Novel methods for exploring fractional solitary waves in the improved mKdV equation with time-fractional derivative and space-time dispersion

This paper introduces a new approach for obtaining solitary wave solutions, specifically periodic or kink types, for nonlinear evolution problems. The proposed method formulates solutions in a rational form, with the numerator and denominator being linear combinations of either sine and cosine functions or sinh and cosh functions. The effectiveness of this method is evaluated through examining solutions to the improved modified Korteweg-de Vries (mKdV) equation, which includes a space-time dispersion term and Atangana conformable fractional derivative. Graphical analysis, including 2D and 3D plots, is conducted to understand the propagation of the obtained solutions and how it is affected by changing the value of the fractional derivative. The results obtained in this work are anticipated to have significant implications in the field of nonlinear physics and its applications in science and engineering.