Abstract

We consider the critical generalized Zakharov–Kuznetsov (ZK) equation, $$u_t + \partial _{x_1}(\Delta u + u^3) = 0, (x_1,x_2) \in {\mathbb {R}}^2$$ . 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 by Martel and Merle (GAFA Geom Funct Anal 11:74–123, 2001) in their study of the critical generalized Kortweg–de 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.

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