On instability of solitons in the 2d cubic Zakharov-Kuznetsov equation

We consider the critical generalized Zakharov-Kuznetsov (ZK) equation, u(t) + partial derivative(x1)(Delta u + u(3)) = 0, (x(1), x(2)) is an element of R-2. In Farah et al. (Instability of solitons in the 2d cubic Zakharov-Kuznetsov equation, arXiv:1711.05907, 2017), we proved that solitons are unstable for this equation following the strategy byMartel and Merle (GAFA Geom Funct Anal 11:74-123, 2001) in their study of the critical generalized Kortwegde Vries equation. The main ingredient used in Farah et al. (Instability of solitons in the 2d cubic Zakharov-Kuznetsov equation, arXiv:1711.05907, 2017) was the new pointwise decay estimates in two dimensions together with monotonicity properties of solutions. In this paper, we show that using only monotonicity properties and not relying on pointwise estimates, thus, greatly simplifying the approach, we can prove an instability of solitons, though a slightly weaker version.