Affine lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction

A generalization of the Drinfel'd-Sokolov hierarchies is proposed using the framework of Sato-Wilson dressing method. In the case of the $A_1^{(1)}$ -type affine Lie group, we obtain the hierarchy that includes the derivative nonlinear Schrödinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition. The fourth Painlevé equation and its Weyl group symmetry are obtained as a similarity reduction. We also discuss the connection between the reduced system and monodromy-preserving deformations.