Dynamical behaviors of different wave structures to the Korteweg–de Vries equation with the Hirota bilinear technique

This work uses the new exponential rational function and a Hirota bilinear technique to solve the Korteweg–de Vries equation. This method generates several exact solutions, most of which are new in various forms of solitons. Some of the obtained soliton solutions are used to find the breather solutions of the equation via an exponential function. The linear stability technique is used to study the stability of the derived solutions using the stability analysis. All of the obtained solutions are stable and exact solutions that have also been put into the equation to ensure their existence which are graphically shown as well. It makes constructive contributions to science by formulating nonlinear wave distribution and the dynamic behavior of wave systems, which are a part of real life. From both a mathematical and physical standpoint, the derived wave function via the new homoclinic approach has been examined and discussed. This research also involves an in-depth examination of nonlinear longwave propagation using the newly discovered soliton. The effects of forces acting on a solitary wave, which has elastic dispersion, on nonlinear advection, dispersion and collision mechanisms are discussed. In this study, which allows making predictions about the environment in which the wave propagates, the physical interpretations of the dynamics affecting the solitaire are supported by graphics. The acquired findings indicate the generality and effectiveness of the used approach.