Analytical Study of Refractive Index Variation with Light Intensity and the Propagation of Optical Soliton Wave Structures

In the present paper, the Riccati–Bernoulli sub-ODE and modified tanh expansion techniques are applied to the Estévez–Mansfield–Clarkson (EMC) equation, which can model shallow water waves where the wave amplitude significantly impacts the wave speed and represent the propagation of light pulses in a medium where the refractive index changes with the intensity of the light. One can obtain the solutions of hyperbolic, trigonometric, exponential, algebraic, and rational functions from these methods. The effectiveness of these methods is illustrated by the following derived families of solutions for the EMC equation. Moreover, we discuss the physical implications of the soliton solutions about this theme and their application in describing stable wave patterns. To strengthen the findings of the study, Maple 2024 is used to solve the ODE of the problem and determine the set of solutions, as well as Mathematica 13.3.1 to demonstrate the graphical representation of solutions such as 2D plots, 3D plots, and contour plots. The present study presents the effect of the parameters involved in the EMC equation by employing analytical and graphical analysis. We give a detailed discussion of the solutions to these methods. These results extend our understanding of the related dynamics and indicate that the Riccati–Bernoulli sub-ODE approach can be used to solve other nonlinear evolution equations of the same kind.