Instability of travelling waves for weakly coupled KdV systems

In this paper we consider the weakly coupled KdV system equation(0.1){ut=(−uxx+f(u)+ϵFu(u,v))xvt=(−vxx+g(v)+ϵFv(u,v))x, where ϵϵ is a small parameter. Assuming that for ϵ=0ϵ=0 the decoupled system has a travelling wave, we first give a sufficient condition for the persistence of such a wave. Secondly, assuming that each wave for the unperturbed system is stable and under a generic condition, we show that, at least for a one-sided interval for the parameter ϵϵ, the persistent wave is unstable.