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(a –d) 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(a –j ) 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(a –h)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(a –d) 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(e –h) 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(a –j) 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.