The KdV approximation for a system with unstable resonances

Abstract

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. By working in spaces of analytic functions, the approach will allow us to handle more complicated systems without a detailed discussion of the resonances and without finding a suitable energy.