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 （N, 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.
