Exploration of the Soliton Solution of Nonlinear Equations by Modified Simple Equation Method

Abstract

This study addresses the challenge of derivig exact and soliton solutions for nonlinear evolution equations, which are essential for understanding complex phenomena in science, applied mathematics, and mathematical physics. Nonlinear evolution equations such as the ubiquitous Korteweg-de Vries equation and the Hirota-Ramani equation were studied due to their significant applications in modeling wave propagation, fluid dynamics, optics, plasma physics, and nonlinear dynamical systems. The modified simple equation method was employed, a strong method known for its consistency, efficiency, and effectiveness in deriving traveling wave solutions. Using this method, we obtained various solution types, including bell-shaped solitons, anti-bell-shaped solitons, kink-shaped solutions, pulse-shaped solitons, and soliton solutions. These results enhance our ability to predict system behavior under diverse conditions and extend the understanding of nonlinear systems. The novelty of this work lies in the improved applicability and performance of the modified simple equation method compared to existing methods, offering a more comprehensive framework for analyzing nonlinear evolution equations and advancing prior research in the field.