Instability of solitons in the 2d cubic Zakharov-Kuznetsov equation

TL;DR

This paper proves the instability of solitons in the two-dimensional cubic Zakharov-Kuznetsov equation, filling a gap in understanding the critical case where nonlinearities are at the threshold of stability.

Contribution

The authors establish the instability of solitons in the 2D cubic ZK equation, using new virial-type quantities and decay estimates to handle the complexities of the two-dimensional case.

Findings

Solitons are unstable for the cubic ZK equation.

Development of new virial-type quantities for 2D analysis.

Introduction of pointwise decay estimates applicable to similar problems.

Abstract

We consider the two dimensional generalization of the Korteweg-de Vries 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 $p < 3$ and unstable for $p > 3$, which was established by Anne de Bouard in [5] generalizing the arguments of Bona-Souganidis-Strauss in [1] for the gKdV equation. The $L^2$-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 in [22]. While the general strategy follows [22], 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.