The KdV approximation for a system with unstable resonances

The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. We expect that the approach will allow to handle more complicated systems without a detailed discussion of the resonances and without finding a suitable energy.