Fractional Soliton and Semirational Solutions of a Fractional Two-Component Generalized Hirota Equation

The Darboux transformation (DT) and generalized DT (GDT) have played important roles in constructing multisoliton solutions, rogue wave solutions, and semirational solutions of integrable systems. The main purpose of this article is to extend the DT and GDT to a conformable fractional two-component generalized Hirota (TCGH) equation for revealing novel dynamic characteristics of fractional soliton and semirational solutions. As for the main contributions, specifically, we propose a fractional form of the TCGH equation, provide the associated fractional Lax pair, and obtain fractional soliton and semirational solutions of the fractional TCGH equation by constructing its fractional DT and GDT. In addition, we find that the dominant role of fractional order leads to new dynamic characteristics of the obtained fractional soliton and semirational solutions, mainly including a certain degree of tilt of wave crests and the variations in velocities and wave widths over time during propagation, which are not possessed by the corresponding integer-order TCGH equation. Meanwhile, this study predicts the deceleration propagation of solitons in fractional dimensional media and brings the possibility of exploring the asymmetric regulation mechanism of rogue waves from the perspective of fractional-order dominance.