Remark 1. For the case of conformable derivative
Consider the time fractional (2+1)-dimensional KMN equation [33–38]:
\(iD_{t}^{\tau}Q\ +\ pQ_{\text{xy}}\ +\ \text{iqQ}\ \left(QQ_{x}^{*}\ -\ Q^{*}Q_{x}\right)=\ 0,i=\sqrt{-1}\), (3.4)
where \(D_{t}^{\tau}\) denotes the conformable derivative of fractional order \(\tau\) with respect to \(t\), and \(0<\tau\ \leq\ 1\). When we substitute \(\tau=1\) in Eq. (3.4), the time fractional KMN equation is converted to the integer order KMN equation specified by Eq. (1.1). Assume the following complex transformation to the time fractional (2+1)-D KMN equation:
\(Q(x,\ y,\ t;\tau)\ =\ U(\xi)e^{\text{iη}(x,y,t)}\),\(\xi=\ l_{1}x\ +\ l_{2}y\ \ v\frac{t^{\tau}}{\tau}\) and\(\eta=-h_{1}x-h_{2}y+\omega\frac{t^{\tau}}{\tau}+\ \theta_{0}\). (3.5)
With the help of Khalil’s definition [40], feasible properties [41] of the conformable derivative, and a complex transformation specified by Eq. (3.5), Eq. (3.4) is converted to the identical ODE specified by Eq. (1.3). Then, the gKM and NAEM are applied to the time fractional KMN equation. Sixty optical wave solutions are explored, which are new in the sense of conformable fractional derivative. Some applications of conformable derivative to NPDEs are available in refs. [42–44]. For the sake of straightforwardness, we have included four solutions obtained via the gKM and the NAEM in the sense of conformable fractional derivative. Optical wave solutions of the time fractional KMN equation by the gKM are given below:
\(Q_{1,2}\left(x,\ y,\ t;\tau\right)\ =\ \frac{\mp\left(\frac{1}{2}\frac{pb_{1}l_{1}l_{2}\ \ln\left(a\right)}{\sqrt{\text{pq}h_{1}l_{1}l_{2}}}\frac{1}{\left(1+da^{\xi}\right)}-b_{1}\ln\left(a\right)\ \sqrt{\frac{pl_{1}l_{2}}{qh_{1}}}\frac{1}{\left(1+da^{\xi}\right)^{2}}\right)}{\frac{b_{1}}{\left(1+da^{\xi}\right)}}\times e^{i\left(-h_{1}x\ -h_{2}y\ -\frac{1}{2}p\left(l_{1}l_{2}\left(\ln\left(a\right)\right)^{2}+2h_{1}h_{2}\right)\frac{t^{\tau}}{\tau}+\ \theta_{0}\right)}\), (3.6)
where\(\xi=\ l_{1}x\ +\ l_{2}y+p\ (l_{1}h_{2}\ +\ l_{2}h_{1})\frac{t^{\tau}}{\tau}\).
Optical wave solutions of the time fractional KMN equation by the NAEM are given below:
For Family-I : When \(\beta^{2}-4\alpha\sigma\ <0\) and\(\sigma\neq 0\),
\(Q_{1,2}\left(x,\ y,\ t;\tau\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}+\frac{\sqrt{4\text{ασ}-\beta^{2}}}{2\sigma}\tan\left(\frac{\sqrt{4\text{ασ}-\beta^{2}}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\text{ασ}l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)\frac{t^{\tau}}{\tau}\ +\ \theta_{0}\right)}\), (3.7)
where\(\xi=\ l_{1}x\ +\ l_{2}y+p\ (l_{1}h_{2}\ +\ l_{2}h_{1})\frac{t^{\tau}}{\tau}\).