Affine Lie group approach to a derivative nonlinear Schr\"odinger equatoin and its similarity reduction
Abstract
The generalized Drinfel'd-Sokolov hierarchies studied by de Groot-Hollowood-Miramontes are extended from the viewpoint of Sato-Wilson dressing method. In the A1(1) case, we obtain the hierarchy that include the derivative nonlinear Schr\"odinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition of the A1(1) affine Lie group. The fourth Painlev\'e equation and their Weyl group symmetry are obtained as a similarity reduction. We also clarify the connection between these systems and monodromy preserving deformations.
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