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 $H^s({\mathbb{R}}^2)$, for $s\in [1,2]$, on the background of an $L^{\mathrm{\infty }}({\mathbb{R}}^3)$-function $\mathrm{\Psi }(t,x,y)$, with $\mathrm{\Psi }(t,x,y)$ 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 $H^1({\mathbb{R}}^2)$.