Remark 2. For the case of conformable derivative and
oblique wave solutions
Assume the following oblique wave transformation [45]:
\(Q(x,\ y,\ t;\tau)\ =\ U(\xi)e^{\text{iη}(x,y,t)}\), (3.8)
where\(\xi=\ \cos\left(\theta\right)x\ +\sin\left(\theta\right)y\ \ v\frac{t^{\tau}}{\tau}\), and\(\eta=-k\cos\left(\theta\right)x-\operatorname{k\ sin}\left(\theta\right)y+\omega\frac{t^{\tau}}{\tau}+\ \theta_{0}\)with\(\cos^{2}\left(\theta\right)+\sin^{2}\left(\theta\right)=1\) .
Utilizing the oblique wave transformation specified by Eq. (3.8) into
conformable time fractional KMN equation specified by (3.4), the real
and imaginary parts of the equation can be put in the following form:
\(\alpha\sin\left(\theta\right)U^{\prime\prime}\ -\ (\omega+\ \alpha k^{2}\sin\left(\theta\right))\ U\ \ 2\beta\text{kU}^{3}\ =\ 0\ \)(3.9)
and \(v\ =\ -pk\sin\left(2\theta\right)\), (3.10)
where a prime denotes the ordinary derivative with respect to \(\xi\)and \(\theta\) presents the wave obliqueness.
Recently, some scholars [45–47] adopted the oblique wave
transformation to the fractional PDEs for determining oblique wave
solutions. After using an oblique wave transformation specified by Eq.
(3.8) into the KMN equation, the gKM and NAEM are applied to Eq. (3.9).
As outcomes, sixty optical wave solutions are produced in total, which
are new in the sense of obliqueness. Recently, Hafez et al. [46]
investigated the plane wave propagation in the xy-plane by a
simple schematic diagram, where they emphasized the importance of
obliqueness with fractional temporal evolution to some PDEs.
To explain the physical behaviors of fractionality and wave obliqueness,
four oblique wave solutions are included by means of the gKM and NAEM.
Oblique optical wave solutions to the time fractional (2+1)-D KMN
equation are explored via the gKM as
\(Q_{1,2}\left(x,\ y,\ t;\tau,\theta\right)\ =\ \frac{\mp\left(\frac{1}{2}\frac{pb_{1}\cos\left(\theta\right)\sin\left(\theta\right)\ \ln\left(a\right)}{\sqrt{\text{pq}\operatorname{kcos}\left(\theta\right)\cos\left(\theta\right)\sin\left(\theta\right)}}\frac{1}{\left(1+da^{\xi}\right)}-b_{1}\ln\left(a\right)\ \sqrt{\frac{p\sin\left(\theta\right)}{\text{qk}}}\frac{1}{\left(1+da^{\xi}\right)^{2}}\right)}{\frac{b_{1}}{\left(1+da^{\xi}\right)}}\times\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e^{i\left(-\operatorname{kcos}\left(\theta\right)x\ -k\sin\left(\theta\right)y\ -\frac{1}{2}p\cos\left(\theta\right)\sin\left(\theta\right)\left(\left(\ln\left(a\right)\right)^{2}+2k^{2}\right)\frac{t^{\tau}}{\tau}+\ \theta_{0}\right)}\).
(3.11)
On the other hand, oblique optical wave solutions to the time fractional
(2+1)-D KMN equation are explored via the NAEM as follows for Family-I:
When \(\beta^{2}-4\alpha\sigma\ <0\) and \(\sigma\neq 0\),
\(Q_{1,2}\left(x,\ y,\ t;\tau,\theta\right)\ =\ \pm\frac{1}{2q\operatorname{kcos}\left(\theta\right)}\sqrt{\text{pqk}\cos^{2}\left(\theta\right)\sin\left(\theta\right)}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}+\frac{\sqrt{4\text{ασ}-\beta^{2}}}{2\sigma}\tan\left(\frac{\sqrt{4\text{ασ}-\beta^{2}}}{2}\xi\right)\right)\right)\times e^{i\left(-k\cos\left(\theta\right)x\ -k\sin\left(\theta\right)y\ +\cos\left(\theta\right)\sin\left(\theta\right)\ \left(\frac{p}{2}\left(4\text{ασ}-\beta^{2}-2k^{2}\right)\right)\frac{t^{\tau}}{\tau}\ +\ \theta_{0}\right)}\),
(3.12)
where\(\xi=\ \cos\left(\theta\right)x\ +\ \sin\left(\theta\right)y+pk\sin\left(2\theta\right)\frac{t^{\tau}}{\tau}\)for the both solutions given by Eqs. (3.11) and (3.12).
Remark 3. It is remarkable to note here that all the obtained
optical solutions of the studied equation are verified by putting them
back to the original equation and found them accurate.