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Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

In this paper, we considered the nonlinear Schrodinger equation modeling cubic optical solitons in a polarization-preserving fiber with Kerr law. Using the improved t a n (φ / 2) -expansion method, a powerful and effective method, we constructed exact solutions including the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The obtained solutions are entirely new and can be considered as generalization of the existing results in the ordinary derivative case. [ABSTRACT FROM AUTHOR]

In this paper, we studied a nonlinear wave equation that models the propagation of optical solutions in a weakly nonlocal and parabolic competing nonlinear medium. The exact traveling wave type solutions formulated in hyperbolic functions, rational and trigonometric functions multiplied by some exponential functions to the governing equation are determined explicitly by the extended form of the Kudryashov method. We have examined the behavior of the modulation instability (MI) growth rate. To substantiate the stability of the obtained dark and bell-shaped solitons, we use the split-step Fourier method. In addition, the conservation laws describing significant physical concepts of this equation are examined. Compared to the obtained results with Younis et al. (J Nonlinear Opt Phys Mater 24(04):1550049, 2015), Zhou et al. (Optik 124(22):5683–5686, 2013; Proc Rom Acad Ser A 16(2):152–159, 2015) and Akinyemi et al. (Optik 230:166281, 2021), we have shown the propagation of the solitonic waves which sometime tilt from right to left with stable shape. [ABSTRACT FROM AUTHOR]