In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.

In the present paper, high-order nonlinear Schrödinger equations in non-Kerr law media with different laws of nonlinearities are studied. In this respect, after considering a complex envelope and distinguishing the real and imaginary portions of the models, describing the propagation of solitons through nonlinear optical fibers, their soliton solutions are obtained using the well-organized new Kudryashov method. It is believed that the new Kudryashov method provides an effective mathematical tool to look for soliton solutions of high-order nonlinear Schrödinger equations.