Spectral property for the 2D Zakharov–Kuznetsov equation

We discuss a spectral property for the virial operator of the 2D Zakharov–Kuznetsov (ZK) equation. This is a crucial ingredient to establish blow-up or asymptotic stability of solitary waves in higher-dimensional problems. This model in 3D setting was originally introduced by Zakharov and Kuznetsov in plasma physics, and is also a higher-dimensional generalization of the well-known Korteweg–de Vries (KdV) equation. The problem of stability of solitary waves in ZK equation or stable blow-up in modified ZK (or KdV-type) equation is an important physical question, for which virial operators and their spectral properties are the essential elements of the analysis. In this paper we investigate this problem analytically and reduce it to verifying numerically only some signs of inner products and certain eigenvalues.