Scaling symmetry reductions of coupled KdV systems

Abstract In this paper we discuss the Painlevé reductions of coupled KdV systems. We start by comparing the procedure with that of stationary reductions . Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the t 2 flow of the KdV and dispersive water wave hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical structure to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case.