Instability of Solitons in the 2d Cubic Zakharov-Kuznetsov Equation

Abstract

We consider the two dimensional generalization of the Korteweg-de Vries (KdV) equation, the generalized Zakharov-Kuznetsov (ZK) equation \(u_t + \partial _{x_1}(\Delta u + u^p) = 0, (x_1,x_2) \in \mathbb R^2\). It is known that solitons are stable for nonlinearities p 3, which was established by de Bouard (Proc R Soc Edinb Sect A 126:89–112, 1996) generalizing the arguments of Bona et al. (Proc R Soc Lond 411:395–412, 1987) for the gKdV equation. The L2-critical case with p = 3 has been open and in this paper we prove that solitons are unstable in the cubic ZK equation. This matches the situation with the critical gKdV equation, proved in 2001 by Martel and Merle (Geom Funct Anal 11:74–123, 2001). While the general strategy follows (Martel and Merle, Geom Funct Anal 11:74–123, 2001), the two dimensional case creates several difficulties and to deal with them, we design a new virial-type quantity, revisit monotonicity properties and, most importantly, develop new pointwise decay estimates, which can be useful in other contexts.