Darboux transformation, soliton solutions of a generalized variable coefficients Hirota equation

It is known that the varible coefficients Hirota equations have been widely studied in the propagating pulses amplificat or absorpt and the yield of supercontinuum in inhomogeneous optical fibers. In this paper, a new generalized case is considered, i.e. iu t + α(t)u xx + iβ(t)u xxx + 3iβ(t)γ|u| 2 u x + α(t)γ|u| 2 u + δ(t)u = 0 , where i = √ −1 indicates the imaginary unit and u is a complex function with the variables (t, x) . In particular, for α, β, γ are all constants, δ = 0 , the classical Hirota equation will be recovered. Firstly, we constructed the classical and generalized Darboux transformations of the equation, respectively. Next, multisoliton solutions are obtained based on classical Darboux transformation, as well as, rogue wave solutions are found by generalized Darboux transformation. Finally, we discussed the evolutions of solitons.