In this paper, considering the Hirota and the Maxwell-Bloch (H-MB) equations which are governed by femtosecond pulse propagation through a two-level doped fiber system, we construct the Darboux transformation of this system through a linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions (both bright and dark breathers) of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.
In this paper,we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB)equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant representation of Darboux transformation is used to derive soliton solutions,positon solutions to the IH-MB equations.
The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy(BTH).As a specific example of exponential solutions of the BTH,the authors consider a regular solution for the(1,2)-BTH with a 3×3-sized Lax matrix,and discuss some geometric structures of the solution from which the difference between the(1,2)-BTH and the original Toda hierarchy is shown.After this,the authors construct another kind of Lax representation of(N,1)-BTH which does not use the fractional operator of Lax operator.Then the authors introduce the lattice Miura transformation of(AT,1)-BTH which leads to equations depending on one field,and meanwhile some specific examples which contain the Volterra lattice equation(a useful ecological competition model)are given.
