Advanced techniques for analyzing solitary waves in circular rods: a sensitivity visualization study

This study evaluates the effectiveness of the New Sub-Equation Method and the Modified Khater Method in solving the longitudinal wave equation (LWE), a critical nonlinear partial differential equation in mathematical physics. In mathematical physics, the longitudinal wave equation arises with dispersion caused by transverse Poisson’s effect in a circular rod. Employ wave transformation to reformulate the model into ordinary differential equation. The research successfully identifies a range of exact traveling wave solutions, including kink, lump, anti-kink, combined bright-dark, periodic, and U-shaped solitons. Detailed graphical visualizations, produced using Wolfram Mathematica and MATLAB, offer in-depth insights into the internal structures and dynamic behaviors of these solutions. These visualizations reveal the impact of various parametric values and wave velocities on the wave profiles, presented through three-dimensional, two-dimensional, and contour plots. Sensitivity analysis further elucidates the impact of parameter variations on system behavior, revealing the methods’ robustness and adaptability. These types of solitary waves are very important due to their flexibility in the long-distance optical communication. The study demonstrates the methods’ efficacy in solving complex nonlinear evolution equations and enhances the understanding of solitary wave solitons in diverse contexts.