Complex and Real Optical Soliton Properties of the Paraxial Non-linear Schrödinger Equation in Kerr Media With M-Fractional

In this paper, we use the modified exponential function method in terms of 𝐾𝑓(𝑥) instead of 𝑒𝑓(𝑥) and the extended sinh-Gordon method to find some new family solution of the M-fractional paraxial nonlinear Schrödinger equation. The novel complex and real optical soliton solutions are plotted in 2-D, 3-D with a contour plot. Moreover, the dark exact solution, singular soliton solutions, kink-type soliton solution and periodic dark-singular soliton solutions for M-fractional paraxial nonlinear Schrödinger equation are constructed. We guarantee that all solutions are new and verified the main equation of the M-fractional paraxial wave equation. For existence, the constraint condition is also added.