5. Discussion and concluding remarks
As mentioned earlier in the literature section that some researchers have reported bright, dark, singular soliton solutions to the KMN equation through diverse methods [19, 20, 33–39]. Consequently, the researchers picked up only the bright, dark, singular soliton solutions to an integer order KMN equation without considering obliqueness. However, in this article, we have explored some new types of bright, dark, U-shaped periodic, and singular shaped soliton solutions with different amplitudes to an integer and fractional KMN equation considering wave obliqueness via the gKM and NAEM. The employed methods also have extracted some new oblique wave solitons to the studied equation. It has been demonstrated that the wave profile is changed with the changing of obliqueness and fractionality. The 3D graphical illustrations and 2D cross sectional graphics for different values of various parameters are represented to understand the effects of the changing values of the parameters over the solutions. In comparison with the attained solutions [19, 20, 33–39], to the best of authors’ knowledge, the generated bright, dark, U-shaped periodic, and singular soliton wave solutions are new in conformable derivative and obliqueness senses, which are not reported in previously published articles. It is remarkable to perceive that most of the investigated solutions in this article have diverse structures over the solutions available in the literature in the wave propagation obliquely and the executed methods are completely new for the studied equation. Therefore, the acquired optical solutions may illuminate the researchers for further studies to explain pragmatic phenomena of the wave approaching obliqueness in the field of fiber optics and optical communications. This work provides evidence that the gKM and NAEM are suitable for acquiring new optical soliton features in any physical system with or without fractional and obliqueness conditions.