For a long time the Luttinger liquid has served as the standard low energy description of gapless one-dimensional quantum systems. More recently, using an effective mobile impurity model has allowed progress to be made beyond the linearity of the spectrum. In this thesis we show that the depleton model of mobile impurities can be extended to become equivalent to the effective impurity model calculate the dynamic response functions. We also use exact Bethe ansatz solutions to conorm predictions of the effective model to demonstrate the reliability of both the depleton model and the response functions.