Extremely short pulses in Maxwell-Duffing model

We analyze the reduced Maxwell-Duffing model relevant for ultra short pulse propagation in non-resonant media. We find a wide class of localized pulse solution, through a novel method. It is observed that, these solutions are related to non-linear Schr\"{o}dinger equation with a source. We also find continuous wave solutions. Some of the solutions are singular, indicating self-focusing effect. Modulation instability and stability analysis of the solutions have also been studied.