Solitonic Dynamics in the Westervelt Equation for Ultrasound Imaging Through the Exponential Expansion Method

ABSTRACT Ultrasonic propagation, which transmits the properties of molecules or particles through mechanical vibrations, is the focus of this work. Over the past two decades, significant advancements have been made in the numerical modeling of medical ultrasound. This progress has created numerous opportunities for related technologies, including medical ultrasound imaging (MUI), ultrasound imaging (UI), photoacoustic imaging, and therapeutic ultrasound (TU). These medical ultrasound applications have been studied using various new mathematical models and algorithms. Furthermore, modeling acoustic wave transmission in three‐dimensional large‐scale domains has become feasible due to rapid advancements in computing technology. This study presents a comprehensive literature review and technical assessment of current developments in medical ultrasound modeling. We will also discuss emerging challenges. We employ the exponential expansion method (EEM) to derive analytical solutions. The solutions obtained include topological kinks, singular kinks, rational solutions, non‐topological bell‐type solutions, solitons, periodic solutions, and solitary wave solutions. Additionally, this article presents a proof of an existence theorem for the mathematical model under investigation.