Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov's law of refractive index using the first integral method

The Biswas–Milovic equation in polarization preserving fibres is investigated in this study, along with Kudryashov's law and nonlinear perturbation terms. This nonlinear model may be thought of as a more precise approximation to the nonlinear Schrödinger equation when it comes to explaining wave propagation in the ocean and optical fibres. By applying the first integral method to this nonlinear model, we obtain abundant explicit exact solutions. These exact solutions are adequately presented in the form of distinct complex wave-structures of solutions like dark solitons, singular solitons, and periodic solutions. The obtained soliton solutions are proven to be effective for understanding the dynamics of the exact solutions of this model and demonstrate the authenticity and also the efficiency of the suggested technique. The existence conditions for such solitons are provided. Furthermore, three-dimensional, two-dimensional, and related contour plots are also provided to demonstrate the wave pattern of the acquired solutions.