Researchers have developed new mathematical methods to analyze the solitary wave solutions of the time-fractional equal width (EW) equation. This work addresses the complexity of modeling nonlinear phenomena with memory and non-local effects, which are intrinsic characteristics of systems described by fractional differential equations. The EW equation is a fundamental model in physics for describing wave propagation in nonlinear dispersive media, and its extension to fractional derivatives allows for a more precise description of complex systems.

The study focused on applying techniques such as the improved exponential function method and the extended tanh-coth function method to obtain exact analytical solutions. These methods allow the fractional partial differential equation to be transformed into a nonlinear ordinary differential equation, which can then be solved to find the solitary waveform solutions. The solutions obtained include bright and dark solitary waves, which are crucial for understanding the dynamics of energy and information in various physical systems.

The relevance of this work lies in its ability to provide a more solid theoretical foundation for the study of wave phenomena in fields such as nonlinear optics, plasma physics, and fluid mechanics, where fractional effects can play a significant role. Understanding these fractional solitary waves is essential for the design and optimization of devices and systems that exploit wave propagation, as well as for the interpretation of experimental observations in complex environments.