Exact Solutions Of Nonlinear Fractional Cahn–Allen Equation Aries In Different Nonlinear Physical Phenomenon Using Unified Technique

The nonlinear fractional Cahn–Allen (NFCA) equation provides insight into phase modifications and pattern elaboration in natural structures, clarifying how various states of matter have changed throughout time. By using a unified technique, this study aims to present novel, precise solutions to the NFCA problem. This method has led to the discovery of a broad spectrum of soliton solutions, ranging from dark compactons and dark solitons to periodic kink behaviors, rough wave behaviors, and periodic patterns featuring peaked crests and troughs. It has also unveiled anti-kink periodic behaviors, periodic wave behaviors, bright-dark single solitons, periodic patterns displaying anti-peaked crests and anti-troughs, bright solitons, and compound soliton phenomena. Notably, these accomplishments have been made possible through the utilization of Maple software. These findings are important for fractional nonlinear dynamical models (FNLDMs). In order to clarify many dynamic behaviors that solutions of the NFCA equation display in other scientific fields, such as quantum mechanics, mathematical biology, and plasma physics, contour plots and 3D surface representations of these exact solutions are also featured. Through our investigation using the unified technique, we have unveiled a plethora of novel discoveries. Employing this method has revealed a total of 54 fresh findings. In contrast, Javeed et al. [S. Javeed, S. Saif and D. Baleanu, New exact solutions of fractional Cahn–Allen equation and fractional DSW system, Adv. Differential Equations 2018 (2018) 459] applied the first integral technique, resulting in only eight outcomes. Upon comparing their results with ours, our approach has yielded an additional four out of six innovative results. As far as we are aware, the application of our technique to the NFCA equation has not been previously documented. We anticipate that future research will expand this methodology to include other FNLDMs.