Stability, modulation instability, and analytical study of the confirmable time fractional Westervelt equation and the Wazwaz Kaur Boussinesq equation

This study delves into the analytical exploration of two pivotal equations, the confirmable Westervelt equation, relevant in acoustic nonlinear phenomena for applications like medical imaging and therapy, and the (2 + 1)-dimensional Wazwaz Kaur Boussinesq equation, providing insights into the unique characteristics of solitons and enriching our understanding of wave dynamics across various optical systems. Utilizing the potent ( G^'/G , 1/G )-expansion analytical method, we construct diverse wave structures and unveil a spectrum of soliton solutions, ranging from trigonometric and hyperbolic functions to rational expressions. Extensive validation using Mathematica software guarantees precision, while dynamic visual representations vividly portray a spectrum of soliton solutions. These solutions encompass a variety of patterns, such as bright solitons, kink solitons with periodic patterns, bell-shaped structures, parabolic structures, and hyperbolic formations. These solutions hold importance in acoustic image processing and the study of wave dynamics across different optical systems. They aid in comprehending the propagation of light in optical systems, thereby providing valuable insights that drive advancements in optical technology and communication. We also investigate modulation instability of the Wazwaz Kaur Boussinesq equation and stability analysis of the confirmable Westervelt equation. Our mentioned expansion scheme proves versatile and applicable across a diverse array of mathematical and physical challenges, showcasing its utility in producing such solutions.