Rogue waves of the Hirota and the Maxwell-Bloch equations

In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch(H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fibre and further generalize it to the matrix form of the $n$-fold Darboux transformation of this system. This $n$-fold Darboux transformation implies the determinant representation of $n$-th new solutions of $(E^{[n]},p^{[n]}, \eta^{[n]})$ generated from known solution of $(E, p,\eta)$. The determinant representation of $(E^{[n]},p^{[n]} ,\eta^{[n]})$ provides soliton solutions, positon solutions, and breather solutions (both bright and dark breathers) of the H-MB system. From the breather solutions, we also construct bright and dark rogue wave solutions for the H-MB system, which is currently one of the hottest topics in mathematics and physics. Surprisingly, the rogue wave solution for $p\, and\, \eta$ has two peaks because of the order of the numerator and denominator of them. Meanwhile, after fixing time and spatial parameters and changing other two unknown parameters $\alpha$ and $\beta$, we generate a rogue wave shape for the first time.