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 fiber 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 solutions of $(E^{\left[n\right]},p^{\left[n\right]},{\eta }^{\left[n\right]})$ generated from the known solution of $(E,p,\eta )$. The determinant representation of $(E^{\left[n\right]},p^{\left[n\right]},{\eta }^{\left[n\right]})$ 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 a bright and dark rogue wave solution 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 the time and spatial parameters and changing two other unknown parameters $\alpha$ and $\beta$, we generate a rogue wave shape.

• Received 26 October 2012
• Corrected 6 May 2013

DOI:https://doi.org/10.1103/PhysRevE.87.012913