Nonintegrable reductions of the self-dual Yang–Mills equations in a metric of plane wave type

Abstract

Symmetry reductions of the self-dual Yang–Mills equations for SL(2,C) bundles with the background metric ds2=2 du dv−dx2+f2(u)dy2 are considered. One of the field components in the reduced equations can be cast into Jordan normal form after gauge transformations. The reduced equations for the two possible normal forms are equivalent, respectively, to certain generalizations of the Korteweg–de Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation. It is shown that the generalized KdV and NLS equations fail the Painlevé test except when the metric is flat. The generalized KdV equation is transformed to a simple form in the case when f(u)=ua and it is shown that one may obtain either the KdV equation or the cylindrical KdV equation by this method only when the metric is flat.