Instability of solitons - revisited, II: the supercritical Zakharov-Kuznetsov equation

TL;DR

This paper investigates the instability of solitons in the two-dimensional supercritical Zakharov-Kuznetsov equation, offering a simplified proof method that could be applicable to other related stability problems.

Contribution

It introduces a new approach using truncation and monotonicity to prove soliton instability, simplifying previous methods and broadening applicability.

Findings

Confirmed soliton instability for p > 3 in 2D ZK equation

Developed a simplified proof technique using truncation and monotonicity

Potential applicability to other stability analyses in KdV-type equations

Abstract

We revisit the phenomenon of instability of solitons in 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 unstable in this two dimensional equation for nonlinearities $p > 3$. This was shown by Anne de Bouard in [4] generalizing the arguments of Bona-Souganidis-Strauss in [1] for the generalized KdV equation. In this paper, we use a different method to obtain the instability of solitons, namely, truncation and monotonicity properties. Not only does this approach simplify the proof, but it can also be useful for studying various other stability questions in the ZK equation as well as other generalizations of the KdV equation.