This paper explore the new solitary wave solutions of the Biswas–Milovic equation and resonant nonlinear Schr \(\ddot{o}\) dinger equations with Kerr–Law nonlinearity. In mathematical physics, the Biswas–Milovic equation plays an important role. The dynamics of these solitons are well known to propagate over large distances in a few femto-seconds through these fibres. These dynamics are governed by the nonlinear Schrödinger’s equation. The resonant nonlinear Schrödinger equation can be used to model the propagation of waves in fiber optics. The powerful analytical approach known as the extended simple equation method is employed to explore the solitary wave solutions of the Biswas–Milovic and resonant nonlinear Schr \(\ddot{o}\) dinger equations. These equations are primarily explored in the realm of solitons in nonlinear fiber optics. New dark, kink, anti-kink and singular periodic solitons are secured. Furthermore, 3D surface graphs, contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica. The constructed solutions will helpful to understand the dynamical framework of nonlinear Schr \(\ddot{o}\) dinger equations in the related physical phenomena.
