\) =\(0.25,\ \ 0.50,\ \ 0.75\) and \(1\), respectively. The variations of wave propagation along the \(x\)-axis and \(t\)-axis with different values of \(\tau\), keeping \(\theta=45\) as constant are displayed in Fig. 6(e, f), respectively. It can be seen fromFigs. 6(e) and 6(f) that the surface wave profiles along the \(x\)-axis and \(t\)-axis are changed for\(\tau=0.25\ \text{and}\ 0.50\) and almost unchanged for\(\tau=0.75\ \text{and}\ 1.\) Amplitudes of the wave profiles are found to be nearly identical, but its positions are moved along the\(x\)-axis for the distinct values of \(\tau=0.25,\ \ 0.50,\ \ 0.75\)and \(1\).
Figure 7(aj ) presents the oblique wave propagation of the gKM attained solution, specified by (3.11) by their 3D surface and 2D cross sectional line plots. Figure 7(ah)exhibits the oblique wave propagation of the solution having the time fractional derivative for distinct values of the wave obliqueness\(\theta=15,\ 30{,\ 45},\ 75,105,\ 120{,\ 135},\)and \(165\), respectively, with the fractional parameter\(\tau=0.75\), space \(y=1,\) and the fixed values of the remaining parameters, namely \(p=1\), \(q=1\), \(d=1\), \(k=1\),\(a=3.5\), \(b_{1}=1\) and \(\theta_{0}=0\). It is observed fromFig. 7(ad) that the oblique waves are propagating in the same direction in which the amplitudes are increasing with the increase of \(\theta\) for \(0<\theta\leq 80\), whereas the amplitudes are decreasing with the increase of \(\theta\) for\(80<\theta<90\). On the other hand, Fig. 7(eh) shows the propagation of oblique waves in the opposite direction with the increase of \(\theta\) for\(90<\theta<180\). In such cases, the wave amplitudes are increasing with the increase of \(\theta\) for\(90<\theta<180\). Amplitudes of the oblique wave profiles are clarified with the 2D cross sectional line plots (see Fig. 7(i, j )). However, the variation in the oblique wave propagation with wave obliqueness of \(\theta\) for \(75<\theta<90\) and\(165<\theta<180\) are not included in this paper for the sake of brevity. However, it can be observed from Fig. 7(aj) that the wave profiles have been changed significantly with the increase of obliqueness. It is clearly visible fromFig. 7(i, j) that the oblique wave amplitude is maximum at\(\theta=75\) and \(\theta=135\) along with its \(x\)-axis (\(-5\leq x\leq 5\ \)) and \(t\)-axis (\(0\leq t\leq 3\)) within the wave directions \(0<\theta<90\) and\(90<\theta<180\), respectively. Such types of wave phenomena are known as fission-fusion interaction phenomena.
In order to examine the dependence of the wave obliqueness with the axis\(t\) \((0<t\leq 5)\), 3D and 2D graphs are prepared for\(\left|Q_{1}(x=1,y=1,t;\tau=0.75,\theta)\ \right|\)for different values of obliqueness (\(\theta\)), fractional parameter (\(\tau=0.75\)) and space (\(x=1,y=1\)) and are displayed inFig. 8(a, b) . It is seen from Fig. 8(b) that the wave amplitude attains its maximum value at \(\theta=75\) within\(0<\theta<90\) and that is maximum again at \(\theta=135\)within \(90<\theta<180\) along with its \(t\) axis (\(0\leq t\leq 3\)). In order to show the effects of fractional value (\(\tau\)), 3D and 2D graphs of\(\left|Q_{1}(x=1,y=1,t=2;\tau,\theta)\right|\) are constructed within the wave obliqueness \(0<\theta<180\) and are displayed inFig. 8(c, d) . It can be perceived from Fig. 8(d) that the amplitudes are varied for \(\tau\) \(=0.25\ \text{and\ }0.50\ \)and stable for \(\tau=\) \(0.75\ \text{and}\ 0.95\ \) that is mentioned earlier in this section. Thus, it is reasonable to say from Fig. 8(c, d) that the obtained solution is varied highly at \(\tau=0.25\)among the values of the fractional parameter, namely\(0.25,\ 0.50,\ 075\) and \(0.95\). In order to ensure the effects of fractional parameter on NAEM extracted solutions, the 3D graphs are constructed for\(\left|Q_{1}(x,y=1,t;\tau,\theta=45)\ \right|\) in thext plane with fractionality\(\tau=0.25,\ \ 0.5,\ \ 0.75,\ \ 1\) and are pictured Fig. 9(a-d) , respectively. 2D line plots are also constructed to present the variability of the solution presented through Fig. 9(a-d) along the\(x\)-axis at \(t=2\), and along the \(t\)-axis at \(x=1\) and are exposed in Fig. 9(e, f), respectively. In such cases, the identical phenomena have been observed as that of the gKM attained solutions. However, the modulus plot of the NAEM obtained solution,\(\left|Q_{1}(x,y=1,t;\tau,\theta)\ \right|\) represents a periodic soliton.
In a similar way to show the effectiveness of the oblique wave parameter on the NAEM attained solution, the 3D graphs of\(\left|Q_{1}(x,y=1,t;\tau=0.75,\theta)\ \right|\) are prepared in the xt plane under the wave obliqueness of\(\theta=15,\ 30{,\ 45},\ 75,105,\ 120{,\ 135}\)and \(165\), respectively, keeping the other free parameters remain fixed, which are indicated in Fig. 10(a-h) . The 3D graphs show U-shaped periodic solitons. The numbers of U-shaped periodic wave are decreasing with the increase of wave obliqueness\(\theta=15,\ 30{,\ 45}\), and \(75\) within\(0<\theta<90\), as illustrated in Fig. 10(a-d) , respectively, whereas the numbers of U-shaped periodic wave are increasing with the increase of the wave obliqueness\(\theta=105,\ 120{,\ 135}\) and \(165\) within\(90<\theta<180\), as exposed in Fig. 10(e-h),respectively. The 2D cross sectional line plots of\(\left|Q_{1}(x,y=1,t;\tau=0.75,\theta)\ \right|\) along the\(x\)-axis at \(t=2\), and along the \(t\)-axis at \(x=1\) are presented in Figs. 10(i) and 10(j), respectively, to show the numbers of above U-shaped periodic wave behaviors. Moreover,Figs. 9 and 10 show the identical phenomena as that of the gKM obtained solutions. However, the NAEM obtained solution\(\left|Q_{1}(x,y=1,t;\tau,\theta)\ \right|\) represents a U-shaped periodic wave soliton. It is suggestive that the results presented in this article would be extremely helpful for analyzing the nature of the plane wave phenomena in nonlinear optical fiber communication systems, and telecommunication engineering.