1.2. Objective of the study
The aim of this study is to adopt the gKM and newly developed NAEM to secure some new optical solutions, namely dark, bright, periodic U-shaped and singular soliton solutions to the KMN equation that can be of great importance in the field of fiber optics and optical communications. Furthermore, our intension is to implement the conformable derivative and wave oblique complex transform in coordination with the mentioned methods of interest to the KMN equation for obtaining new optical solutions in the sense of fractional derivative and wave obliqueness.
1.3. Governing equation
The dimensionless form of (2+1)-dimensional KMN equation is [33–38]
\(iQ_{t}\ +\ pQ_{\text{xy}}\ +\ iqQ\ (QQ_{x}^{*}\ -\ Q^{*}Q_{x})\ =\ 0.\)(1.1)
The KMN equation specified by (1.1) was introduced by Kundu et al. [30], which is a new extension of the well-known NLS equation. In Eq. (1.1), \(x\) and \(y\) stand for the spatial variables while \(t\)designates the temporal variable. The dependent variable\(Q(x,\ y,\ t)\) represents nonlinear wave envelope, where the asterisk denotes the complex conjugate of \(Q\). The first term in Eq. (1.1) stands for denoting the temporal evolution of the wave followed by the dispersion term that is given by the coefficient of \(p\). The constant\(q\) ensures the existence of the different case of nonlinearity media which does not fall into the conventional Kerr law nonlinearity or any known non-Kerr law media [31]. The nonlinear term in this equation accounts for “current–like” nonlinearity that stems from chirality [33].