Local well-posedness for the Zakharov-Kuznetsov equation on the background of a bounded function

We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in Hˢ( R ²), for s∈ [1,2], on the background of an L^∞( R ³)-function Ψ(t,x,y), with Ψ(t,x,y) satisfying some natural extra conditions. This result not only gives us a framework to solve the ZK equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of periodic solutions. Additionally, we show the global well-posedness in the energy space H¹( R ²).